diff options
110 files changed, 2694 insertions, 2695 deletions
@@ -41,7 +41,7 @@ DIRS += MenhirLib COQINCLUDES += -R MenhirLib MenhirLib endif -COQCOPTS ?= -w -undeclared-scope -w -omega-is-deprecated +COQCOPTS ?= -w -undeclared-scope COQC="$(COQBIN)coqc" -q $(COQINCLUDES) $(COQCOPTS) COQDEP="$(COQBIN)coqdep" $(COQINCLUDES) COQDOC="$(COQBIN)coqdoc" diff --git a/aarch64/Asm.v b/aarch64/Asm.v index 346cb649..b5f4c838 100644 --- a/aarch64/Asm.v +++ b/aarch64/Asm.v @@ -1293,7 +1293,7 @@ Ltac Equalities := split. auto. intros. destruct B; auto. subst. auto. - (* trace length *) red; intros. inv H; simpl. - omega. + lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. - (* initial states *) diff --git a/aarch64/Asmgenproof.v b/aarch64/Asmgenproof.v index d3515a96..dc0bc509 100644 --- a/aarch64/Asmgenproof.v +++ b/aarch64/Asmgenproof.v @@ -67,7 +67,7 @@ Lemma transf_function_no_overflow: transf_function f = OK tf -> list_length_z tf.(fn_code) <= Ptrofs.max_unsigned. Proof. intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z x.(fn_code))); inv EQ0. - omega. + lia. Qed. Lemma exec_straight_exec: @@ -424,8 +424,8 @@ Proof. split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Ptrofs.unsigned_repr. replace (pos' - 0) with pos' in Q. - auto. omega. - generalize (transf_function_no_overflow _ _ H0). omega. + auto. lia. + generalize (transf_function_no_overflow _ _ H0). lia. intros. apply Pregmap.gso; auto. Qed. @@ -949,10 +949,10 @@ Local Transparent destroyed_by_op. rewrite <- (sp_val _ _ _ AG). rewrite F. reflexivity. reflexivity. eexact U. } - exploit exec_straight_steps_2; eauto using functions_transl. omega. constructor. + exploit exec_straight_steps_2; eauto using functions_transl. lia. constructor. intros (ofs' & X & Y). left; exists (State rs3 m3'); split. - eapply exec_straight_steps_1; eauto. omega. constructor. + eapply exec_straight_steps_1; eauto. lia. constructor. econstructor; eauto. rewrite X; econstructor; eauto. apply agree_exten with rs2; eauto with asmgen. @@ -981,7 +981,7 @@ Local Transparent destroyed_at_function_entry. simpl. - (* return *) inv STACKS. simpl in *. - right. split. omega. split. auto. + right. split. lia. split. auto. rewrite <- ATPC in H5. econstructor; eauto. congruence. Qed. diff --git a/aarch64/Asmgenproof1.v b/aarch64/Asmgenproof1.v index d95376d2..5f27f6bf 100644 --- a/aarch64/Asmgenproof1.v +++ b/aarch64/Asmgenproof1.v @@ -81,8 +81,8 @@ Local Opaque Zzero_ext. induction N as [ | N]; simpl; intros. - constructor. - set (frag := Zzero_ext 16 (Z.shiftr n p)) in *. destruct (Z.eqb frag 0). -+ apply IHN. omega. -+ econstructor. reflexivity. omega. apply IHN; omega. ++ apply IHN. lia. ++ econstructor. reflexivity. lia. apply IHN; lia. Qed. Fixpoint recompose_int (accu: Z) (l: list (Z * Z)) : Z := @@ -100,43 +100,43 @@ Lemma decompose_int_correct: if zlt i p then Z.testbit accu i else Z.testbit n i). Proof. induction N as [ | N]; intros until accu; intros PPOS ABOVE i RANGE. -- simpl. rewrite zlt_true; auto. xomega. +- simpl. rewrite zlt_true; auto. extlia. - rewrite inj_S in RANGE. simpl. set (frag := Zzero_ext 16 (Z.shiftr n p)). assert (FRAG: forall i, p <= i < p + 16 -> Z.testbit n i = Z.testbit frag (i - p)). - { unfold frag; intros. rewrite Zzero_ext_spec by omega. rewrite zlt_true by omega. - rewrite Z.shiftr_spec by omega. f_equal; omega. } + { unfold frag; intros. rewrite Zzero_ext_spec by lia. rewrite zlt_true by lia. + rewrite Z.shiftr_spec by lia. f_equal; lia. } destruct (Z.eqb_spec frag 0). + rewrite IHN. -* destruct (zlt i p). rewrite zlt_true by omega. auto. +* destruct (zlt i p). rewrite zlt_true by lia. auto. destruct (zlt i (p + 16)); auto. - rewrite ABOVE by omega. rewrite FRAG by omega. rewrite e, Z.testbit_0_l. auto. -* omega. -* intros; apply ABOVE; omega. -* xomega. + rewrite ABOVE by lia. rewrite FRAG by lia. rewrite e, Z.testbit_0_l. auto. +* lia. +* intros; apply ABOVE; lia. +* extlia. + simpl. rewrite IHN. * destruct (zlt i (p + 16)). -** rewrite Zinsert_spec by omega. unfold proj_sumbool. - rewrite zlt_true by omega. +** rewrite Zinsert_spec by lia. unfold proj_sumbool. + rewrite zlt_true by lia. destruct (zlt i p). - rewrite zle_false by omega. auto. - rewrite zle_true by omega. simpl. symmetry; apply FRAG; omega. -** rewrite Z.ldiff_spec, Z.shiftl_spec by omega. - change 65535 with (two_p 16 - 1). rewrite Ztestbit_two_p_m1 by omega. - rewrite zlt_false by omega. rewrite zlt_false by omega. apply andb_true_r. -* omega. -* intros. rewrite Zinsert_spec by omega. unfold proj_sumbool. - rewrite zle_true by omega. rewrite zlt_false by omega. simpl. - apply ABOVE. omega. -* xomega. + rewrite zle_false by lia. auto. + rewrite zle_true by lia. simpl. symmetry; apply FRAG; lia. +** rewrite Z.ldiff_spec, Z.shiftl_spec by lia. + change 65535 with (two_p 16 - 1). rewrite Ztestbit_two_p_m1 by lia. + rewrite zlt_false by lia. rewrite zlt_false by lia. apply andb_true_r. +* lia. +* intros. rewrite Zinsert_spec by lia. unfold proj_sumbool. + rewrite zle_true by lia. rewrite zlt_false by lia. simpl. + apply ABOVE. lia. +* extlia. Qed. Corollary decompose_int_eqmod: forall N n, eqmod (two_power_nat (N * 16)%nat) (recompose_int 0 (decompose_int N n 0)) n. Proof. intros; apply eqmod_same_bits; intros. - rewrite decompose_int_correct. apply zlt_false; omega. - omega. intros; apply Z.testbit_0_l. xomega. + rewrite decompose_int_correct. apply zlt_false; lia. + lia. intros; apply Z.testbit_0_l. extlia. Qed. Corollary decompose_notint_eqmod: forall N n, @@ -145,8 +145,8 @@ Corollary decompose_notint_eqmod: forall N n, Proof. intros; apply eqmod_same_bits; intros. rewrite Z.lnot_spec, decompose_int_correct. - rewrite zlt_false by omega. rewrite Z.lnot_spec by omega. apply negb_involutive. - omega. intros; apply Z.testbit_0_l. xomega. omega. + rewrite zlt_false by lia. rewrite Z.lnot_spec by lia. apply negb_involutive. + lia. intros; apply Z.testbit_0_l. extlia. lia. Qed. Lemma negate_decomposition_wf: @@ -156,7 +156,7 @@ Proof. instantiate (1 := (Z.lnot m)). apply equal_same_bits; intros. rewrite H. change 65535 with (two_p 16 - 1). - rewrite Z.lxor_spec, !Zzero_ext_spec, Z.lnot_spec, Ztestbit_two_p_m1 by omega. + rewrite Z.lxor_spec, !Zzero_ext_spec, Z.lnot_spec, Ztestbit_two_p_m1 by lia. destruct (zlt i 16). apply xorb_true_r. auto. @@ -167,7 +167,7 @@ Lemma Zinsert_eqmod: eqmod (two_power_nat n) x1 x2 -> eqmod (two_power_nat n) (Zinsert x1 y p l) (Zinsert x2 y p l). Proof. - intros. apply eqmod_same_bits; intros. rewrite ! Zinsert_spec by omega. + intros. apply eqmod_same_bits; intros. rewrite ! Zinsert_spec by lia. destruct (zle p i && zlt i (p + l)); auto. apply same_bits_eqmod with n; auto. Qed. @@ -178,12 +178,12 @@ Lemma Zinsert_0_l: Z.shiftl (Zzero_ext l y) p = Zinsert 0 (Zzero_ext l y) p l. Proof. intros. apply equal_same_bits; intros. - rewrite Zinsert_spec by omega. unfold proj_sumbool. - destruct (zlt i p); [rewrite zle_false by omega|rewrite zle_true by omega]; simpl. + rewrite Zinsert_spec by lia. unfold proj_sumbool. + destruct (zlt i p); [rewrite zle_false by lia|rewrite zle_true by lia]; simpl. - rewrite Z.testbit_0_l, Z.shiftl_spec_low by auto. auto. -- rewrite Z.shiftl_spec by omega. +- rewrite Z.shiftl_spec by lia. destruct (zlt i (p + l)); auto. - rewrite Zzero_ext_spec, zlt_false, Z.testbit_0_l by omega. auto. + rewrite Zzero_ext_spec, zlt_false, Z.testbit_0_l by lia. auto. Qed. Lemma recompose_int_negated: @@ -193,12 +193,12 @@ Proof. induction 1; intros accu; simpl. - auto. - rewrite <- IHwf_decomposition. f_equal. apply equal_same_bits; intros. - rewrite Z.lnot_spec, ! Zinsert_spec, Z.lxor_spec, Z.lnot_spec by omega. + rewrite Z.lnot_spec, ! Zinsert_spec, Z.lxor_spec, Z.lnot_spec by lia. unfold proj_sumbool. destruct (zle p i); simpl; auto. destruct (zlt i (p + 16)); simpl; auto. change 65535 with (two_p 16 - 1). - rewrite Ztestbit_two_p_m1 by omega. rewrite zlt_true by omega. + rewrite Ztestbit_two_p_m1 by lia. rewrite zlt_true by lia. apply xorb_true_r. Qed. @@ -219,7 +219,7 @@ Proof. (Zinsert accu n p 16)) as (rs' & P & Q & R). Simpl. rewrite ACCU. simpl. f_equal. apply Int.eqm_samerepr. - apply Zinsert_eqmod. auto. omega. apply Int.eqm_sym; apply Int.eqm_unsigned_repr. + apply Zinsert_eqmod. auto. lia. apply Int.eqm_sym; apply Int.eqm_unsigned_repr. exists rs'; split. eapply exec_straight_opt_step_opt. simpl; eauto. auto. exact P. split. exact Q. intros; Simpl. rewrite R by auto. Simpl. @@ -244,7 +244,7 @@ Proof. unfold rs1; Simpl. exists rs2; split. eapply exec_straight_opt_step; eauto. - simpl. unfold rs1. do 5 f_equal. unfold accu0. rewrite H. apply Zinsert_0_l; omega. + simpl. unfold rs1. do 5 f_equal. unfold accu0. rewrite H. apply Zinsert_0_l; lia. reflexivity. split. exact Q. intros. rewrite R by auto. unfold rs1; Simpl. @@ -272,7 +272,7 @@ Proof. exists rs2; split. eapply exec_straight_opt_step; eauto. simpl. unfold rs1. do 5 f_equal. - unfold accu0. f_equal. rewrite H. apply Zinsert_0_l; omega. + unfold accu0. f_equal. rewrite H. apply Zinsert_0_l; lia. reflexivity. split. unfold accu0 in Q; rewrite recompose_int_negated in Q by auto. exact Q. intros. rewrite R by auto. unfold rs1; Simpl. @@ -302,8 +302,8 @@ Proof. apply Int.eqm_samerepr. apply decompose_notint_eqmod. apply Int.repr_unsigned. } destruct Nat.leb. -+ rewrite <- A. apply exec_loadimm_z_w. apply decompose_int_wf; omega. -+ rewrite <- B. apply exec_loadimm_n_w. apply decompose_int_wf; omega. ++ rewrite <- A. apply exec_loadimm_z_w. apply decompose_int_wf; lia. ++ rewrite <- B. apply exec_loadimm_n_w. apply decompose_int_wf; lia. Qed. Lemma exec_loadimm_k_x: @@ -323,7 +323,7 @@ Proof. (Zinsert accu n p 16)) as (rs' & P & Q & R). Simpl. rewrite ACCU. simpl. f_equal. apply Int64.eqm_samerepr. - apply Zinsert_eqmod. auto. omega. apply Int64.eqm_sym; apply Int64.eqm_unsigned_repr. + apply Zinsert_eqmod. auto. lia. apply Int64.eqm_sym; apply Int64.eqm_unsigned_repr. exists rs'; split. eapply exec_straight_opt_step_opt. simpl; eauto. auto. exact P. split. exact Q. intros; Simpl. rewrite R by auto. Simpl. @@ -348,7 +348,7 @@ Proof. unfold rs1; Simpl. exists rs2; split. eapply exec_straight_opt_step; eauto. - simpl. unfold rs1. do 5 f_equal. unfold accu0. rewrite H. apply Zinsert_0_l; omega. + simpl. unfold rs1. do 5 f_equal. unfold accu0. rewrite H. apply Zinsert_0_l; lia. reflexivity. split. exact Q. intros. rewrite R by auto. unfold rs1; Simpl. @@ -376,7 +376,7 @@ Proof. exists rs2; split. eapply exec_straight_opt_step; eauto. simpl. unfold rs1. do 5 f_equal. - unfold accu0. f_equal. rewrite H. apply Zinsert_0_l; omega. + unfold accu0. f_equal. rewrite H. apply Zinsert_0_l; lia. reflexivity. split. unfold accu0 in Q; rewrite recompose_int_negated in Q by auto. exact Q. intros. rewrite R by auto. unfold rs1; Simpl. @@ -406,8 +406,8 @@ Proof. apply Int64.eqm_samerepr. apply decompose_notint_eqmod. apply Int64.repr_unsigned. } destruct Nat.leb. -+ rewrite <- A. apply exec_loadimm_z_x. apply decompose_int_wf; omega. -+ rewrite <- B. apply exec_loadimm_n_x. apply decompose_int_wf; omega. ++ rewrite <- A. apply exec_loadimm_z_x. apply decompose_int_wf; lia. ++ rewrite <- B. apply exec_loadimm_n_x. apply decompose_int_wf; lia. Qed. (** Add immediate *) @@ -426,14 +426,14 @@ Lemma exec_addimm_aux_32: Proof. intros insn sem SEM ASSOC; intros. unfold addimm_aux. set (nlo := Zzero_ext 12 (Int.unsigned n)). set (nhi := Int.unsigned n - nlo). - assert (E: Int.unsigned n = nhi + nlo) by (unfold nhi; omega). + assert (E: Int.unsigned n = nhi + nlo) by (unfold nhi; lia). rewrite <- (Int.repr_unsigned n). destruct (Z.eqb_spec nhi 0); [|destruct (Z.eqb_spec nlo 0)]. - econstructor; split. apply exec_straight_one. apply SEM. Simpl. - split. Simpl. do 3 f_equal; omega. + split. Simpl. do 3 f_equal; lia. intros; Simpl. - econstructor; split. apply exec_straight_one. apply SEM. Simpl. - split. Simpl. do 3 f_equal; omega. + split. Simpl. do 3 f_equal; lia. intros; Simpl. - econstructor; split. eapply exec_straight_two. apply SEM. apply SEM. Simpl. Simpl. @@ -484,14 +484,14 @@ Lemma exec_addimm_aux_64: Proof. intros insn sem SEM ASSOC; intros. unfold addimm_aux. set (nlo := Zzero_ext 12 (Int64.unsigned n)). set (nhi := Int64.unsigned n - nlo). - assert (E: Int64.unsigned n = nhi + nlo) by (unfold nhi; omega). + assert (E: Int64.unsigned n = nhi + nlo) by (unfold nhi; lia). rewrite <- (Int64.repr_unsigned n). destruct (Z.eqb_spec nhi 0); [|destruct (Z.eqb_spec nlo 0)]. - econstructor; split. apply exec_straight_one. apply SEM. Simpl. - split. Simpl. do 3 f_equal; omega. + split. Simpl. do 3 f_equal; lia. intros; Simpl. - econstructor; split. apply exec_straight_one. apply SEM. Simpl. - split. Simpl. do 3 f_equal; omega. + split. Simpl. do 3 f_equal; lia. intros; Simpl. - econstructor; split. eapply exec_straight_two. apply SEM. apply SEM. Simpl. Simpl. diff --git a/aarch64/ConstpropOpproof.v b/aarch64/ConstpropOpproof.v index deab7cd4..7f5f1e06 100644 --- a/aarch64/ConstpropOpproof.v +++ b/aarch64/ConstpropOpproof.v @@ -391,7 +391,7 @@ Proof. Int.bit_solve. destruct (zlt i0 n0). replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)). rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto. - rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto. + rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto. rewrite Int.bits_not by auto. apply negb_involutive. rewrite H6 by auto. auto. econstructor; split; eauto. auto. diff --git a/aarch64/Conventions1.v b/aarch64/Conventions1.v index 4873dd91..cfcbcbf1 100644 --- a/aarch64/Conventions1.v +++ b/aarch64/Conventions1.v @@ -274,33 +274,33 @@ Proof. assert (ALP: forall ofs ty, ofs >= 0 -> align ofs (typesize ty) >= 0). { intros. assert (ofs <= align ofs (typesize ty)) by (apply align_le; apply typesize_pos). - omega. } + lia. } assert (ALD: forall ofs ty, ofs >= 0 -> (typealign ty | align ofs (typesize ty))). { intros. apply Z.divide_trans with (typesize ty). apply typealign_typesize. apply align_divides. apply typesize_pos. } assert (ALP2: forall ofs, ofs >= 0 -> align ofs 2 >= 0). { intros. - assert (ofs <= align ofs 2) by (apply align_le; omega). - omega. } + assert (ofs <= align ofs 2) by (apply align_le; lia). + lia. } assert (ALD2: forall ofs ty, ofs >= 0 -> (typealign ty | align ofs 2)). { intros. eapply Z.divide_trans with 2. exists (2 / typealign ty). destruct ty; reflexivity. - apply align_divides. omega. } + apply align_divides. lia. } assert (STK: forall tyl ofs, ofs >= 0 -> OK (loc_arguments_stack tyl ofs)). { induction tyl as [ | ty tyl]; intros ofs OO; red; simpl; intros. - contradiction. - destruct H. + subst p. split. auto. simpl. apply Z.divide_1_l. - + apply IHtyl with (ofs := ofs + 2). omega. auto. + + apply IHtyl with (ofs := ofs + 2). lia. auto. } assert (A: forall ty ri rf ofs f, OKF f -> ofs >= 0 -> OK (stack_arg ty ri rf ofs f)). { intros until f; intros OF OO; red; unfold stack_arg; intros. destruct Archi.abi; destruct H. - subst p; simpl; auto. - - eapply OF; [|eauto]. apply ALP2 in OO. omega. + - eapply OF; [|eauto]. apply ALP2 in OO. lia. - subst p; simpl; auto. - - eapply OF; [|eauto]. apply (ALP ofs ty) in OO. generalize (typesize_pos ty). omega. + - eapply OF; [|eauto]. apply (ALP ofs ty) in OO. generalize (typesize_pos ty). lia. } assert (B: forall ty ri rf ofs f, OKF f -> ofs >= 0 -> OK (int_arg ty ri rf ofs f)). @@ -332,7 +332,7 @@ Lemma loc_arguments_acceptable: In p (loc_arguments s) -> forall_rpair loc_argument_acceptable p. Proof. unfold loc_arguments; intros. - eapply loc_arguments_rec_charact; eauto. omega. + eapply loc_arguments_rec_charact; eauto. lia. Qed. Hint Resolve loc_arguments_acceptable: locs. diff --git a/aarch64/Op.v b/aarch64/Op.v index a7483d56..f8d2510e 100644 --- a/aarch64/Op.v +++ b/aarch64/Op.v @@ -957,25 +957,25 @@ End SHIFT_AMOUNT. Program Definition mk_amount32 (n: int): amount32 := {| a32_amount := Int.zero_ext 5 n |}. Next Obligation. - apply mk_amount_range. omega. reflexivity. + apply mk_amount_range. lia. reflexivity. Qed. Lemma mk_amount32_eq: forall n, Int.ltu n Int.iwordsize = true -> a32_amount (mk_amount32 n) = n. Proof. - intros. eapply mk_amount_eq; eauto. omega. reflexivity. + intros. eapply mk_amount_eq; eauto. lia. reflexivity. Qed. Program Definition mk_amount64 (n: int): amount64 := {| a64_amount := Int.zero_ext 6 n |}. Next Obligation. - apply mk_amount_range. omega. reflexivity. + apply mk_amount_range. lia. reflexivity. Qed. Lemma mk_amount64_eq: forall n, Int.ltu n Int64.iwordsize' = true -> a64_amount (mk_amount64 n) = n. Proof. - intros. eapply mk_amount_eq; eauto. omega. reflexivity. + intros. eapply mk_amount_eq; eauto. lia. reflexivity. Qed. (** Recognition of move operations. *) diff --git a/aarch64/SelectLongproof.v b/aarch64/SelectLongproof.v index b051369c..aee09b12 100644 --- a/aarch64/SelectLongproof.v +++ b/aarch64/SelectLongproof.v @@ -225,8 +225,8 @@ Proof. intros. unfold Int.ltu; apply zlt_true. apply Int.ltu_inv in H. apply Int.ltu_inv in H0. change (Int.unsigned Int64.iwordsize') with Int64.zwordsize in *. - unfold Int.sub; rewrite Int.unsigned_repr. omega. - assert (Int64.zwordsize < Int.max_unsigned) by reflexivity. omega. + unfold Int.sub; rewrite Int.unsigned_repr. lia. + assert (Int64.zwordsize < Int.max_unsigned) by reflexivity. lia. Qed. Theorem eval_shrluimm: @@ -242,13 +242,13 @@ Local Opaque Int64.zwordsize. + destruct (Int.ltu n a) eqn:L2. * assert (L3: Int.ltu (Int.sub a n) Int64.iwordsize' = true). { apply sub_shift_amount; auto using a64_range. - apply Int.ltu_inv in L2. omega. } + apply Int.ltu_inv in L2. lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount64_eq, L3, a64_range by auto. simpl. rewrite L. rewrite Int64.shru'_shl', L2 by auto using a64_range. auto. * assert (L3: Int.ltu (Int.sub n a) Int64.iwordsize' = true). { apply sub_shift_amount; auto using a64_range. - unfold Int.ltu in L2. destruct zlt in L2; discriminate || omega. } + unfold Int.ltu in L2. destruct zlt in L2; discriminate || lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount64_eq, L3, a64_range by auto. simpl. rewrite L. rewrite Int64.shru'_shl', L2 by auto using a64_range. auto. @@ -261,11 +261,11 @@ Local Opaque Int64.zwordsize. * econstructor; split. EvalOp. rewrite mk_amount64_eq by auto. destruct v1; simpl; auto. rewrite ! L; simpl. set (s' := s - Int.unsigned n). - replace s with (s' + Int.unsigned n) by (unfold s'; omega). - rewrite Int64.shru'_zero_ext. auto. unfold s'; omega. + replace s with (s' + Int.unsigned n) by (unfold s'; lia). + rewrite Int64.shru'_zero_ext. auto. unfold s'; lia. * econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite ! L; simpl. - rewrite Int64.shru'_zero_ext_0 by omega. auto. + rewrite Int64.shru'_zero_ext_0 by lia. auto. + econstructor; eauto using eval_shrluimm_base. - intros; TrivialExists. Qed. @@ -290,13 +290,13 @@ Proof. + destruct (Int.ltu n a) eqn:L2. * assert (L3: Int.ltu (Int.sub a n) Int64.iwordsize' = true). { apply sub_shift_amount; auto using a64_range. - apply Int.ltu_inv in L2. omega. } + apply Int.ltu_inv in L2. lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount64_eq, L3, a64_range by auto. simpl. rewrite L. rewrite Int64.shr'_shl', L2 by auto using a64_range. auto. * assert (L3: Int.ltu (Int.sub n a) Int64.iwordsize' = true). { apply sub_shift_amount; auto using a64_range. - unfold Int.ltu in L2. destruct zlt in L2; discriminate || omega. } + unfold Int.ltu in L2. destruct zlt in L2; discriminate || lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount64_eq, L3, a64_range by auto. simpl. rewrite L. rewrite Int64.shr'_shl', L2 by auto using a64_range. auto. @@ -309,8 +309,8 @@ Proof. * InvBooleans. econstructor; split. EvalOp. rewrite mk_amount64_eq by auto. destruct v1; simpl; auto. rewrite ! L; simpl. set (s' := s - Int.unsigned n). - replace s with (s' + Int.unsigned n) by (unfold s'; omega). - rewrite Int64.shr'_sign_ext. auto. unfold s'; omega. unfold s'; omega. + replace s with (s' + Int.unsigned n) by (unfold s'; lia). + rewrite Int64.shr'_sign_ext. auto. unfold s'; lia. unfold s'; lia. * econstructor; split; [|eauto]. apply eval_shrlimm_base; auto. EvalOp. + econstructor; eauto using eval_shrlimm_base. - intros; TrivialExists. @@ -392,7 +392,7 @@ Proof. - TrivialExists. - destruct (zlt (Int.unsigned a0) sz). + econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite a64_range; simpl. - apply Val.lessdef_same. f_equal. rewrite Int64.shl'_zero_ext by omega. f_equal. omega. + apply Val.lessdef_same. f_equal. rewrite Int64.shl'_zero_ext by lia. f_equal. lia. + TrivialExists. - TrivialExists. Qed. diff --git a/aarch64/SelectOpproof.v b/aarch64/SelectOpproof.v index 625a0c14..ccc4c0f1 100644 --- a/aarch64/SelectOpproof.v +++ b/aarch64/SelectOpproof.v @@ -243,8 +243,8 @@ Remark sub_shift_amount: Proof. intros. unfold Int.ltu; apply zlt_true. rewrite Int.unsigned_repr_wordsize. apply Int.ltu_iwordsize_inv in H. apply Int.ltu_iwordsize_inv in H0. - unfold Int.sub; rewrite Int.unsigned_repr. omega. - generalize Int.wordsize_max_unsigned; omega. + unfold Int.sub; rewrite Int.unsigned_repr. lia. + generalize Int.wordsize_max_unsigned; lia. Qed. Theorem eval_shruimm: @@ -260,13 +260,13 @@ Local Opaque Int.zwordsize. + destruct (Int.ltu n a) eqn:L2. * assert (L3: Int.ltu (Int.sub a n) Int.iwordsize = true). { apply sub_shift_amount; auto using a32_range. - apply Int.ltu_inv in L2. omega. } + apply Int.ltu_inv in L2. lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount32_eq, L3, a32_range by auto. simpl. rewrite L. rewrite Int.shru_shl, L2 by auto using a32_range. auto. * assert (L3: Int.ltu (Int.sub n a) Int.iwordsize = true). { apply sub_shift_amount; auto using a32_range. - unfold Int.ltu in L2. destruct zlt in L2; discriminate || omega. } + unfold Int.ltu in L2. destruct zlt in L2; discriminate || lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount32_eq, L3, a32_range by auto. simpl. rewrite L. rewrite Int.shru_shl, L2 by auto using a32_range. auto. @@ -279,11 +279,11 @@ Local Opaque Int.zwordsize. * econstructor; split. EvalOp. rewrite mk_amount32_eq by auto. destruct v1; simpl; auto. rewrite ! L; simpl. set (s' := s - Int.unsigned n). - replace s with (s' + Int.unsigned n) by (unfold s'; omega). - rewrite Int.shru_zero_ext. auto. unfold s'; omega. + replace s with (s' + Int.unsigned n) by (unfold s'; lia). + rewrite Int.shru_zero_ext. auto. unfold s'; lia. * econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite ! L; simpl. - rewrite Int.shru_zero_ext_0 by omega. auto. + rewrite Int.shru_zero_ext_0 by lia. auto. + econstructor; eauto using eval_shruimm_base. - intros; TrivialExists. Qed. @@ -308,13 +308,13 @@ Proof. + destruct (Int.ltu n a) eqn:L2. * assert (L3: Int.ltu (Int.sub a n) Int.iwordsize = true). { apply sub_shift_amount; auto using a32_range. - apply Int.ltu_inv in L2. omega. } + apply Int.ltu_inv in L2. lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount32_eq, L3, a32_range by auto. simpl. rewrite L. rewrite Int.shr_shl, L2 by auto using a32_range. auto. * assert (L3: Int.ltu (Int.sub n a) Int.iwordsize = true). { apply sub_shift_amount; auto using a32_range. - unfold Int.ltu in L2. destruct zlt in L2; discriminate || omega. } + unfold Int.ltu in L2. destruct zlt in L2; discriminate || lia. } econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite mk_amount32_eq, L3, a32_range by auto. simpl. rewrite L. rewrite Int.shr_shl, L2 by auto using a32_range. auto. @@ -327,8 +327,8 @@ Proof. * InvBooleans. econstructor; split. EvalOp. rewrite mk_amount32_eq by auto. destruct v1; simpl; auto. rewrite ! L; simpl. set (s' := s - Int.unsigned n). - replace s with (s' + Int.unsigned n) by (unfold s'; omega). - rewrite Int.shr_sign_ext. auto. unfold s'; omega. unfold s'; omega. + replace s with (s' + Int.unsigned n) by (unfold s'; lia). + rewrite Int.shr_sign_ext. auto. unfold s'; lia. unfold s'; lia. * econstructor; split; [|eauto]. apply eval_shrimm_base; auto. EvalOp. + econstructor; eauto using eval_shrimm_base. - intros; TrivialExists. @@ -399,20 +399,20 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.signed i * Int.signed i0) 32)). - unfold Int.mulhs; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhs; f_equal. rewrite Zshiftr_div_two_p by lia. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. - assert (N1: 0 <= n < 64) by omega. + assert (N1: 0 <= n < 64) by lia. rewrite Int64.bits_loword by auto. rewrite Int64.bits_shr' by auto. change (Int.unsigned (Int.repr 32)) with 32. change Int64.zwordsize with 64. - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite Int.testbit_repr by auto. - unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; omega). + unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; lia). transitivity (Z.testbit (Int.signed i * Int.signed i0) (n + 32)). - rewrite Z.shiftr_spec by omega. auto. + rewrite Z.shiftr_spec by lia. auto. apply Int64.same_bits_eqm. apply Int64.eqm_mult; apply Int64.eqm_unsigned_repr. - change Int64.zwordsize with 64; omega. + change Int64.zwordsize with 64; lia. Qed. Theorem eval_mulhu: binary_constructor_sound mulhu Val.mulhu. @@ -425,20 +425,20 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.unsigned i * Int.unsigned i0) 32)). - unfold Int.mulhu; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhu; f_equal. rewrite Zshiftr_div_two_p by lia. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. - assert (N1: 0 <= n < 64) by omega. + assert (N1: 0 <= n < 64) by lia. rewrite Int64.bits_loword by auto. rewrite Int64.bits_shru' by auto. change (Int.unsigned (Int.repr 32)) with 32. change Int64.zwordsize with 64. - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite Int.testbit_repr by auto. - unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; omega). + unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; lia). transitivity (Z.testbit (Int.unsigned i * Int.unsigned i0) (n + 32)). - rewrite Z.shiftr_spec by omega. auto. + rewrite Z.shiftr_spec by lia. auto. apply Int64.same_bits_eqm. apply Int64.eqm_mult; apply Int64.eqm_unsigned_repr. - change Int64.zwordsize with 64; omega. + change Int64.zwordsize with 64; lia. Qed. (** Integer conversions *) @@ -451,7 +451,7 @@ Proof. - TrivialExists. - destruct (zlt (Int.unsigned a0) sz). + econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite a32_range; simpl. - apply Val.lessdef_same. f_equal. rewrite Int.shl_zero_ext by omega. f_equal. omega. + apply Val.lessdef_same. f_equal. rewrite Int.shl_zero_ext by lia. f_equal. lia. + TrivialExists. - TrivialExists. Qed. @@ -464,29 +464,29 @@ Proof. - TrivialExists. - destruct (zlt (Int.unsigned a0) sz). + econstructor; split. EvalOp. destruct v1; simpl; auto. rewrite a32_range; simpl. - apply Val.lessdef_same. f_equal. rewrite Int.shl_sign_ext by omega. f_equal. omega. + apply Val.lessdef_same. f_equal. rewrite Int.shl_sign_ext by lia. f_equal. lia. + TrivialExists. - TrivialExists. Qed. Theorem eval_cast8signed: unary_constructor_sound cast8signed (Val.sign_ext 8). Proof. - apply eval_sign_ext; omega. + apply eval_sign_ext; lia. Qed. Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8). Proof. - apply eval_zero_ext; omega. + apply eval_zero_ext; lia. Qed. Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16). Proof. - apply eval_sign_ext; omega. + apply eval_sign_ext; lia. Qed. Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16). Proof. - apply eval_zero_ext; omega. + apply eval_zero_ext; lia. Qed. (** Bitwise not, and, or, xor *) diff --git a/aarch64/Stacklayout.v b/aarch64/Stacklayout.v index 86ba9f45..cdbc64d5 100644 --- a/aarch64/Stacklayout.v +++ b/aarch64/Stacklayout.v @@ -67,13 +67,13 @@ Local Opaque Z.add Z.mul sepconj range. set (ostkdata := align (ol + 4 * b.(bound_local)) 8). change (size_chunk Mptr) with 8. generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + 8 <= oretaddr) by (unfold oretaddr; omega). - assert (oretaddr + 8 <= ocs) by (unfold ocs; omega). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + 8 <= oretaddr) by (unfold oretaddr; lia). + assert (oretaddr + 8 <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). (* Reorder as: outgoing back link @@ -86,11 +86,11 @@ Local Opaque Z.add Z.mul sepconj range. rewrite sep_swap45. (* Apply range_split and range_split2 repeatedly *) unfold fe_ofs_arg. - apply range_split_2. fold olink; omega. omega. - apply range_split. omega. - apply range_split. omega. - apply range_split_2. fold ol. omega. omega. - apply range_drop_right with ostkdata. omega. + apply range_split_2. fold olink; lia. lia. + apply range_split. lia. + apply range_split. lia. + apply range_split_2. fold ol. lia. lia. + apply range_drop_right with ostkdata. lia. eapply sep_drop2. eexact H. Qed. @@ -106,14 +106,14 @@ Proof. set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + 8 <= oretaddr) by (unfold oretaddr; omega). - assert (oretaddr + 8 <= ocs) by (unfold ocs; omega). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + 8 <= oretaddr) by (unfold oretaddr; lia). + assert (oretaddr + 8 <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). - split. omega. apply align_le. omega. + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). + split. lia. apply align_le. lia. Qed. Lemma frame_env_aligned: @@ -133,8 +133,8 @@ Proof. set (ostkdata := align (ol + 4 * b.(bound_local)) 8). change (align_chunk Mptr) with 8. split. apply Z.divide_0_r. - split. apply align_divides; omega. - split. apply align_divides; omega. - split. apply align_divides; omega. - apply Z.divide_add_r. apply align_divides; omega. apply Z.divide_refl. + split. apply align_divides; lia. + split. apply align_divides; lia. + split. apply align_divides; lia. + apply Z.divide_add_r. apply align_divides; lia. apply Z.divide_refl. Qed. @@ -1004,7 +1004,7 @@ Ltac Equalities := split. auto. intros. destruct B; auto. subst. auto. (* trace length *) red; intros; inv H; simpl. - omega. + lia. inv H3; eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. (* initial states *) diff --git a/arm/Asmgenproof.v b/arm/Asmgenproof.v index f60f4b48..93e0c6c2 100644 --- a/arm/Asmgenproof.v +++ b/arm/Asmgenproof.v @@ -68,7 +68,7 @@ Lemma transf_function_no_overflow: forall f tf, transf_function f = OK tf -> list_length_z (fn_code tf) <= Ptrofs.max_unsigned. Proof. - intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z (fn_code x))); inv EQ0. omega. + intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z (fn_code x))); inv EQ0. lia. Qed. Lemma exec_straight_exec: @@ -122,13 +122,13 @@ Proof. case (is_label lbl a). intro EQ; injection EQ; intro; subst c'. exists (pos + 1). split. auto. split. - replace (pos + 1 - pos) with (0 + 1) by omega. constructor. constructor. - rewrite list_length_z_cons. generalize (list_length_z_pos c). omega. + replace (pos + 1 - pos) with (0 + 1) by lia. constructor. constructor. + rewrite list_length_z_cons. generalize (list_length_z_pos c). lia. intros. generalize (IHc (pos + 1) c' H). intros [pos' [A [B C]]]. exists pos'. split. auto. split. - replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by omega. + replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by lia. constructor. auto. - rewrite list_length_z_cons. omega. + rewrite list_length_z_cons. lia. Qed. (** The following lemmas show that the translation from Mach to ARM @@ -378,8 +378,8 @@ Proof. split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Ptrofs.unsigned_repr. replace (pos' - 0) with pos' in Q. - auto. omega. - generalize (transf_function_no_overflow _ _ H0). omega. + auto. lia. + generalize (transf_function_no_overflow _ _ H0). lia. intros. apply Pregmap.gso; auto. Qed. @@ -903,11 +903,11 @@ Opaque loadind. simpl; reflexivity. reflexivity. } (* After the function prologue is the code for the function body *) - exploit exec_straight_steps_2; eauto using functions_transl. omega. constructor. + exploit exec_straight_steps_2; eauto using functions_transl. lia. constructor. intros (ofsbody & U & V). (* Conclusions *) left; exists (State rs4 m3'); split. - eapply exec_straight_steps_1; eauto. omega. constructor. + eapply exec_straight_steps_1; eauto. lia. constructor. econstructor; eauto. rewrite U. econstructor; eauto. apply agree_nextinstr. apply agree_undef_regs2 with rs2. @@ -934,7 +934,7 @@ Opaque loadind. - (* return *) inv STACKS. simpl in *. - right. split. omega. split. auto. + right. split. lia. split. auto. rewrite <- ATPC in H5. econstructor; eauto. congruence. Qed. diff --git a/arm/Asmgenproof1.v b/arm/Asmgenproof1.v index 807e069d..fce9d4a6 100644 --- a/arm/Asmgenproof1.v +++ b/arm/Asmgenproof1.v @@ -352,15 +352,15 @@ Proof. apply exec_straight_one. simpl; eauto. auto. split; intros; Simpl. econstructor; split. eapply exec_straight_two. simpl; reflexivity. simpl; reflexivity. auto. auto. - split; intros; Simpl. simpl. f_equal. rewrite Int.zero_ext_and by omega. + split; intros; Simpl. simpl. f_equal. rewrite Int.zero_ext_and by lia. rewrite Int.and_assoc. change 65535 with (two_p 16 - 1). rewrite Int.and_idem. apply Int.same_bits_eq; intros. rewrite Int.bits_or, Int.bits_and, Int.bits_shl, Int.testbit_repr by auto. - rewrite Ztestbit_two_p_m1 by omega. change (Int.unsigned (Int.repr 16)) with 16. + rewrite Ztestbit_two_p_m1 by lia. change (Int.unsigned (Int.repr 16)) with 16. destruct (zlt i 16). rewrite andb_true_r, orb_false_r; auto. - rewrite andb_false_r; simpl. rewrite Int.bits_shru by omega. - change (Int.unsigned (Int.repr 16)) with 16. rewrite zlt_true by omega. f_equal; omega. + rewrite andb_false_r; simpl. rewrite Int.bits_shru by lia. + change (Int.unsigned (Int.repr 16)) with 16. rewrite zlt_true by lia. f_equal; lia. } destruct (Nat.leb l1 l2). { (* mov - orr* *) @@ -696,10 +696,10 @@ Lemma int_not_lt: Proof. intros. unfold Int.lt. rewrite int_signed_eq. unfold proj_sumbool. destruct (zlt (Int.signed y) (Int.signed x)). - rewrite zlt_false. rewrite zeq_false. auto. omega. omega. + rewrite zlt_false. rewrite zeq_false. auto. lia. lia. destruct (zeq (Int.signed x) (Int.signed y)). - rewrite zlt_false. auto. omega. - rewrite zlt_true. auto. omega. + rewrite zlt_false. auto. lia. + rewrite zlt_true. auto. lia. Qed. Lemma int_lt_not: @@ -713,10 +713,10 @@ Lemma int_not_ltu: Proof. intros. unfold Int.ltu, Int.eq. destruct (zlt (Int.unsigned y) (Int.unsigned x)). - rewrite zlt_false. rewrite zeq_false. auto. omega. omega. + rewrite zlt_false. rewrite zeq_false. auto. lia. lia. destruct (zeq (Int.unsigned x) (Int.unsigned y)). - rewrite zlt_false. auto. omega. - rewrite zlt_true. auto. omega. + rewrite zlt_false. auto. lia. + rewrite zlt_true. auto. lia. Qed. Lemma int_ltu_not: @@ -1279,16 +1279,16 @@ Local Transparent destroyed_by_op. rewrite Int.unsigned_repr. apply zlt_true. assert (Int.unsigned i <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned i). rewrite H1; reflexivity. } - omega. + lia. change (Int.unsigned (Int.repr 31)) with (Int.zwordsize - 1) in H0. - generalize Int.wordsize_max_unsigned; omega. + generalize Int.wordsize_max_unsigned; lia. } assert (LTU'': Int.ltu i Int.iwordsize = true). { generalize (Int.ltu_inv _ _ LTU). intros. unfold Int.ltu. rewrite Int.unsigned_repr_wordsize. apply zlt_true. change (Int.unsigned (Int.repr 31)) with (Int.zwordsize - 1) in H0. - omega. + lia. } set (j := Int.sub Int.iwordsize i) in *. set (rs1 := nextinstr_nf (rs#IR14 <- (Val.shr (Vint i0) (Vint (Int.repr 31))))). diff --git a/arm/ConstpropOpproof.v b/arm/ConstpropOpproof.v index a4f5c29c..cd0afb7a 100644 --- a/arm/ConstpropOpproof.v +++ b/arm/ConstpropOpproof.v @@ -451,7 +451,7 @@ Proof. Int.bit_solve. destruct (zlt i0 n0). replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)). rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto. - rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto. + rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto. rewrite Int.bits_not by auto. apply negb_involutive. rewrite H6 by auto. auto. econstructor; split; eauto. auto. diff --git a/arm/Conventions1.v b/arm/Conventions1.v index fe49a781..3bd2b553 100644 --- a/arm/Conventions1.v +++ b/arm/Conventions1.v @@ -309,7 +309,7 @@ Remark loc_arguments_hf_charact: In p (loc_arguments_hf tyl ir fr ofs) -> forall_rpair (loc_argument_charact ofs) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_charact ofs1 l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition lia. } assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_charact ofs2) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_charact ofs1) p). { destruct p; simpl; intuition eauto. } induction tyl; simpl loc_arguments_hf; intros. @@ -319,40 +319,40 @@ Proof. destruct (zlt ir 4); destruct H. subst. apply ireg_param_caller_save. eapply IHtyl; eauto. - subst. split; [omega | auto]. - eapply Y; eauto. omega. + subst. split; [lia | auto]. + eapply Y; eauto. lia. - (* float *) destruct (zlt fr 8); destruct H. subst. apply freg_param_caller_save. eapply IHtyl; eauto. - subst. split. apply Z.le_ge. apply align_le. omega. auto. - eapply Y; eauto. apply Z.le_trans with (align ofs 2). apply align_le; omega. omega. + subst. split. apply Z.le_ge. apply align_le. lia. auto. + eapply Y; eauto. apply Z.le_trans with (align ofs 2). apply align_le; lia. lia. - (* long *) set (ir' := align ir 2) in *. - assert (ofs <= align ofs 2) by (apply align_le; omega). + assert (ofs <= align ofs 2) by (apply align_le; lia). destruct (zlt ir' 4). destruct H. subst p. split; apply ireg_param_caller_save. eapply IHtyl; eauto. - destruct H. subst p. split; destruct Archi.big_endian; (split; [ omega | auto ]). - eapply Y. eapply IHtyl; eauto. omega. + destruct H. subst p. split; destruct Archi.big_endian; (split; [ lia | auto ]). + eapply Y. eapply IHtyl; eauto. lia. - (* single *) destruct (zlt fr 8); destruct H. subst. apply freg_param_caller_save. eapply IHtyl; eauto. - subst. split; [omega|auto]. - eapply Y; eauto. omega. + subst. split; [lia|auto]. + eapply Y; eauto. lia. - (* any32 *) destruct (zlt ir 4); destruct H. subst. apply ireg_param_caller_save. eapply IHtyl; eauto. - subst. split; [omega | auto]. - eapply Y; eauto. omega. + subst. split; [lia | auto]. + eapply Y; eauto. lia. - (* any64 *) destruct (zlt fr 8); destruct H. subst. apply freg_param_caller_save. eapply IHtyl; eauto. - subst. split. apply Z.le_ge. apply align_le. omega. auto. - eapply Y; eauto. apply Z.le_trans with (align ofs 2). apply align_le; omega. omega. + subst. split. apply Z.le_ge. apply align_le. lia. auto. + eapply Y; eauto. apply Z.le_trans with (align ofs 2). apply align_le; lia. lia. Qed. Remark loc_arguments_sf_charact: @@ -360,7 +360,7 @@ Remark loc_arguments_sf_charact: In p (loc_arguments_sf tyl ofs) -> forall_rpair (loc_argument_charact (Z.max 0 ofs)) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_charact (Z.max 0 ofs2) l -> ofs1 <= ofs2 -> loc_argument_charact (Z.max 0 ofs1) l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition xomega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition extlia. } assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_charact (Z.max 0 ofs2)) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_charact (Z.max 0 ofs1)) p). { destruct p; simpl; intuition eauto. } induction tyl; simpl loc_arguments_sf; intros. @@ -370,44 +370,44 @@ Proof. destruct H. destruct (zlt ofs 0); subst p. apply ireg_param_caller_save. - split; [xomega|auto]. - eapply Y; eauto. omega. + split; [extlia|auto]. + eapply Y; eauto. lia. - (* float *) set (ofs' := align ofs 2) in *. - assert (ofs <= ofs') by (apply align_le; omega). + assert (ofs <= ofs') by (apply align_le; lia). destruct H. destruct (zlt ofs' 0); subst p. apply freg_param_caller_save. - split; [xomega|auto]. - eapply Y. eapply IHtyl; eauto. omega. + split; [extlia|auto]. + eapply Y. eapply IHtyl; eauto. lia. - (* long *) set (ofs' := align ofs 2) in *. - assert (ofs <= ofs') by (apply align_le; omega). + assert (ofs <= ofs') by (apply align_le; lia). destruct H. destruct (zlt ofs' 0); subst p. split; apply ireg_param_caller_save. - split; destruct Archi.big_endian; (split; [xomega|auto]). - eapply Y. eapply IHtyl; eauto. omega. + split; destruct Archi.big_endian; (split; [extlia|auto]). + eapply Y. eapply IHtyl; eauto. lia. - (* single *) destruct H. destruct (zlt ofs 0); subst p. apply freg_param_caller_save. - split; [xomega|auto]. - eapply Y; eauto. omega. + split; [extlia|auto]. + eapply Y; eauto. lia. - (* any32 *) destruct H. destruct (zlt ofs 0); subst p. apply ireg_param_caller_save. - split; [xomega|auto]. - eapply Y; eauto. omega. + split; [extlia|auto]. + eapply Y; eauto. lia. - (* any64 *) set (ofs' := align ofs 2) in *. - assert (ofs <= ofs') by (apply align_le; omega). + assert (ofs <= ofs') by (apply align_le; lia). destruct H. destruct (zlt ofs' 0); subst p. apply freg_param_caller_save. - split; [xomega|auto]. - eapply Y. eapply IHtyl; eauto. omega. + split; [extlia|auto]. + eapply Y. eapply IHtyl; eauto. lia. Qed. Lemma loc_arguments_acceptable: diff --git a/arm/NeedOp.v b/arm/NeedOp.v index c70c7e40..23e8f047 100644 --- a/arm/NeedOp.v +++ b/arm/NeedOp.v @@ -198,8 +198,8 @@ Lemma operation_is_redundant_sound: vagree v arg1' nv. Proof. intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst. -- apply sign_ext_redundant_sound; auto. omega. -- apply sign_ext_redundant_sound; auto. omega. +- apply sign_ext_redundant_sound; auto. lia. +- apply sign_ext_redundant_sound; auto. lia. - apply andimm_redundant_sound; auto. - apply orimm_redundant_sound; auto. Qed. @@ -527,10 +527,10 @@ End SOUNDNESS. Program Definition mk_shift_amount (n: int) : shift_amount := {| s_amount := Int.modu n Int.iwordsize; s_range := _ |}. Next Obligation. - assert (0 <= Z.modulo (Int.unsigned n) 32 < 32). apply Z_mod_lt. omega. + assert (0 <= Z.modulo (Int.unsigned n) 32 < 32). apply Z_mod_lt. lia. unfold Int.ltu, Int.modu. change (Int.unsigned Int.iwordsize) with 32. - rewrite Int.unsigned_repr. apply zlt_true. omega. - assert (32 < Int.max_unsigned). compute; auto. omega. + rewrite Int.unsigned_repr. apply zlt_true. lia. + assert (32 < Int.max_unsigned). compute; auto. lia. Qed. Lemma mk_shift_amount_eq: diff --git a/arm/SelectOpproof.v b/arm/SelectOpproof.v index 70f8f191..bd9f01b1 100644 --- a/arm/SelectOpproof.v +++ b/arm/SelectOpproof.v @@ -754,7 +754,7 @@ Qed. Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8). Proof. red; intros until x. unfold cast8unsigned. - rewrite Val.zero_ext_and. apply eval_andimm. omega. + rewrite Val.zero_ext_and. apply eval_andimm. lia. Qed. Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16). @@ -767,7 +767,7 @@ Qed. Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16). Proof. red; intros until x. unfold cast8unsigned. - rewrite Val.zero_ext_and. apply eval_andimm. omega. + rewrite Val.zero_ext_and. apply eval_andimm. lia. Qed. Theorem eval_singleoffloat: unary_constructor_sound singleoffloat Val.singleoffloat. diff --git a/arm/Stacklayout.v b/arm/Stacklayout.v index 462d83ad..f6e01e0c 100644 --- a/arm/Stacklayout.v +++ b/arm/Stacklayout.v @@ -72,12 +72,12 @@ Local Opaque Z.add Z.mul sepconj range. set (ocs := ol + 4 * b.(bound_local)); set (ostkdata := align (size_callee_save_area b ocs) 8). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= olink) by (unfold olink; omega). - assert (olink <= ora) by (unfold ora; omega). - assert (ora + 4 <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ocs) by (unfold ocs; omega). + assert (0 <= olink) by (unfold olink; lia). + assert (olink <= ora) by (unfold ora; lia). + assert (ora + 4 <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by apply size_callee_save_area_incr. - assert (size_callee_save_area b ocs <= ostkdata) by (apply align_le; omega). + assert (size_callee_save_area b ocs <= ostkdata) by (apply align_le; lia). (* Reorder as: outgoing back link @@ -89,11 +89,11 @@ Local Opaque Z.add Z.mul sepconj range. rewrite sep_swap34. (* Apply range_split and range_split2 repeatedly *) unfold fe_ofs_arg. - apply range_split. omega. - apply range_split. omega. - apply range_split_2. fold ol; omega. omega. - apply range_split. omega. - apply range_drop_right with ostkdata. omega. + apply range_split. lia. + apply range_split. lia. + apply range_split_2. fold ol; lia. lia. + apply range_split. lia. + apply range_drop_right with ostkdata. lia. eapply sep_drop2. eexact H. Qed. @@ -109,13 +109,13 @@ Proof. set (ocs := ol + 4 * b.(bound_local)); set (ostkdata := align (size_callee_save_area b ocs) 8). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= olink) by (unfold olink; omega). - assert (olink <= ora) by (unfold ora; omega). - assert (ora + 4 <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ocs) by (unfold ocs; omega). + assert (0 <= olink) by (unfold olink; lia). + assert (olink <= ora) by (unfold ora; lia). + assert (ora + 4 <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by apply size_callee_save_area_incr. - assert (size_callee_save_area b ocs <= ostkdata) by (apply align_le; omega). - split. omega. apply align_le; omega. + assert (size_callee_save_area b ocs <= ostkdata) by (apply align_le; lia). + split. lia. apply align_le; lia. Qed. Lemma frame_env_aligned: @@ -134,7 +134,7 @@ Proof. set (ocs := ol + 4 * b.(bound_local)); set (ostkdata := align (size_callee_save_area b ocs) 8). split. apply Z.divide_0_r. - split. apply align_divides; omega. - split. apply align_divides; omega. + split. apply align_divides; lia. + split. apply align_divides; lia. unfold ora, olink; auto using Z.divide_mul_l, Z.divide_add_r, Z.divide_refl. Qed. diff --git a/backend/Allocproof.v b/backend/Allocproof.v index 51755912..3fdbacbe 100644 --- a/backend/Allocproof.v +++ b/backend/Allocproof.v @@ -548,7 +548,7 @@ Proof. unfold select_reg_l; intros. destruct H. red in H. congruence. rewrite Pos.leb_le in *. red in H. destruct H as [A | [A B]]. - red in A. zify; omega. + red in A. zify; lia. rewrite <- A; auto. Qed. @@ -560,7 +560,7 @@ Proof. unfold select_reg_h; intros. destruct H. red in H. congruence. rewrite Pos.leb_le in *. red in H. destruct H as [A | [A B]]. - red in A. zify; omega. + red in A. zify; lia. rewrite A; auto. Qed. @@ -568,7 +568,7 @@ Remark select_reg_charact: forall r q, select_reg_l r q = true /\ select_reg_h r q = true <-> ereg q = r. Proof. unfold select_reg_l, select_reg_h; intros; split. - rewrite ! Pos.leb_le. unfold reg; zify; omega. + rewrite ! Pos.leb_le. unfold reg; zify; lia. intros. rewrite H. rewrite ! Pos.leb_refl; auto. Qed. diff --git a/backend/Asmgenproof0.v b/backend/Asmgenproof0.v index 3638c465..5e8acd6f 100644 --- a/backend/Asmgenproof0.v +++ b/backend/Asmgenproof0.v @@ -473,7 +473,7 @@ Inductive code_tail: Z -> code -> code -> Prop := Lemma code_tail_pos: forall pos c1 c2, code_tail pos c1 c2 -> pos >= 0. Proof. - induction 1. omega. omega. + induction 1. lia. lia. Qed. Lemma find_instr_tail: @@ -484,8 +484,8 @@ Proof. induction c1; simpl; intros. inv H. destruct (zeq pos 0). subst pos. - inv H. auto. generalize (code_tail_pos _ _ _ H4). intro. omegaContradiction. - inv H. congruence. replace (pos0 + 1 - 1) with pos0 by omega. + inv H. auto. generalize (code_tail_pos _ _ _ H4). intro. extlia. + inv H. congruence. replace (pos0 + 1 - 1) with pos0 by lia. eauto. Qed. @@ -494,8 +494,8 @@ Remark code_tail_bounds_1: code_tail ofs fn c -> 0 <= ofs <= list_length_z fn. Proof. induction 1; intros; simpl. - generalize (list_length_z_pos c). omega. - rewrite list_length_z_cons. omega. + generalize (list_length_z_pos c). lia. + rewrite list_length_z_cons. lia. Qed. Remark code_tail_bounds_2: @@ -505,8 +505,8 @@ Proof. assert (forall ofs fn c, code_tail ofs fn c -> forall i c', c = i :: c' -> 0 <= ofs < list_length_z fn). induction 1; intros; simpl. - rewrite H. rewrite list_length_z_cons. generalize (list_length_z_pos c'). omega. - rewrite list_length_z_cons. generalize (IHcode_tail _ _ H0). omega. + rewrite H. rewrite list_length_z_cons. generalize (list_length_z_pos c'). lia. + rewrite list_length_z_cons. generalize (IHcode_tail _ _ H0). lia. eauto. Qed. @@ -531,7 +531,7 @@ Lemma code_tail_next_int: Proof. intros. rewrite Ptrofs.add_unsigned, Ptrofs.unsigned_one. rewrite Ptrofs.unsigned_repr. apply code_tail_next with i; auto. - generalize (code_tail_bounds_2 _ _ _ _ H0). omega. + generalize (code_tail_bounds_2 _ _ _ _ H0). lia. Qed. (** [transl_code_at_pc pc fb f c ep tf tc] holds if the code pointer [pc] points @@ -654,7 +654,7 @@ Opaque transl_instr. exists (Ptrofs.repr ofs). red; intros. rewrite Ptrofs.unsigned_repr. congruence. exploit code_tail_bounds_1; eauto. - apply transf_function_len in TF. omega. + apply transf_function_len in TF. lia. + exists Ptrofs.zero; red; intros. congruence. Qed. @@ -663,7 +663,7 @@ End RETADDR_EXISTS. Remark code_tail_no_bigger: forall pos c1 c2, code_tail pos c1 c2 -> (length c2 <= length c1)%nat. Proof. - induction 1; simpl; omega. + induction 1; simpl; lia. Qed. Remark code_tail_unique: @@ -671,8 +671,8 @@ Remark code_tail_unique: code_tail pos fn c -> code_tail pos' fn c -> pos = pos'. Proof. induction fn; intros until pos'; intros ITA CT; inv ITA; inv CT; auto. - generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; omega. - generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; omega. + generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; lia. + generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; lia. f_equal. eauto. Qed. @@ -713,13 +713,13 @@ Proof. case (is_label lbl a). intro EQ; injection EQ; intro; subst c'. exists (pos + 1). split. auto. split. - replace (pos + 1 - pos) with (0 + 1) by omega. constructor. constructor. - rewrite list_length_z_cons. generalize (list_length_z_pos c). omega. + replace (pos + 1 - pos) with (0 + 1) by lia. constructor. constructor. + rewrite list_length_z_cons. generalize (list_length_z_pos c). lia. intros. generalize (IHc (pos + 1) c' H). intros [pos' [A [B C]]]. exists pos'. split. auto. split. - replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by omega. + replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by lia. constructor. auto. - rewrite list_length_z_cons. omega. + rewrite list_length_z_cons. lia. Qed. (** Helper lemmas to reason about diff --git a/backend/Bounds.v b/backend/Bounds.v index fa695234..4231d861 100644 --- a/backend/Bounds.v +++ b/backend/Bounds.v @@ -163,7 +163,7 @@ Proof. intros until valu. unfold max_over_list. assert (forall l z, fold_left (fun x y => Z.max x (valu y)) l z >= z). induction l; simpl; intros. - omega. apply Zge_trans with (Z.max z (valu a)). + lia. apply Zge_trans with (Z.max z (valu a)). auto. apply Z.le_ge. apply Z.le_max_l. auto. Qed. @@ -307,7 +307,7 @@ Proof. let f := fold_left (fun x y => Z.max x (valu y)) c z in z <= f /\ (In x c -> valu x <= f)). induction c; simpl; intros. - split. omega. tauto. + split. lia. tauto. elim (IHc (Z.max z (valu a))); intros. split. apply Z.le_trans with (Z.max z (valu a)). apply Z.le_max_l. auto. intro H1; elim H1; intro. @@ -446,12 +446,12 @@ Lemma size_callee_save_area_rec_incr: Proof. Local Opaque mreg_type. induction l as [ | r l]; intros; simpl. -- omega. +- lia. - eapply Z.le_trans. 2: apply IHl. generalize (AST.typesize_pos (mreg_type r)); intros. apply Z.le_trans with (align ofs (AST.typesize (mreg_type r))). apply align_le; auto. - omega. + lia. Qed. Lemma size_callee_save_area_incr: diff --git a/backend/CSEproof.v b/backend/CSEproof.v index 03c7ecfc..a2a1b461 100644 --- a/backend/CSEproof.v +++ b/backend/CSEproof.v @@ -128,9 +128,9 @@ Proof. exists valu2; splitall. + constructor; simpl; intros. * constructor; simpl; intros. - apply wf_equation_incr with (num_next n). eauto with cse. xomega. + apply wf_equation_incr with (num_next n). eauto with cse. extlia. rewrite PTree.gsspec in H0. destruct (peq r0 r). - inv H0; xomega. + inv H0; extlia. apply Plt_trans_succ; eauto with cse. rewrite PMap.gsspec in H0. destruct (peq v (num_next n)). replace r0 with r by (simpl in H0; intuition). rewrite PTree.gss. subst; auto. @@ -142,8 +142,8 @@ Proof. rewrite peq_false. eauto with cse. apply Plt_ne; eauto with cse. + unfold valu2. rewrite peq_true; auto. + auto. -+ xomega. -+ xomega. ++ extlia. ++ extlia. Qed. Lemma valnum_regs_holds: @@ -158,7 +158,7 @@ Lemma valnum_regs_holds: /\ Ple n.(num_next) n'.(num_next). Proof. induction rl; simpl; intros. -- inv H0. exists valu1; splitall; auto. red; auto. simpl; tauto. xomega. +- inv H0. exists valu1; splitall; auto. red; auto. simpl; tauto. extlia. - destruct (valnum_reg n a) as [n1 v1] eqn:V1. destruct (valnum_regs n1 rl) as [n2 vs] eqn:V2. inv H0. @@ -169,9 +169,9 @@ Proof. exists valu3; splitall. + auto. + simpl; f_equal; auto. rewrite R; auto. - + red; intros. transitivity (valu2 v); auto. apply R. xomega. - + simpl; intros. destruct H0; auto. subst v1; xomega. - + xomega. + + red; intros. transitivity (valu2 v); auto. apply R. extlia. + + simpl; intros. destruct H0; auto. subst v1; extlia. + + extlia. Qed. Lemma find_valnum_rhs_charact: @@ -327,11 +327,11 @@ Proof. { red; intros. unfold valu2. apply peq_false. apply Plt_ne; auto. } exists valu2; constructor; simpl; intros. + constructor; simpl; intros. - * destruct H3. inv H3. simpl; split. xomega. + * destruct H3. inv H3. simpl; split. extlia. red; intros. apply Plt_trans_succ; eauto. - apply wf_equation_incr with (num_next n). eauto with cse. xomega. + apply wf_equation_incr with (num_next n). eauto with cse. extlia. * rewrite PTree.gsspec in H3. destruct (peq r rd). - inv H3. xomega. + inv H3. extlia. apply Plt_trans_succ; eauto with cse. * apply update_reg_charact; eauto with cse. + destruct H3. inv H3. @@ -495,10 +495,10 @@ Lemma store_normalized_range_sound: Proof. intros. unfold Val.load_result; remember Archi.ptr64 as ptr64. destruct chunk; simpl in *; destruct v; auto. -- inv H. rewrite is_sgn_sign_ext in H4 by omega. rewrite H4; auto. -- inv H. rewrite is_uns_zero_ext in H4 by omega. rewrite H4; auto. -- inv H. rewrite is_sgn_sign_ext in H4 by omega. rewrite H4; auto. -- inv H. rewrite is_uns_zero_ext in H4 by omega. rewrite H4; auto. +- inv H. rewrite is_sgn_sign_ext in H4 by lia. rewrite H4; auto. +- inv H. rewrite is_uns_zero_ext in H4 by lia. rewrite H4; auto. +- inv H. rewrite is_sgn_sign_ext in H4 by lia. rewrite H4; auto. +- inv H. rewrite is_uns_zero_ext in H4 by lia. rewrite H4; auto. - destruct ptr64; auto. - destruct ptr64; auto. - destruct ptr64; auto. @@ -557,7 +557,7 @@ Proof. simpl. rewrite negb_false_iff in H8. eapply Mem.load_storebytes_other. eauto. - rewrite H6. rewrite Z2Nat.id by omega. + rewrite H6. rewrite Z2Nat.id by lia. eapply pdisjoint_sound. eauto. unfold aaddressing. apply match_aptr_of_aval. eapply eval_static_addressing_sound; eauto. erewrite <- regs_valnums_sound by eauto. eauto with va. @@ -578,39 +578,39 @@ Proof. set (n1 := i - ofs1). set (n2 := size_chunk chunk). set (n3 := sz - (n1 + n2)). - replace sz with (n1 + (n2 + n3)) in H by (unfold n3, n2, n1; omega). + replace sz with (n1 + (n2 + n3)) in H by (unfold n3, n2, n1; lia). exploit Mem.loadbytes_split; eauto. - unfold n1; omega. - unfold n3, n2, n1; omega. + unfold n1; lia. + unfold n3, n2, n1; lia. intros (bytes1 & bytes23 & LB1 & LB23 & EQ). clear H. exploit Mem.loadbytes_split; eauto. - unfold n2; omega. - unfold n3, n2, n1; omega. + unfold n2; lia. + unfold n3, n2, n1; lia. intros (bytes2 & bytes3 & LB2 & LB3 & EQ'). subst bytes23; subst bytes. exploit Mem.load_loadbytes; eauto. intros (bytes2' & A & B). assert (bytes2' = bytes2). - { replace (ofs1 + n1) with i in LB2 by (unfold n1; omega). unfold n2 in LB2. congruence. } + { replace (ofs1 + n1) with i in LB2 by (unfold n1; lia). unfold n2 in LB2. congruence. } subst bytes2'. exploit Mem.storebytes_split; eauto. intros (m1 & SB1 & SB23). clear H0. exploit Mem.storebytes_split; eauto. intros (m2 & SB2 & SB3). clear SB23. assert (L1: Z.of_nat (length bytes1) = n1). - { erewrite Mem.loadbytes_length by eauto. apply Z2Nat.id. unfold n1; omega. } + { erewrite Mem.loadbytes_length by eauto. apply Z2Nat.id. unfold n1; lia. } assert (L2: Z.of_nat (length bytes2) = n2). - { erewrite Mem.loadbytes_length by eauto. apply Z2Nat.id. unfold n2; omega. } + { erewrite Mem.loadbytes_length by eauto. apply Z2Nat.id. unfold n2; lia. } rewrite L1 in *. rewrite L2 in *. assert (LB': Mem.loadbytes m2 b2 (ofs2 + n1) n2 = Some bytes2). { rewrite <- L2. eapply Mem.loadbytes_storebytes_same; eauto. } assert (LB'': Mem.loadbytes m' b2 (ofs2 + n1) n2 = Some bytes2). { rewrite <- LB'. eapply Mem.loadbytes_storebytes_other; eauto. - unfold n2; omega. - right; left; omega. } + unfold n2; lia. + right; left; lia. } exploit Mem.load_valid_access; eauto. intros [P Q]. rewrite B. apply Mem.loadbytes_load. - replace (i + (ofs2 - ofs1)) with (ofs2 + n1) by (unfold n1; omega). + replace (i + (ofs2 - ofs1)) with (ofs2 + n1) by (unfold n1; lia). exact LB''. apply Z.divide_add_r; auto. Qed. @@ -655,9 +655,9 @@ Proof with (try discriminate). Mem.loadv chunk m (Vptr sp ofs) = Some v -> Mem.loadv chunk m' (Vptr sp (Ptrofs.repr j)) = Some v). { - simpl; intros. rewrite Ptrofs.unsigned_repr by omega. + simpl; intros. rewrite Ptrofs.unsigned_repr by lia. unfold j, delta. eapply load_memcpy; eauto. - apply Zmod_divide; auto. generalize (align_chunk_pos chunk); omega. + apply Zmod_divide; auto. generalize (align_chunk_pos chunk); lia. } inv H2. + inv H3. exploit eval_addressing_Ainstack_inv; eauto. intros [E1 E2]. diff --git a/backend/CleanupLabelsproof.v b/backend/CleanupLabelsproof.v index e92be2b4..fb8e57b7 100644 --- a/backend/CleanupLabelsproof.v +++ b/backend/CleanupLabelsproof.v @@ -286,7 +286,7 @@ Proof. constructor. econstructor; eauto with coqlib. (* eliminated *) - right. split. simpl. omega. split. auto. econstructor; eauto with coqlib. + right. split. simpl. lia. split. auto. econstructor; eauto with coqlib. (* Lgoto *) left; econstructor; split. econstructor. eapply find_label_translated; eauto. red; auto. diff --git a/backend/Cminor.v b/backend/Cminor.v index 91a4c104..cf0ba314 100644 --- a/backend/Cminor.v +++ b/backend/Cminor.v @@ -588,7 +588,7 @@ Proof. exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]]. exists (Returnstate vres2 k m2). econstructor; eauto. (* trace length *) - red; intros; inv H; simpl; try omega; eapply external_call_trace_length; eauto. + red; intros; inv H; simpl; try lia; eapply external_call_trace_length; eauto. Qed. (** This semantics is determinate. *) @@ -645,7 +645,7 @@ Proof. intros (A & B). split; intros; auto. apply B in H; destruct H; congruence. - (* single event *) - red; simpl. destruct 1; simpl; try omega; + red; simpl. destruct 1; simpl; try lia; eapply external_call_trace_length; eauto. - (* initial states *) inv H; inv H0. unfold ge0, ge1 in *. congruence. diff --git a/backend/CminorSel.v b/backend/CminorSel.v index 96cb8ae6..5cbdc249 100644 --- a/backend/CminorSel.v +++ b/backend/CminorSel.v @@ -522,9 +522,9 @@ Lemma insert_lenv_lookup1: nth_error le' n = Some v. Proof. induction 1; intros. - omegaContradiction. + extlia. destruct n; simpl; simpl in H0. auto. - apply IHinsert_lenv. auto. omega. + apply IHinsert_lenv. auto. lia. Qed. Lemma insert_lenv_lookup2: @@ -536,8 +536,8 @@ Lemma insert_lenv_lookup2: Proof. induction 1; intros. simpl. assumption. - simpl. destruct n. omegaContradiction. - apply IHinsert_lenv. exact H0. omega. + simpl. destruct n. extlia. + apply IHinsert_lenv. exact H0. lia. Qed. Lemma eval_lift_expr: diff --git a/backend/Constpropproof.v b/backend/Constpropproof.v index a5d08a0f..a3592c4d 100644 --- a/backend/Constpropproof.v +++ b/backend/Constpropproof.v @@ -364,7 +364,7 @@ Proof. - (* Inop, skipped over *) assert (s0 = pc') by congruence. subst s0. - right; exists n; split. omega. split. auto. + right; exists n; split. lia. split. auto. apply match_states_intro; auto. - (* Iop *) @@ -528,7 +528,7 @@ Opaque builtin_strength_reduction. - (* Icond, skipped over *) rewrite H1 in H; inv H. - right; exists n; split. omega. split. auto. + right; exists n; split. lia. split. auto. econstructor; eauto. - (* Ijumptable *) diff --git a/backend/Conventions.v b/backend/Conventions.v index 14ffb587..8910ee49 100644 --- a/backend/Conventions.v +++ b/backend/Conventions.v @@ -60,9 +60,9 @@ Remark fold_max_outgoing_above: forall l n, fold_left max_outgoing_2 l n >= n. Proof. assert (A: forall n l, max_outgoing_1 n l >= n). - { intros; unfold max_outgoing_1. destruct l as [_ | []]; xomega. } + { intros; unfold max_outgoing_1. destruct l as [_ | []]; extlia. } induction l; simpl; intros. - - omega. + - lia. - eapply Zge_trans. eauto. destruct a; simpl. apply A. eapply Zge_trans; eauto. Qed. @@ -80,14 +80,14 @@ Lemma loc_arguments_bounded: Proof. intros until ty. assert (A: forall n l, n <= max_outgoing_1 n l). - { intros; unfold max_outgoing_1. destruct l as [_ | []]; xomega. } + { intros; unfold max_outgoing_1. destruct l as [_ | []]; extlia. } assert (B: forall p n, In (S Outgoing ofs ty) (regs_of_rpair p) -> ofs + typesize ty <= max_outgoing_2 n p). { intros. destruct p; simpl in H; intuition; subst; simpl. - - xomega. - - eapply Z.le_trans. 2: apply A. xomega. - - xomega. } + - extlia. + - eapply Z.le_trans. 2: apply A. extlia. + - extlia. } assert (C: forall l n, In (S Outgoing ofs ty) (regs_of_rpairs l) -> ofs + typesize ty <= fold_left max_outgoing_2 l n). @@ -168,7 +168,7 @@ Proof. unfold loc_argument_acceptable. destruct l; intros. auto. destruct sl; try contradiction. destruct H1. generalize (loc_arguments_bounded _ _ _ H0). - generalize (typesize_pos ty). omega. + generalize (typesize_pos ty). lia. Qed. diff --git a/backend/Deadcodeproof.v b/backend/Deadcodeproof.v index 2edc0395..7aa6ff88 100644 --- a/backend/Deadcodeproof.v +++ b/backend/Deadcodeproof.v @@ -67,7 +67,7 @@ Lemma mextends_agree: forall m1 m2 P, Mem.extends m1 m2 -> magree m1 m2 P. Proof. intros. destruct H. destruct mext_inj. constructor; intros. -- replace ofs with (ofs + 0) by omega. eapply mi_perm; eauto. auto. +- replace ofs with (ofs + 0) by lia. eapply mi_perm; eauto. auto. - eauto. - exploit mi_memval; eauto. unfold inject_id; eauto. rewrite Z.add_0_r. auto. @@ -99,15 +99,15 @@ Proof. induction n; intros; simpl. constructor. rewrite Nat2Z.inj_succ in H. constructor. - apply H. omega. - apply IHn. intros; apply H; omega. + apply H. lia. + apply IHn. intros; apply H; lia. } Local Transparent Mem.loadbytes. unfold Mem.loadbytes; intros. destruct H. destruct (Mem.range_perm_dec m1 b ofs (ofs + n) Cur Readable); inv H0. rewrite pred_dec_true. econstructor; split; eauto. apply GETN. intros. rewrite Z_to_nat_max in H. - assert (ofs <= i < ofs + n) by xomega. + assert (ofs <= i < ofs + n) by extlia. apply ma_memval0; auto. red; intros; eauto. Qed. @@ -146,11 +146,11 @@ Proof. (ZMap.get q (Mem.setN bytes2 p c2))). { induction 1; intros; simpl. - - apply H; auto. simpl. omega. + - apply H; auto. simpl. lia. - simpl length in H1; rewrite Nat2Z.inj_succ in H1. apply IHlist_forall2; auto. intros. rewrite ! ZMap.gsspec. destruct (ZIndexed.eq i p). auto. - apply H1; auto. unfold ZIndexed.t in *; omega. + apply H1; auto. unfold ZIndexed.t in *; lia. } intros. destruct (Mem.range_perm_storebytes m2 b ofs bytes2) as [m2' ST2]. @@ -211,8 +211,8 @@ Proof. - rewrite (Mem.storebytes_mem_contents _ _ _ _ _ H0). rewrite PMap.gsspec. destruct (peq b0 b). + subst b0. rewrite Mem.setN_outside. eapply ma_memval; eauto. eapply Mem.perm_storebytes_2; eauto. - destruct (zlt ofs0 ofs); auto. destruct (zle (ofs + Z.of_nat (length bytes1)) ofs0); try omega. - elim (H1 ofs0). omega. auto. + destruct (zlt ofs0 ofs); auto. destruct (zle (ofs + Z.of_nat (length bytes1)) ofs0); try lia. + elim (H1 ofs0). lia. auto. + eapply ma_memval; eauto. eapply Mem.perm_storebytes_2; eauto. - rewrite (Mem.nextblock_storebytes _ _ _ _ _ H0). eapply ma_nextblock; eauto. @@ -966,7 +966,7 @@ Ltac UseTransfer := intros. eapply nlive_remove; eauto. unfold adst, vanalyze; rewrite AN; eapply aaddr_arg_sound_1; eauto. erewrite Mem.loadbytes_length in H1 by eauto. - rewrite Z2Nat.id in H1 by omega. auto. + rewrite Z2Nat.id in H1 by lia. auto. eauto. intros (tm' & A & B). econstructor; split. @@ -993,7 +993,7 @@ Ltac UseTransfer := intros (bc & A & B & C). intros. eapply nlive_contains; eauto. erewrite Mem.loadbytes_length in H0 by eauto. - rewrite Z2Nat.id in H0 by omega. auto. + rewrite Z2Nat.id in H0 by lia. auto. + (* annot *) destruct (transfer_builtin_args (kill_builtin_res res ne, nm) _x2) as (ne1, nm1) eqn:TR. InvSoundState. diff --git a/backend/Inlining.v b/backend/Inlining.v index f7ee4166..7eb0f0fa 100644 --- a/backend/Inlining.v +++ b/backend/Inlining.v @@ -71,12 +71,12 @@ Inductive sincr (s1 s2: state) : Prop := Remark sincr_refl: forall s, sincr s s. Proof. - intros; constructor; xomega. + intros; constructor; extlia. Qed. Lemma sincr_trans: forall s1 s2 s3, sincr s1 s2 -> sincr s2 s3 -> sincr s1 s3. Proof. - intros. inv H; inv H0. constructor; xomega. + intros. inv H; inv H0. constructor; extlia. Qed. (** Dependently-typed state monad, ensuring that the final state is @@ -111,7 +111,7 @@ Program Definition set_instr (pc: node) (i: instruction): mon unit := (mkstate s.(st_nextreg) s.(st_nextnode) (PTree.set pc i s.(st_code)) s.(st_stksize)) _. Next Obligation. - intros; constructor; simpl; xomega. + intros; constructor; simpl; extlia. Qed. Program Definition add_instr (i: instruction): mon node := @@ -121,7 +121,7 @@ Program Definition add_instr (i: instruction): mon node := (mkstate s.(st_nextreg) (Pos.succ pc) (PTree.set pc i s.(st_code)) s.(st_stksize)) _. Next Obligation. - intros; constructor; simpl; xomega. + intros; constructor; simpl; extlia. Qed. Program Definition reserve_nodes (numnodes: positive): mon positive := @@ -130,7 +130,7 @@ Program Definition reserve_nodes (numnodes: positive): mon positive := (mkstate s.(st_nextreg) (Pos.add s.(st_nextnode) numnodes) s.(st_code) s.(st_stksize)) _. Next Obligation. - intros; constructor; simpl; xomega. + intros; constructor; simpl; extlia. Qed. Program Definition reserve_regs (numregs: positive): mon positive := @@ -139,7 +139,7 @@ Program Definition reserve_regs (numregs: positive): mon positive := (mkstate (Pos.add s.(st_nextreg) numregs) s.(st_nextnode) s.(st_code) s.(st_stksize)) _. Next Obligation. - intros; constructor; simpl; xomega. + intros; constructor; simpl; extlia. Qed. Program Definition request_stack (sz: Z): mon unit := @@ -148,7 +148,7 @@ Program Definition request_stack (sz: Z): mon unit := (mkstate s.(st_nextreg) s.(st_nextnode) s.(st_code) (Z.max s.(st_stksize) sz)) _. Next Obligation. - intros; constructor; simpl; xomega. + intros; constructor; simpl; extlia. Qed. Program Definition ptree_mfold {A: Type} (f: positive -> A -> mon unit) (t: PTree.t A): mon unit := diff --git a/backend/Inliningproof.v b/backend/Inliningproof.v index cc84b1cc..0434a4a4 100644 --- a/backend/Inliningproof.v +++ b/backend/Inliningproof.v @@ -67,21 +67,21 @@ Qed. Remark sreg_below_diff: forall ctx r r', Plt r' ctx.(dreg) -> sreg ctx r <> r'. Proof. - intros. zify. unfold sreg; rewrite shiftpos_eq. xomega. + intros. zify. unfold sreg; rewrite shiftpos_eq. extlia. Qed. Remark context_below_diff: forall ctx1 ctx2 r1 r2, context_below ctx1 ctx2 -> Ple r1 ctx1.(mreg) -> sreg ctx1 r1 <> sreg ctx2 r2. Proof. - intros. red in H. zify. unfold sreg; rewrite ! shiftpos_eq. xomega. + intros. red in H. zify. unfold sreg; rewrite ! shiftpos_eq. extlia. Qed. Remark context_below_lt: forall ctx1 ctx2 r, context_below ctx1 ctx2 -> Ple r ctx1.(mreg) -> Plt (sreg ctx1 r) ctx2.(dreg). Proof. intros. red in H. unfold Plt; zify. unfold sreg; rewrite shiftpos_eq. - xomega. + extlia. Qed. (* @@ -89,7 +89,7 @@ Remark context_below_le: forall ctx1 ctx2 r, context_below ctx1 ctx2 -> Ple r ctx1.(mreg) -> Ple (sreg ctx1 r) ctx2.(dreg). Proof. intros. red in H. unfold Ple; zify. unfold sreg; rewrite shiftpos_eq. - xomega. + extlia. Qed. *) @@ -105,7 +105,7 @@ Definition val_reg_charact (F: meminj) (ctx: context) (rs': regset) (v: val) (r: Remark Plt_Ple_dec: forall p q, {Plt p q} + {Ple q p}. Proof. - intros. destruct (plt p q). left; auto. right; xomega. + intros. destruct (plt p q). left; auto. right; extlia. Qed. Lemma agree_val_reg_gen: @@ -149,7 +149,7 @@ Proof. repeat rewrite Regmap.gsspec. destruct (peq r0 r). subst r0. rewrite peq_true. auto. rewrite peq_false. auto. apply shiftpos_diff; auto. - rewrite Regmap.gso. auto. xomega. + rewrite Regmap.gso. auto. extlia. Qed. Lemma agree_set_reg_undef: @@ -184,7 +184,7 @@ Proof. unfold agree_regs; intros. destruct H. split; intros. rewrite H0. auto. apply shiftpos_above. - eapply Pos.lt_le_trans. apply shiftpos_below. xomega. + eapply Pos.lt_le_trans. apply shiftpos_below. extlia. apply H1; auto. Qed. @@ -272,7 +272,7 @@ Lemma range_private_invariant: range_private F1 m1 m1' sp lo hi. Proof. intros; red; intros. exploit H; eauto. intros [A B]. split; auto. - intros; red; intros. exploit H0; eauto. omega. intros [P Q]. + intros; red; intros. exploit H0; eauto. lia. intros [P Q]. eelim B; eauto. Qed. @@ -293,12 +293,12 @@ Lemma range_private_alloc_left: range_private F1 m1 m' sp' (base + Z.max sz 0) hi. Proof. intros; red; intros. - exploit (H ofs). generalize (Z.le_max_r sz 0). omega. intros [A B]. + exploit (H ofs). generalize (Z.le_max_r sz 0). lia. intros [A B]. split; auto. intros; red; intros. exploit Mem.perm_alloc_inv; eauto. destruct (eq_block b sp); intros. subst b. rewrite H1 in H4; inv H4. - rewrite Zmax_spec in H3. destruct (zlt 0 sz); omega. + rewrite Zmax_spec in H3. destruct (zlt 0 sz); lia. rewrite H2 in H4; auto. eelim B; eauto. Qed. @@ -313,21 +313,21 @@ Proof. intros; red; intros. destruct (zlt ofs (base + Z.max sz 0)) as [z|z]. red; split. - replace ofs with ((ofs - base) + base) by omega. + replace ofs with ((ofs - base) + base) by lia. eapply Mem.perm_inject; eauto. eapply Mem.free_range_perm; eauto. - rewrite Zmax_spec in z. destruct (zlt 0 sz); omega. + rewrite Zmax_spec in z. destruct (zlt 0 sz); lia. intros; red; intros. destruct (eq_block b b0). subst b0. rewrite H1 in H4; inv H4. - eelim Mem.perm_free_2; eauto. rewrite Zmax_spec in z. destruct (zlt 0 sz); omega. + eelim Mem.perm_free_2; eauto. rewrite Zmax_spec in z. destruct (zlt 0 sz); lia. exploit Mem.mi_no_overlap; eauto. apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable; auto with mem. eapply Mem.free_range_perm. eauto. - instantiate (1 := ofs - base). rewrite Zmax_spec in z. destruct (zlt 0 sz); omega. + instantiate (1 := ofs - base). rewrite Zmax_spec in z. destruct (zlt 0 sz); lia. eapply Mem.perm_free_3; eauto. - intros [A | A]. congruence. omega. + intros [A | A]. congruence. lia. - exploit (H ofs). omega. intros [A B]. split. auto. + exploit (H ofs). lia. intros [A B]. split. auto. intros; red; intros. eelim B; eauto. eapply Mem.perm_free_3; eauto. Qed. @@ -607,39 +607,39 @@ Proof. (* cons *) apply match_stacks_cons with (fenv := fenv) (ctx := ctx); auto. eapply match_stacks_inside_invariant; eauto. - intros; eapply INJ; eauto; xomega. - intros; eapply PERM1; eauto; xomega. - intros; eapply PERM2; eauto; xomega. - intros; eapply PERM3; eauto; xomega. + intros; eapply INJ; eauto; extlia. + intros; eapply PERM1; eauto; extlia. + intros; eapply PERM2; eauto; extlia. + intros; eapply PERM3; eauto; extlia. eapply agree_regs_incr; eauto. eapply range_private_invariant; eauto. (* untailcall *) apply match_stacks_untailcall with (ctx := ctx); auto. eapply match_stacks_inside_invariant; eauto. - intros; eapply INJ; eauto; xomega. - intros; eapply PERM1; eauto; xomega. - intros; eapply PERM2; eauto; xomega. - intros; eapply PERM3; eauto; xomega. + intros; eapply INJ; eauto; extlia. + intros; eapply PERM1; eauto; extlia. + intros; eapply PERM2; eauto; extlia. + intros; eapply PERM3; eauto; extlia. eapply range_private_invariant; eauto. induction 1; intros. (* base *) eapply match_stacks_inside_base; eauto. eapply match_stacks_invariant; eauto. - intros; eapply INJ; eauto; xomega. - intros; eapply PERM1; eauto; xomega. - intros; eapply PERM2; eauto; xomega. - intros; eapply PERM3; eauto; xomega. + intros; eapply INJ; eauto; extlia. + intros; eapply PERM1; eauto; extlia. + intros; eapply PERM2; eauto; extlia. + intros; eapply PERM3; eauto; extlia. (* inlined *) apply match_stacks_inside_inlined with (fenv := fenv) (ctx' := ctx'); auto. apply IHmatch_stacks_inside; auto. - intros. apply RS. red in BELOW. xomega. + intros. apply RS. red in BELOW. extlia. apply agree_regs_incr with F; auto. apply agree_regs_invariant with rs'; auto. - intros. apply RS. red in BELOW. xomega. + intros. apply RS. red in BELOW. extlia. eapply range_private_invariant; eauto. - intros. split. eapply INJ; eauto. xomega. eapply PERM1; eauto. xomega. - intros. eapply PERM2; eauto. xomega. + intros. split. eapply INJ; eauto. extlia. eapply PERM1; eauto. extlia. + intros. eapply PERM2; eauto. extlia. Qed. Lemma match_stacks_empty: @@ -668,7 +668,7 @@ Lemma match_stacks_inside_set_reg: match_stacks_inside F m m' stk stk' f' ctx sp' (rs'#(sreg ctx r) <- v). Proof. intros. eapply match_stacks_inside_invariant; eauto. - intros. apply Regmap.gso. zify. unfold sreg; rewrite shiftpos_eq. xomega. + intros. apply Regmap.gso. zify. unfold sreg; rewrite shiftpos_eq. extlia. Qed. Lemma match_stacks_inside_set_res: @@ -717,11 +717,11 @@ Proof. subst b1. rewrite H1 in H4. inv H4. eelim Plt_strict; eauto. (* inlined *) eapply match_stacks_inside_inlined; eauto. - eapply IHmatch_stacks_inside; eauto. destruct SBELOW. omega. + eapply IHmatch_stacks_inside; eauto. destruct SBELOW. lia. eapply agree_regs_incr; eauto. eapply range_private_invariant; eauto. intros. exploit Mem.perm_alloc_inv; eauto. destruct (eq_block b0 b); intros. - subst b0. rewrite H2 in H5; inv H5. elimtype False; xomega. + subst b0. rewrite H2 in H5; inv H5. elimtype False; extlia. rewrite H3 in H5; auto. Qed. @@ -753,25 +753,25 @@ Lemma min_alignment_sound: Proof. intros; red; intros. unfold min_alignment in H. assert (2 <= sz -> (2 | n)). intros. - destruct (zle sz 1). omegaContradiction. + destruct (zle sz 1). extlia. destruct (zle sz 2). auto. destruct (zle sz 4). apply Z.divide_trans with 4; auto. exists 2; auto. apply Z.divide_trans with 8; auto. exists 4; auto. assert (4 <= sz -> (4 | n)). intros. - destruct (zle sz 1). omegaContradiction. - destruct (zle sz 2). omegaContradiction. + destruct (zle sz 1). extlia. + destruct (zle sz 2). extlia. destruct (zle sz 4). auto. apply Z.divide_trans with 8; auto. exists 2; auto. assert (8 <= sz -> (8 | n)). intros. - destruct (zle sz 1). omegaContradiction. - destruct (zle sz 2). omegaContradiction. - destruct (zle sz 4). omegaContradiction. + destruct (zle sz 1). extlia. + destruct (zle sz 2). extlia. + destruct (zle sz 4). extlia. auto. destruct chunk; simpl in *; auto. apply Z.divide_1_l. apply Z.divide_1_l. - apply H2; omega. - apply H2; omega. + apply H2; lia. + apply H2; lia. Qed. (** Preservation by external calls *) @@ -803,19 +803,19 @@ Proof. inv MG. constructor; intros; eauto. destruct (F1 b1) as [[b2' delta']|] eqn:?. exploit INCR; eauto. intros EQ; rewrite H0 in EQ; inv EQ. eapply IMAGE; eauto. - exploit SEP; eauto. intros [A B]. elim B. red. xomega. + exploit SEP; eauto. intros [A B]. elim B. red. extlia. eapply match_stacks_cons; eauto. - eapply match_stacks_inside_extcall; eauto. xomega. + eapply match_stacks_inside_extcall; eauto. extlia. eapply agree_regs_incr; eauto. - eapply range_private_extcall; eauto. red; xomega. - intros. apply SSZ2; auto. apply MAXPERM'; auto. red; xomega. + eapply range_private_extcall; eauto. red; extlia. + intros. apply SSZ2; auto. apply MAXPERM'; auto. red; extlia. eapply match_stacks_untailcall; eauto. - eapply match_stacks_inside_extcall; eauto. xomega. - eapply range_private_extcall; eauto. red; xomega. - intros. apply SSZ2; auto. apply MAXPERM'; auto. red; xomega. + eapply match_stacks_inside_extcall; eauto. extlia. + eapply range_private_extcall; eauto. red; extlia. + intros. apply SSZ2; auto. apply MAXPERM'; auto. red; extlia. induction 1; intros. eapply match_stacks_inside_base; eauto. - eapply match_stacks_extcall; eauto. xomega. + eapply match_stacks_extcall; eauto. extlia. eapply match_stacks_inside_inlined; eauto. eapply agree_regs_incr; eauto. eapply range_private_extcall; eauto. @@ -829,7 +829,7 @@ Lemma align_unchanged: forall n amount, amount > 0 -> (amount | n) -> align n amount = n. Proof. intros. destruct H0 as [p EQ]. subst n. unfold align. decEq. - apply Zdiv_unique with (b := amount - 1). omega. omega. + apply Zdiv_unique with (b := amount - 1). lia. lia. Qed. Lemma match_stacks_inside_inlined_tailcall: @@ -849,10 +849,10 @@ Proof. (* inlined *) assert (dstk ctx <= dstk ctx'). rewrite H1. apply align_le. apply min_alignment_pos. eapply match_stacks_inside_inlined; eauto. - red; intros. destruct (zlt ofs (dstk ctx)). apply PAD; omega. apply H3. inv H4. xomega. + red; intros. destruct (zlt ofs (dstk ctx)). apply PAD; lia. apply H3. inv H4. extlia. congruence. - unfold context_below in *. xomega. - unfold context_stack_call in *. omega. + unfold context_below in *. extlia. + unfold context_stack_call in *. lia. Qed. (** ** Relating states *) @@ -1014,12 +1014,12 @@ Proof. + (* inlined *) assert (EQ: fd = Internal f0) by (eapply find_inlined_function; eauto). subst fd. - right; split. simpl; omega. split. auto. + right; split. simpl; lia. split. auto. econstructor; eauto. eapply match_stacks_inside_inlined; eauto. - red; intros. apply PRIV. inv H13. destruct H16. xomega. + red; intros. apply PRIV. inv H13. destruct H16. extlia. apply agree_val_regs_gen; auto. - red; intros; apply PRIV. destruct H16. omega. + red; intros; apply PRIV. destruct H16. lia. - (* tailcall *) exploit match_stacks_inside_globalenvs; eauto. intros [bound G]. @@ -1032,9 +1032,9 @@ Proof. assert (X: { m1' | Mem.free m'0 sp' 0 (fn_stacksize f') = Some m1'}). apply Mem.range_perm_free. red; intros. destruct (zlt ofs f.(fn_stacksize)). - replace ofs with (ofs + dstk ctx) by omega. eapply Mem.perm_inject; eauto. - eapply Mem.free_range_perm; eauto. omega. - inv FB. eapply range_private_perms; eauto. xomega. + replace ofs with (ofs + dstk ctx) by lia. eapply Mem.perm_inject; eauto. + eapply Mem.free_range_perm; eauto. lia. + inv FB. eapply range_private_perms; eauto. extlia. destruct X as [m1' FREE]. left; econstructor; split. eapply plus_one. eapply exec_Itailcall; eauto. @@ -1045,12 +1045,12 @@ Proof. intros. eapply Mem.perm_free_3; eauto. intros. eapply Mem.perm_free_1; eauto with ordered_type. intros. eapply Mem.perm_free_3; eauto. - erewrite Mem.nextblock_free; eauto. red in VB; xomega. + erewrite Mem.nextblock_free; eauto. red in VB; extlia. eapply agree_val_regs; eauto. eapply Mem.free_right_inject; eauto. eapply Mem.free_left_inject; eauto. (* show that no valid location points into the stack block being freed *) - intros. rewrite DSTK in PRIV'. exploit (PRIV' (ofs + delta)). omega. intros [P Q]. - eelim Q; eauto. replace (ofs + delta - delta) with ofs by omega. + intros. rewrite DSTK in PRIV'. exploit (PRIV' (ofs + delta)). lia. intros [P Q]. + eelim Q; eauto. replace (ofs + delta - delta) with ofs by lia. apply Mem.perm_max with k. apply Mem.perm_implies with p; auto with mem. + (* turned into a call *) left; econstructor; split. @@ -1065,7 +1065,7 @@ Proof. + (* inlined *) assert (EQ: fd = Internal f0) by (eapply find_inlined_function; eauto). subst fd. - right; split. simpl; omega. split. auto. + right; split. simpl; lia. split. auto. econstructor; eauto. eapply match_stacks_inside_inlined_tailcall; eauto. eapply match_stacks_inside_invariant; eauto. @@ -1074,7 +1074,7 @@ Proof. eapply Mem.free_left_inject; eauto. red; intros; apply PRIV'. assert (dstk ctx <= dstk ctx'). red in H14; rewrite H14. apply align_le. apply min_alignment_pos. - omega. + lia. - (* builtin *) exploit tr_funbody_inv; eauto. intros TR; inv TR. @@ -1124,10 +1124,10 @@ Proof. assert (X: { m1' | Mem.free m'0 sp' 0 (fn_stacksize f') = Some m1'}). apply Mem.range_perm_free. red; intros. destruct (zlt ofs f.(fn_stacksize)). - replace ofs with (ofs + dstk ctx) by omega. eapply Mem.perm_inject; eauto. - eapply Mem.free_range_perm; eauto. omega. + replace ofs with (ofs + dstk ctx) by lia. eapply Mem.perm_inject; eauto. + eapply Mem.free_range_perm; eauto. lia. inv FB. eapply range_private_perms; eauto. - generalize (Zmax_spec (fn_stacksize f) 0). destruct (zlt 0 (fn_stacksize f)); omega. + generalize (Zmax_spec (fn_stacksize f) 0). destruct (zlt 0 (fn_stacksize f)); lia. destruct X as [m1' FREE]. left; econstructor; split. eapply plus_one. eapply exec_Ireturn; eauto. @@ -1137,19 +1137,19 @@ Proof. intros. eapply Mem.perm_free_3; eauto. intros. eapply Mem.perm_free_1; eauto with ordered_type. intros. eapply Mem.perm_free_3; eauto. - erewrite Mem.nextblock_free; eauto. red in VB; xomega. + erewrite Mem.nextblock_free; eauto. red in VB; extlia. destruct or; simpl. apply agree_val_reg; auto. auto. eapply Mem.free_right_inject; eauto. eapply Mem.free_left_inject; eauto. (* show that no valid location points into the stack block being freed *) intros. inversion FB; subst. assert (PRIV': range_private F m' m'0 sp' (dstk ctx) f'.(fn_stacksize)). rewrite H8 in PRIV. eapply range_private_free_left; eauto. - rewrite DSTK in PRIV'. exploit (PRIV' (ofs + delta)). omega. intros [A B]. - eelim B; eauto. replace (ofs + delta - delta) with ofs by omega. + rewrite DSTK in PRIV'. exploit (PRIV' (ofs + delta)). lia. intros [A B]. + eelim B; eauto. replace (ofs + delta - delta) with ofs by lia. apply Mem.perm_max with k. apply Mem.perm_implies with p; auto with mem. + (* inlined *) - right. split. simpl. omega. split. auto. + right. split. simpl. lia. split. auto. econstructor; eauto. eapply match_stacks_inside_invariant; eauto. intros. eapply Mem.perm_free_3; eauto. @@ -1165,7 +1165,7 @@ Proof. { eapply tr_function_linkorder; eauto. } inversion TR; subst. exploit Mem.alloc_parallel_inject. eauto. eauto. apply Z.le_refl. - instantiate (1 := fn_stacksize f'). inv H1. xomega. + instantiate (1 := fn_stacksize f'). inv H1. extlia. intros [F' [m1' [sp' [A [B [C [D E]]]]]]]. left; econstructor; split. eapply plus_one. eapply exec_function_internal; eauto. @@ -1187,13 +1187,13 @@ Proof. rewrite H5. apply agree_regs_init_regs. eauto. auto. inv H1; auto. congruence. auto. eapply Mem.valid_new_block; eauto. red; intros. split. - eapply Mem.perm_alloc_2; eauto. inv H1; xomega. + eapply Mem.perm_alloc_2; eauto. inv H1; extlia. intros; red; intros. exploit Mem.perm_alloc_inv. eexact H. eauto. destruct (eq_block b stk); intros. - subst. rewrite D in H9; inv H9. inv H1; xomega. + subst. rewrite D in H9; inv H9. inv H1; extlia. rewrite E in H9; auto. eelim Mem.fresh_block_alloc. eexact A. eapply Mem.mi_mappedblocks; eauto. auto. - intros. exploit Mem.perm_alloc_inv; eauto. rewrite dec_eq_true. omega. + intros. exploit Mem.perm_alloc_inv; eauto. rewrite dec_eq_true. lia. - (* internal function, inlined *) inversion FB; subst. @@ -1203,19 +1203,19 @@ Proof. (* sp' is valid *) instantiate (1 := sp'). auto. (* offset is representable *) - instantiate (1 := dstk ctx). generalize (Z.le_max_r (fn_stacksize f) 0). omega. + instantiate (1 := dstk ctx). generalize (Z.le_max_r (fn_stacksize f) 0). lia. (* size of target block is representable *) - intros. right. exploit SSZ2; eauto with mem. inv FB; omega. + intros. right. exploit SSZ2; eauto with mem. inv FB; lia. (* we have full permissions on sp' at and above dstk ctx *) intros. apply Mem.perm_cur. apply Mem.perm_implies with Freeable; auto with mem. - eapply range_private_perms; eauto. xomega. + eapply range_private_perms; eauto. extlia. (* offset is aligned *) - replace (fn_stacksize f - 0) with (fn_stacksize f) by omega. + replace (fn_stacksize f - 0) with (fn_stacksize f) by lia. inv FB. apply min_alignment_sound; auto. (* nobody maps to (sp, dstk ctx...) *) - intros. exploit (PRIV (ofs + delta')); eauto. xomega. + intros. exploit (PRIV (ofs + delta')); eauto. extlia. intros [A B]. eelim B; eauto. - replace (ofs + delta' - delta') with ofs by omega. + replace (ofs + delta' - delta') with ofs by lia. apply Mem.perm_max with k. apply Mem.perm_implies with p; auto with mem. intros [F' [A [B [C D]]]]. exploit tr_moves_init_regs; eauto. intros [rs'' [P [Q R]]]. @@ -1224,7 +1224,7 @@ Proof. econstructor. eapply match_stacks_inside_alloc_left; eauto. eapply match_stacks_inside_invariant; eauto. - omega. + lia. eauto. auto. apply agree_regs_incr with F; auto. auto. auto. auto. @@ -1245,7 +1245,7 @@ Proof. eapply match_stacks_extcall with (F1 := F) (F2 := F1) (m1 := m) (m1' := m'0); eauto. intros; eapply external_call_max_perm; eauto. intros; eapply external_call_max_perm; eauto. - xomega. + extlia. eapply external_call_nextblock; eauto. auto. auto. @@ -1267,14 +1267,14 @@ Proof. eauto. auto. apply agree_set_reg; auto. auto. auto. auto. - red; intros. destruct (zlt ofs (dstk ctx)). apply PAD; omega. apply PRIV; omega. + red; intros. destruct (zlt ofs (dstk ctx)). apply PAD; lia. apply PRIV; lia. auto. auto. - (* return from inlined function *) inv MS0; try congruence. rewrite RET0 in RET; inv RET. unfold inline_return in AT. assert (PRIV': range_private F m m' sp' (dstk ctx' + mstk ctx') f'.(fn_stacksize)). - red; intros. destruct (zlt ofs (dstk ctx)). apply PAD. omega. apply PRIV. omega. + red; intros. destruct (zlt ofs (dstk ctx)). apply PAD. lia. apply PRIV. lia. destruct or. + (* with a result *) left; econstructor; split. diff --git a/backend/Inliningspec.v b/backend/Inliningspec.v index c345c942..477f883a 100644 --- a/backend/Inliningspec.v +++ b/backend/Inliningspec.v @@ -73,7 +73,7 @@ Qed. Lemma shiftpos_eq: forall x y, Zpos (shiftpos x y) = (Zpos x + Zpos y) - 1. Proof. intros. unfold shiftpos. zify. try rewrite Pos2Z.inj_sub. auto. - zify. omega. + zify. lia. Qed. Lemma shiftpos_inj: @@ -82,7 +82,7 @@ Proof. intros. assert (Zpos (shiftpos x n) = Zpos (shiftpos y n)) by congruence. rewrite ! shiftpos_eq in H0. - assert (Z.pos x = Z.pos y) by omega. + assert (Z.pos x = Z.pos y) by lia. congruence. Qed. @@ -95,25 +95,25 @@ Qed. Lemma shiftpos_above: forall x n, Ple n (shiftpos x n). Proof. - intros. unfold Ple; zify. rewrite shiftpos_eq. xomega. + intros. unfold Ple; zify. rewrite shiftpos_eq. extlia. Qed. Lemma shiftpos_not_below: forall x n, Plt (shiftpos x n) n -> False. Proof. - intros. generalize (shiftpos_above x n). xomega. + intros. generalize (shiftpos_above x n). extlia. Qed. Lemma shiftpos_below: forall x n, Plt (shiftpos x n) (Pos.add x n). Proof. - intros. unfold Plt; zify. rewrite shiftpos_eq. omega. + intros. unfold Plt; zify. rewrite shiftpos_eq. lia. Qed. Lemma shiftpos_le: forall x y n, Ple x y -> Ple (shiftpos x n) (shiftpos y n). Proof. - intros. unfold Ple in *; zify. rewrite ! shiftpos_eq. omega. + intros. unfold Ple in *; zify. rewrite ! shiftpos_eq. lia. Qed. @@ -219,9 +219,9 @@ Proof. induction srcs; simpl; intros. monadInv H. auto. destruct dsts; monadInv H. auto. - transitivity (st_code s0)!pc. eapply IHsrcs; eauto. monadInv EQ; simpl. xomega. + transitivity (st_code s0)!pc. eapply IHsrcs; eauto. monadInv EQ; simpl. extlia. monadInv EQ; simpl. apply PTree.gso. - inversion INCR0; simpl in *. xomega. + inversion INCR0; simpl in *. extlia. Qed. Lemma add_moves_spec: @@ -234,13 +234,13 @@ Proof. monadInv H. apply tr_moves_nil; auto. destruct dsts; monadInv H. apply tr_moves_nil; auto. apply tr_moves_cons with x. eapply IHsrcs; eauto. - intros. inversion INCR. apply H0; xomega. + intros. inversion INCR. apply H0; extlia. monadInv EQ. rewrite H0. erewrite add_moves_unchanged; eauto. simpl. apply PTree.gss. - simpl. xomega. - xomega. - inversion INCR; inversion INCR0; simpl in *; xomega. + simpl. extlia. + extlia. + inversion INCR; inversion INCR0; simpl in *; extlia. Qed. (** ** Relational specification of CFG expansion *) @@ -386,9 +386,9 @@ Proof. monadInv H. unfold inline_function in EQ. monadInv EQ. transitivity (s2.(st_code)!pc'). eauto. transitivity (s5.(st_code)!pc'). eapply add_moves_unchanged; eauto. - left. inversion INCR5. inversion INCR3. monadInv EQ1; simpl in *. xomega. + left. inversion INCR5. inversion INCR3. monadInv EQ1; simpl in *. extlia. transitivity (s4.(st_code)!pc'). eapply rec_unchanged; eauto. - simpl. monadInv EQ; simpl. monadInv EQ1; simpl. xomega. + simpl. monadInv EQ; simpl. monadInv EQ1; simpl. extlia. simpl. monadInv EQ1; simpl. auto. monadInv EQ; simpl. monadInv EQ1; simpl. auto. (* tailcall *) @@ -397,9 +397,9 @@ Proof. monadInv H. unfold inline_tail_function in EQ. monadInv EQ. transitivity (s2.(st_code)!pc'). eauto. transitivity (s5.(st_code)!pc'). eapply add_moves_unchanged; eauto. - left. inversion INCR5. inversion INCR3. monadInv EQ1; simpl in *. xomega. + left. inversion INCR5. inversion INCR3. monadInv EQ1; simpl in *. extlia. transitivity (s4.(st_code)!pc'). eapply rec_unchanged; eauto. - simpl. monadInv EQ; simpl. monadInv EQ1; simpl. xomega. + simpl. monadInv EQ; simpl. monadInv EQ1; simpl. extlia. simpl. monadInv EQ1; simpl. auto. monadInv EQ; simpl. monadInv EQ1; simpl. auto. (* return *) @@ -422,7 +422,7 @@ Proof. destruct a as [pc1 instr1]; simpl in *. monadInv H. inv H3. transitivity ((st_code s0)!pc). - eapply IHl; eauto. destruct INCR; xomega. destruct INCR; xomega. + eapply IHl; eauto. destruct INCR; extlia. destruct INCR; extlia. eapply expand_instr_unchanged; eauto. Qed. @@ -438,7 +438,7 @@ Proof. exploit ptree_mfold_spec; eauto. intros [INCR' ITER]. eapply iter_expand_instr_unchanged; eauto. subst s0; auto. - subst s0; simpl. xomega. + subst s0; simpl. extlia. red; intros. exploit list_in_map_inv; eauto. intros [pc1 [A B]]. subst pc. unfold spc in H1. eapply shiftpos_not_below; eauto. apply PTree.elements_keys_norepet. @@ -464,7 +464,7 @@ Remark min_alignment_pos: forall sz, min_alignment sz > 0. Proof. intros; unfold min_alignment. - destruct (zle sz 1). omega. destruct (zle sz 2). omega. destruct (zle sz 4); omega. + destruct (zle sz 1). lia. destruct (zle sz 2). lia. destruct (zle sz 4); lia. Qed. Ltac inv_incr := @@ -501,20 +501,20 @@ Proof. apply tr_call_inlined with (pc1 := x0) (ctx' := ctx') (f := f); auto. eapply BASE; eauto. eapply add_moves_spec; eauto. - intros. rewrite S1. eapply set_instr_other; eauto. unfold node; xomega. - xomega. xomega. + intros. rewrite S1. eapply set_instr_other; eauto. unfold node; extlia. + extlia. extlia. eapply rec_spec; eauto. red; intros. rewrite PTree.grspec in H. destruct (PTree.elt_eq id0 id); try discriminate. auto. - simpl. subst s2; simpl in *; xomega. - simpl. subst s3; simpl in *; xomega. - simpl. xomega. + simpl. subst s2; simpl in *; extlia. + simpl. subst s3; simpl in *; extlia. + simpl. extlia. simpl. apply align_divides. apply min_alignment_pos. - assert (dstk ctx + mstk ctx <= dstk ctx'). simpl. apply align_le. apply min_alignment_pos. omega. - omega. + assert (dstk ctx + mstk ctx <= dstk ctx'). simpl. apply align_le. apply min_alignment_pos. lia. + lia. intros. simpl in H. rewrite S1. - transitivity (s1.(st_code)!pc0). eapply set_instr_other; eauto. unfold node in *; xomega. - eapply add_moves_unchanged; eauto. unfold node in *; xomega. xomega. - red; simpl. subst s2; simpl in *. xomega. + transitivity (s1.(st_code)!pc0). eapply set_instr_other; eauto. unfold node in *; extlia. + eapply add_moves_unchanged; eauto. unfold node in *; extlia. extlia. + red; simpl. subst s2; simpl in *. extlia. red; simpl. split. auto. apply align_le. apply min_alignment_pos. (* tailcall *) destruct (can_inline fe s1) as [|id f P Q]. @@ -532,20 +532,20 @@ Proof. apply tr_tailcall_inlined with (pc1 := x0) (ctx' := ctx') (f := f); auto. eapply BASE; eauto. eapply add_moves_spec; eauto. - intros. rewrite S1. eapply set_instr_other; eauto. unfold node; xomega. xomega. xomega. + intros. rewrite S1. eapply set_instr_other; eauto. unfold node; extlia. extlia. extlia. eapply rec_spec; eauto. red; intros. rewrite PTree.grspec in H. destruct (PTree.elt_eq id0 id); try discriminate. auto. - simpl. subst s3; simpl in *. subst s2; simpl in *. xomega. - simpl. subst s3; simpl in *; xomega. - simpl. xomega. + simpl. subst s3; simpl in *. subst s2; simpl in *. extlia. + simpl. subst s3; simpl in *; extlia. + simpl. extlia. simpl. apply align_divides. apply min_alignment_pos. - assert (dstk ctx <= dstk ctx'). simpl. apply align_le. apply min_alignment_pos. omega. - omega. + assert (dstk ctx <= dstk ctx'). simpl. apply align_le. apply min_alignment_pos. lia. + lia. intros. simpl in H. rewrite S1. - transitivity (s1.(st_code))!pc0. eapply set_instr_other; eauto. unfold node in *; xomega. - eapply add_moves_unchanged; eauto. unfold node in *; xomega. xomega. + transitivity (s1.(st_code))!pc0. eapply set_instr_other; eauto. unfold node in *; extlia. + eapply add_moves_unchanged; eauto. unfold node in *; extlia. extlia. red; simpl. -subst s2; simpl in *; xomega. +subst s2; simpl in *; extlia. red; auto. (* builtin *) eapply tr_builtin; eauto. destruct b; eauto. @@ -577,31 +577,31 @@ Proof. destruct a as [pc1 instr1]; simpl in *. inv H0. monadInv H. inv_incr. assert (A: Ple ctx.(dpc) s0.(st_nextnode)). assert (B: Plt (spc ctx pc) (st_nextnode s)) by eauto. - unfold spc in B. generalize (shiftpos_above pc (dpc ctx)). xomega. + unfold spc in B. generalize (shiftpos_above pc (dpc ctx)). extlia. destruct H9. inv H. (* same pc *) eapply expand_instr_spec; eauto. - omega. + lia. intros. transitivity ((st_code s')!pc'). - apply H7. auto. xomega. + apply H7. auto. extlia. eapply iter_expand_instr_unchanged; eauto. red; intros. rewrite list_map_compose in H9. exploit list_in_map_inv; eauto. intros [[pc0 instr0] [P Q]]. simpl in P. - assert (Plt (spc ctx pc0) (st_nextnode s)) by eauto. xomega. + assert (Plt (spc ctx pc0) (st_nextnode s)) by eauto. extlia. transitivity ((st_code s')!(spc ctx pc)). eapply H8; eauto. eapply iter_expand_instr_unchanged; eauto. - assert (Plt (spc ctx pc) (st_nextnode s)) by eauto. xomega. + assert (Plt (spc ctx pc) (st_nextnode s)) by eauto. extlia. red; intros. rewrite list_map_compose in H. exploit list_in_map_inv; eauto. intros [[pc0 instr0] [P Q]]. simpl in P. assert (pc = pc0) by (eapply shiftpos_inj; eauto). subst pc0. elim H12. change pc with (fst (pc, instr0)). apply List.in_map; auto. (* older pc *) inv_incr. eapply IHl; eauto. - intros. eapply Pos.lt_le_trans. eapply H2. right; eauto. xomega. + intros. eapply Pos.lt_le_trans. eapply H2. right; eauto. extlia. intros; eapply Ple_trans; eauto. - intros. apply H7; auto. xomega. + intros. apply H7; auto. extlia. Qed. Lemma expand_cfg_rec_spec: @@ -629,16 +629,16 @@ Proof. intros. assert (Ple pc0 (max_pc_function f)). eapply max_pc_function_sound. eapply PTree.elements_complete; eauto. - eapply Pos.lt_le_trans. apply shiftpos_below. subst s0; simpl; xomega. + eapply Pos.lt_le_trans. apply shiftpos_below. subst s0; simpl; extlia. subst s0; simpl; auto. - intros. apply H8; auto. subst s0; simpl in H11; xomega. + intros. apply H8; auto. subst s0; simpl in H11; extlia. intros. apply H8. apply shiftpos_above. assert (Ple pc0 (max_pc_function f)). eapply max_pc_function_sound. eapply PTree.elements_complete; eauto. - eapply Pos.lt_le_trans. apply shiftpos_below. inversion i; xomega. + eapply Pos.lt_le_trans. apply shiftpos_below. inversion i; extlia. apply PTree.elements_correct; auto. auto. auto. auto. - inversion INCR0. subst s0; simpl in STKSIZE; xomega. + inversion INCR0. subst s0; simpl in STKSIZE; extlia. Qed. End EXPAND_INSTR. @@ -721,12 +721,12 @@ Opaque initstate. apply funenv_program_compat. eapply expand_cfg_spec with (fe := fenv); eauto. red; auto. - unfold ctx; rewrite <- H1; rewrite <- H2; rewrite <- H3; simpl. xomega. - unfold ctx; rewrite <- H0; rewrite <- H1; simpl. xomega. - simpl. xomega. + unfold ctx; rewrite <- H1; rewrite <- H2; rewrite <- H3; simpl. extlia. + unfold ctx; rewrite <- H0; rewrite <- H1; simpl. extlia. + simpl. extlia. simpl. apply Z.divide_0_r. - simpl. omega. - simpl. omega. + simpl. lia. + simpl. lia. simpl. split; auto. destruct INCR2. destruct INCR1. destruct INCR0. destruct INCR. - simpl. change 0 with (st_stksize initstate). omega. + simpl. change 0 with (st_stksize initstate). lia. Qed. diff --git a/backend/Linearizeproof.v b/backend/Linearizeproof.v index 10a3d8b2..b065238c 100644 --- a/backend/Linearizeproof.v +++ b/backend/Linearizeproof.v @@ -642,7 +642,7 @@ Proof. (* Lbranch *) assert ((reachable f)!!pc = true). apply REACH; simpl; auto. - right; split. simpl; omega. split. auto. simpl. econstructor; eauto. + right; split. simpl; lia. split. auto. simpl. econstructor; eauto. (* Lcond *) assert (REACH1: (reachable f)!!pc1 = true) by (apply REACH; simpl; auto). @@ -659,12 +659,12 @@ Proof. rewrite eval_negate_condition. rewrite H. auto. eauto. rewrite DC. econstructor; eauto. (* cond is false: branch is taken *) - right; split. simpl; omega. split. auto. rewrite <- DC. econstructor; eauto. + right; split. simpl; lia. split. auto. rewrite <- DC. econstructor; eauto. rewrite eval_negate_condition. rewrite H. auto. (* branch if cond is true *) destruct b. (* cond is true: branch is taken *) - right; split. simpl; omega. split. auto. econstructor; eauto. + right; split. simpl; lia. split. auto. econstructor; eauto. (* cond is false: no branch *) left; econstructor; split. apply plus_one. eapply exec_Lcond_false. eauto. eauto. @@ -673,7 +673,7 @@ Proof. (* Ljumptable *) assert (REACH': (reachable f)!!pc = true). apply REACH. simpl. eapply list_nth_z_in; eauto. - right; split. simpl; omega. split. auto. econstructor; eauto. + right; split. simpl; lia. split. auto. econstructor; eauto. (* Lreturn *) left; econstructor; split. diff --git a/backend/Locations.v b/backend/Locations.v index c437df5d..2a3ae1d7 100644 --- a/backend/Locations.v +++ b/backend/Locations.v @@ -157,7 +157,7 @@ Module Loc. forall l, ~(diff l l). Proof. destruct l; unfold diff; auto. - red; intros. destruct H; auto. generalize (typesize_pos ty); omega. + red; intros. destruct H; auto. generalize (typesize_pos ty); lia. Qed. Lemma diff_not_eq: @@ -184,7 +184,7 @@ Module Loc. left; auto. destruct (zle (pos0 + typesize ty0) pos). left; auto. - right; red; intros [P | [P | P]]. congruence. omega. omega. + right; red; intros [P | [P | P]]. congruence. lia. lia. left; auto. Defined. @@ -497,7 +497,7 @@ Module OrderedLoc <: OrderedType. destruct x. eelim Plt_strict; eauto. destruct H. eelim OrderedSlot.lt_not_eq; eauto. red; auto. - destruct H. destruct H0. omega. + destruct H. destruct H0. lia. destruct H0. eelim OrderedTyp.lt_not_eq; eauto. red; auto. Qed. Definition compare : forall x y : t, Compare lt eq x y. @@ -545,18 +545,18 @@ Module OrderedLoc <: OrderedType. { destruct H. apply not_eq_sym. apply Plt_ne; auto. apply Plt_ne; auto. } congruence. - assert (RANGE: forall ty, 1 <= typesize ty <= 2). - { intros; unfold typesize. destruct ty0; omega. } + { intros; unfold typesize. destruct ty0; lia. } destruct H. + destruct H. left. apply not_eq_sym. apply OrderedSlot.lt_not_eq; auto. destruct H. right. - destruct H0. right. generalize (RANGE ty'); omega. + destruct H0. right. generalize (RANGE ty'); lia. destruct H0. assert (ty' = Tint \/ ty' = Tsingle \/ ty' = Tany32). { unfold OrderedTyp.lt in H1. destruct ty'; auto; compute in H1; congruence. } - right. destruct H2 as [E|[E|E]]; subst ty'; simpl typesize; omega. + right. destruct H2 as [E|[E|E]]; subst ty'; simpl typesize; lia. + destruct H. left. apply OrderedSlot.lt_not_eq; auto. destruct H. right. - destruct H0. left; omega. + destruct H0. left; lia. destruct H0. exfalso. destruct ty'; compute in H1; congruence. Qed. @@ -572,14 +572,14 @@ Module OrderedLoc <: OrderedType. - destruct (OrderedSlot.compare sl sl'); auto. destruct H. contradiction. destruct H. - right; right; split; auto. left; omega. + right; right; split; auto. left; lia. left; right; split; auto. assert (EITHER: typesize ty' = 1 /\ OrderedTyp.lt ty' Tany64 \/ typesize ty' = 2). { destruct ty'; compute; auto. } destruct (zlt ofs' (ofs - 1)). left; auto. destruct EITHER as [[P Q] | P]. - right; split; auto. omega. - left; omega. + right; split; auto. lia. + left; lia. Qed. End OrderedLoc. diff --git a/backend/NeedDomain.v b/backend/NeedDomain.v index d9e9e025..d3c6ed75 100644 --- a/backend/NeedDomain.v +++ b/backend/NeedDomain.v @@ -240,9 +240,9 @@ Proof. destruct (zlt i (Int.unsigned n)). - auto. - generalize (Int.unsigned_range n); intros. - apply H. omega. rewrite Int.bits_shru by omega. - replace (i - Int.unsigned n + Int.unsigned n) with i by omega. - rewrite zlt_true by omega. auto. + apply H. lia. rewrite Int.bits_shru by lia. + replace (i - Int.unsigned n + Int.unsigned n) with i by lia. + rewrite zlt_true by lia. auto. Qed. Lemma iagree_shru: @@ -252,9 +252,9 @@ Proof. intros; red; intros. autorewrite with ints; auto. destruct (zlt (i + Int.unsigned n) Int.zwordsize). - generalize (Int.unsigned_range n); intros. - apply H. omega. rewrite Int.bits_shl by omega. - replace (i + Int.unsigned n - Int.unsigned n) with i by omega. - rewrite zlt_false by omega. auto. + apply H. lia. rewrite Int.bits_shl by lia. + replace (i + Int.unsigned n - Int.unsigned n) with i by lia. + rewrite zlt_false by lia. auto. - auto. Qed. @@ -266,7 +266,7 @@ Proof. intros; red; intros. rewrite <- H in H2. rewrite Int.bits_shru in H2 by auto. rewrite ! Int.bits_shr by auto. destruct (zlt (i + Int.unsigned n) Int.zwordsize). -- apply H0; auto. generalize (Int.unsigned_range n); omega. +- apply H0; auto. generalize (Int.unsigned_range n); lia. - discriminate. Qed. @@ -281,11 +281,11 @@ Proof. then i + Int.unsigned n else Int.zwordsize - 1). assert (0 <= j < Int.zwordsize). - { unfold j; destruct (zlt (i + Int.unsigned n) Int.zwordsize); omega. } + { unfold j; destruct (zlt (i + Int.unsigned n) Int.zwordsize); lia. } apply H; auto. autorewrite with ints; auto. apply orb_true_intro. unfold j; destruct (zlt (i + Int.unsigned n) Int.zwordsize). -- left. rewrite zlt_false by omega. - replace (i + Int.unsigned n - Int.unsigned n) with i by omega. +- left. rewrite zlt_false by lia. + replace (i + Int.unsigned n - Int.unsigned n) with i by lia. auto. - right. reflexivity. Qed. @@ -303,7 +303,7 @@ Proof. mod Int.zwordsize) with i. auto. apply eqmod_small_eq with Int.zwordsize; auto. apply eqmod_trans with ((i - Int.unsigned amount) + Int.unsigned amount). - apply eqmod_refl2; omega. + apply eqmod_refl2; lia. eapply eqmod_trans. 2: apply eqmod_mod; auto. apply eqmod_add. apply eqmod_mod; auto. @@ -330,12 +330,12 @@ Lemma eqmod_iagree: Proof. intros. set (p := Z.to_nat (Int.size m)). generalize (Int.size_range m); intros RANGE. - assert (EQ: Int.size m = Z.of_nat p). { symmetry; apply Z2Nat.id. omega. } + assert (EQ: Int.size m = Z.of_nat p). { symmetry; apply Z2Nat.id. lia. } rewrite EQ in H; rewrite <- two_power_nat_two_p in H. red; intros. rewrite ! Int.testbit_repr by auto. destruct (zlt i (Int.size m)). - eapply same_bits_eqmod; eauto. omega. - assert (Int.testbit m i = false) by (eapply Int.bits_size_2; omega). + eapply same_bits_eqmod; eauto. lia. + assert (Int.testbit m i = false) by (eapply Int.bits_size_2; lia). congruence. Qed. @@ -348,11 +348,11 @@ Lemma iagree_eqmod: Proof. intros. set (p := Z.to_nat (Int.size m)). generalize (Int.size_range m); intros RANGE. - assert (EQ: Int.size m = Z.of_nat p). { symmetry; apply Z2Nat.id. omega. } + assert (EQ: Int.size m = Z.of_nat p). { symmetry; apply Z2Nat.id. lia. } rewrite EQ; rewrite <- two_power_nat_two_p. - apply eqmod_same_bits. intros. apply H. omega. - unfold complete_mask. rewrite Int.bits_zero_ext by omega. - rewrite zlt_true by omega. rewrite Int.bits_mone by omega. auto. + apply eqmod_same_bits. intros. apply H. lia. + unfold complete_mask. rewrite Int.bits_zero_ext by lia. + rewrite zlt_true by lia. rewrite Int.bits_mone by lia. auto. Qed. Lemma complete_mask_idem: @@ -363,12 +363,12 @@ Proof. + assert (Int.unsigned m <> 0). { red; intros; elim n. rewrite <- (Int.repr_unsigned m). rewrite H; auto. } assert (0 < Int.size m). - { apply Zsize_pos'. generalize (Int.unsigned_range m); omega. } + { apply Zsize_pos'. generalize (Int.unsigned_range m); lia. } generalize (Int.size_range m); intros. f_equal. apply Int.bits_size_4. tauto. - rewrite Int.bits_zero_ext by omega. rewrite zlt_true by omega. - apply Int.bits_mone; omega. - intros. rewrite Int.bits_zero_ext by omega. apply zlt_false; omega. + rewrite Int.bits_zero_ext by lia. rewrite zlt_true by lia. + apply Int.bits_mone; lia. + intros. rewrite Int.bits_zero_ext by lia. apply zlt_false; lia. Qed. (** ** Abstract operations over value needs. *) @@ -676,12 +676,12 @@ Proof. destruct x; simpl in *. - auto. - unfold Val.zero_ext; InvAgree. - red; intros. autorewrite with ints; try omega. + red; intros. autorewrite with ints; try lia. destruct (zlt i1 n); auto. apply H; auto. - autorewrite with ints; try omega. rewrite zlt_true; auto. + autorewrite with ints; try lia. rewrite zlt_true; auto. - unfold Val.zero_ext; InvAgree; auto. apply Val.lessdef_same. f_equal. - Int.bit_solve; try omega. destruct (zlt i1 n); auto. apply H; auto. - autorewrite with ints; try omega. apply zlt_true; auto. + Int.bit_solve; try lia. destruct (zlt i1 n); auto. apply H; auto. + autorewrite with ints; try lia. apply zlt_true; auto. Qed. Definition sign_ext (n: Z) (x: nval) := @@ -700,25 +700,25 @@ Proof. unfold sign_ext; intros. destruct x; simpl in *. - auto. - unfold Val.sign_ext; InvAgree. - red; intros. autorewrite with ints; try omega. + red; intros. autorewrite with ints; try lia. set (j := if zlt i1 n then i1 else n - 1). assert (0 <= j < Int.zwordsize). - { unfold j; destruct (zlt i1 n); omega. } + { unfold j; destruct (zlt i1 n); lia. } apply H; auto. - autorewrite with ints; try omega. apply orb_true_intro. + autorewrite with ints; try lia. apply orb_true_intro. unfold j; destruct (zlt i1 n). left. rewrite zlt_true; auto. - right. rewrite Int.unsigned_repr. rewrite zlt_false by omega. - replace (n - 1 - (n - 1)) with 0 by omega. reflexivity. - generalize Int.wordsize_max_unsigned; omega. + right. rewrite Int.unsigned_repr. rewrite zlt_false by lia. + replace (n - 1 - (n - 1)) with 0 by lia. reflexivity. + generalize Int.wordsize_max_unsigned; lia. - unfold Val.sign_ext; InvAgree; auto. apply Val.lessdef_same. f_equal. - Int.bit_solve; try omega. + Int.bit_solve; try lia. set (j := if zlt i1 n then i1 else n - 1). assert (0 <= j < Int.zwordsize). - { unfold j; destruct (zlt i1 n); omega. } - apply H; auto. rewrite Int.bits_zero_ext; try omega. + { unfold j; destruct (zlt i1 n); lia. } + apply H; auto. rewrite Int.bits_zero_ext; try lia. rewrite zlt_true. apply Int.bits_mone; auto. - unfold j. destruct (zlt i1 n); omega. + unfold j. destruct (zlt i1 n); lia. Qed. (** The needs of a memory store concerning the value being stored. *) @@ -778,11 +778,11 @@ Proof. - apply sign_ext_sound with (v := Vint i) (w := Vint i0) (x := All) (n := 8). auto. compute; auto. - apply zero_ext_sound with (v := Vint i) (w := Vint i0) (x := All) (n := 8). - auto. omega. + auto. lia. - apply sign_ext_sound with (v := Vint i) (w := Vint i0) (x := All) (n := 16). auto. compute; auto. - apply zero_ext_sound with (v := Vint i) (w := Vint i0) (x := All) (n := 16). - auto. omega. + auto. lia. Qed. (** The needs of a comparison *) @@ -1014,9 +1014,9 @@ Proof. unfold zero_ext_redundant; intros. destruct x; try discriminate. - auto. - simpl in *; InvAgree. simpl. InvBooleans. rewrite <- H. - red; intros; autorewrite with ints; try omega. + red; intros; autorewrite with ints; try lia. destruct (zlt i1 n). apply H0; auto. - rewrite Int.bits_zero_ext in H3 by omega. rewrite zlt_false in H3 by auto. discriminate. + rewrite Int.bits_zero_ext in H3 by lia. rewrite zlt_false in H3 by auto. discriminate. Qed. Definition sign_ext_redundant (n: Z) (x: nval) := @@ -1036,10 +1036,10 @@ Proof. unfold sign_ext_redundant; intros. destruct x; try discriminate. - auto. - simpl in *; InvAgree. simpl. InvBooleans. rewrite <- H. - red; intros; autorewrite with ints; try omega. + red; intros; autorewrite with ints; try lia. destruct (zlt i1 n). apply H0; auto. rewrite Int.bits_or; auto. rewrite H3; auto. - rewrite Int.bits_zero_ext in H3 by omega. rewrite zlt_false in H3 by auto. discriminate. + rewrite Int.bits_zero_ext in H3 by lia. rewrite zlt_false in H3 by auto. discriminate. Qed. (** * Neededness for register environments *) @@ -1300,13 +1300,13 @@ Proof. split; simpl; auto; intros. rewrite PTree.gsspec in H6. destruct (peq id0 id). + inv H6. destruct H3. congruence. destruct gl!id as [iv0|] eqn:NG. - unfold iv'; rewrite ISet.In_add. intros [P|P]. omega. eelim GL; eauto. - unfold iv'; rewrite ISet.In_interval. omega. + unfold iv'; rewrite ISet.In_add. intros [P|P]. lia. eelim GL; eauto. + unfold iv'; rewrite ISet.In_interval. lia. + eauto. - (* Stk ofs *) split; simpl; auto; intros. destruct H3. elim H3. subst b'. eapply bc_stack; eauto. - rewrite ISet.In_add. intros [P|P]. omega. eapply STK; eauto. + rewrite ISet.In_add. intros [P|P]. lia. eapply STK; eauto. Qed. (** Test (conservatively) whether some locations in the range delimited diff --git a/backend/RTL.v b/backend/RTL.v index 9599a24a..a022f55a 100644 --- a/backend/RTL.v +++ b/backend/RTL.v @@ -352,7 +352,7 @@ Proof. exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]]. exists (Returnstate s0 vres2 m2). econstructor; eauto. (* trace length *) - red; intros; inv H; simpl; try omega. + red; intros; inv H; simpl; try lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. Qed. @@ -450,8 +450,8 @@ Proof. rewrite PTree.gempty. congruence. (* inductive case *) intros. rewrite PTree.gsspec in H2. destruct (peq pc k). - inv H2. xomega. - apply Ple_trans with a. auto. xomega. + inv H2. extlia. + apply Ple_trans with a. auto. extlia. Qed. (** Maximum pseudo-register mentioned in a function. All results or arguments @@ -489,9 +489,9 @@ Proof. assert (X: forall l n, Ple m n -> Ple m (fold_left Pos.max l n)). { induction l; simpl; intros. auto. - apply IHl. xomega. } - destruct i; simpl; try (destruct s0); repeat (apply X); try xomega. - destruct o; xomega. + apply IHl. extlia. } + destruct i; simpl; try (destruct s0); repeat (apply X); try extlia. + destruct o; extlia. Qed. Remark max_reg_instr_def: @@ -499,12 +499,12 @@ Remark max_reg_instr_def: Proof. intros. assert (X: forall l n, Ple r n -> Ple r (fold_left Pos.max l n)). - { induction l; simpl; intros. xomega. apply IHl. xomega. } + { induction l; simpl; intros. extlia. apply IHl. extlia. } destruct i; simpl in *; inv H. -- apply X. xomega. -- apply X. xomega. -- destruct s0; apply X; xomega. -- destruct b; inv H1. apply X. simpl. xomega. +- apply X. extlia. +- apply X. extlia. +- destruct s0; apply X; extlia. +- destruct b; inv H1. apply X. simpl. extlia. Qed. Remark max_reg_instr_uses: @@ -514,14 +514,14 @@ Proof. assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)). { induction l; simpl; intros. tauto. - apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. } + apply IHl. destruct H0 as [[A|A]|A]. right; subst; extlia. auto. right; extlia. } destruct i; simpl in *; try (destruct s0); try (apply X; auto). - contradiction. -- destruct H. right; subst; xomega. auto. -- destruct H. right; subst; xomega. auto. -- destruct H. right; subst; xomega. auto. -- intuition. subst; xomega. -- destruct o; simpl in H; intuition. subst; xomega. +- destruct H. right; subst; extlia. auto. +- destruct H. right; subst; extlia. auto. +- destruct H. right; subst; extlia. auto. +- intuition. subst; extlia. +- destruct o; simpl in H; intuition. subst; extlia. Qed. Lemma max_reg_function_def: @@ -539,7 +539,7 @@ Proof. + inv H3. eapply max_reg_instr_def; eauto. + apply Ple_trans with a. auto. apply max_reg_instr_ge. } - unfold max_reg_function. xomega. + unfold max_reg_function. extlia. Qed. Lemma max_reg_function_use: @@ -557,7 +557,7 @@ Proof. + inv H3. eapply max_reg_instr_uses; eauto. + apply Ple_trans with a. auto. apply max_reg_instr_ge. } - unfold max_reg_function. xomega. + unfold max_reg_function. extlia. Qed. Lemma max_reg_function_params: @@ -567,8 +567,8 @@ Proof. assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)). { induction l; simpl; intros. tauto. - apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. } + apply IHl. destruct H0 as [[A|A]|A]. right; subst; extlia. auto. right; extlia. } assert (Y: Ple r (fold_left Pos.max f.(fn_params) 1%positive)). { apply X; auto. } - unfold max_reg_function. xomega. + unfold max_reg_function. extlia. Qed. diff --git a/backend/RTLgenproof.v b/backend/RTLgenproof.v index b94ec22f..88f7fe53 100644 --- a/backend/RTLgenproof.v +++ b/backend/RTLgenproof.v @@ -1145,7 +1145,7 @@ Proof. Qed. Ltac Lt_state := - apply lt_state_intro; simpl; try omega. + apply lt_state_intro; simpl; try lia. Lemma lt_state_wf: well_founded lt_state. diff --git a/backend/SelectDivproof.v b/backend/SelectDivproof.v index c57d3652..9d581ec9 100644 --- a/backend/SelectDivproof.v +++ b/backend/SelectDivproof.v @@ -44,55 +44,55 @@ Proof. set (r := n mod d). intro EUCL. assert (0 <= r <= d - 1). - unfold r. generalize (Z_mod_lt n d d_pos). omega. + unfold r. generalize (Z_mod_lt n d d_pos). lia. assert (0 <= m). apply Zmult_le_0_reg_r with d. auto. - exploit (two_p_gt_ZERO (N + l)). omega. omega. + exploit (two_p_gt_ZERO (N + l)). lia. lia. set (k := m * d - two_p (N + l)). assert (0 <= k <= two_p l). - unfold k; omega. + unfold k; lia. assert ((m * n - two_p (N + l) * q) * d = k * n + two_p (N + l) * r). unfold k. rewrite EUCL. ring. assert (0 <= k * n). - apply Z.mul_nonneg_nonneg; omega. + apply Z.mul_nonneg_nonneg; lia. assert (k * n <= two_p (N + l) - two_p l). apply Z.le_trans with (two_p l * n). - apply Z.mul_le_mono_nonneg_r; omega. - replace (N + l) with (l + N) by omega. + apply Z.mul_le_mono_nonneg_r; lia. + replace (N + l) with (l + N) by lia. rewrite two_p_is_exp. replace (two_p l * two_p N - two_p l) with (two_p l * (two_p N - 1)) by ring. - apply Z.mul_le_mono_nonneg_l. omega. exploit (two_p_gt_ZERO l). omega. omega. - omega. omega. + apply Z.mul_le_mono_nonneg_l. lia. exploit (two_p_gt_ZERO l). lia. lia. + lia. lia. assert (0 <= two_p (N + l) * r). apply Z.mul_nonneg_nonneg. - exploit (two_p_gt_ZERO (N + l)). omega. omega. - omega. + exploit (two_p_gt_ZERO (N + l)). lia. lia. + lia. assert (two_p (N + l) * r <= two_p (N + l) * d - two_p (N + l)). replace (two_p (N + l) * d - two_p (N + l)) with (two_p (N + l) * (d - 1)) by ring. apply Z.mul_le_mono_nonneg_l. - omega. - exploit (two_p_gt_ZERO (N + l)). omega. omega. + lia. + exploit (two_p_gt_ZERO (N + l)). lia. lia. assert (0 <= m * n - two_p (N + l) * q). apply Zmult_le_reg_r with d. auto. - replace (0 * d) with 0 by ring. rewrite H2. omega. + replace (0 * d) with 0 by ring. rewrite H2. lia. assert (m * n - two_p (N + l) * q < two_p (N + l)). - apply Zmult_lt_reg_r with d. omega. + apply Zmult_lt_reg_r with d. lia. rewrite H2. apply Z.le_lt_trans with (two_p (N + l) * d - two_p l). - omega. - exploit (two_p_gt_ZERO l). omega. omega. + lia. + exploit (two_p_gt_ZERO l). lia. lia. symmetry. apply Zdiv_unique with (m * n - two_p (N + l) * q). - ring. omega. + ring. lia. Qed. Lemma Zdiv_unique_2: forall x y q, y > 0 -> 0 < y * q - x <= y -> Z.div x y = q - 1. Proof. intros. apply Zdiv_unique with (x - (q - 1) * y). ring. - replace ((q - 1) * y) with (y * q - y) by ring. omega. + replace ((q - 1) * y) with (y * q - y) by ring. lia. Qed. Lemma Zdiv_mul_opp: @@ -110,42 +110,42 @@ Proof. set (r := n mod d). intro EUCL. assert (0 <= r <= d - 1). - unfold r. generalize (Z_mod_lt n d d_pos). omega. + unfold r. generalize (Z_mod_lt n d d_pos). lia. assert (0 <= m). apply Zmult_le_0_reg_r with d. auto. - exploit (two_p_gt_ZERO (N + l)). omega. omega. + exploit (two_p_gt_ZERO (N + l)). lia. lia. cut (Z.div (- (m * n)) (two_p (N + l)) = -q - 1). - omega. + lia. apply Zdiv_unique_2. - apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. replace (two_p (N + l) * - q - - (m * n)) with (m * n - two_p (N + l) * q) by ring. set (k := m * d - two_p (N + l)). assert (0 < k <= two_p l). - unfold k; omega. + unfold k; lia. assert ((m * n - two_p (N + l) * q) * d = k * n + two_p (N + l) * r). unfold k. rewrite EUCL. ring. split. - apply Zmult_lt_reg_r with d. omega. - replace (0 * d) with 0 by omega. + apply Zmult_lt_reg_r with d. lia. + replace (0 * d) with 0 by lia. rewrite H2. - assert (0 < k * n). apply Z.mul_pos_pos; omega. + assert (0 < k * n). apply Z.mul_pos_pos; lia. assert (0 <= two_p (N + l) * r). - apply Z.mul_nonneg_nonneg. exploit (two_p_gt_ZERO (N + l)); omega. omega. - omega. - apply Zmult_le_reg_r with d. omega. + apply Z.mul_nonneg_nonneg. exploit (two_p_gt_ZERO (N + l)); lia. lia. + lia. + apply Zmult_le_reg_r with d. lia. rewrite H2. assert (k * n <= two_p (N + l)). - rewrite Z.add_comm. rewrite two_p_is_exp; try omega. - apply Z.le_trans with (two_p l * n). apply Z.mul_le_mono_nonneg_r; omega. - apply Z.mul_le_mono_nonneg_l. omega. exploit (two_p_gt_ZERO l). omega. omega. + rewrite Z.add_comm. rewrite two_p_is_exp; try lia. + apply Z.le_trans with (two_p l * n). apply Z.mul_le_mono_nonneg_r; lia. + apply Z.mul_le_mono_nonneg_l. lia. exploit (two_p_gt_ZERO l). lia. lia. assert (two_p (N + l) * r <= two_p (N + l) * d - two_p (N + l)). replace (two_p (N + l) * d - two_p (N + l)) with (two_p (N + l) * (d - 1)) by ring. - apply Z.mul_le_mono_nonneg_l. exploit (two_p_gt_ZERO (N + l)). omega. omega. omega. - omega. + apply Z.mul_le_mono_nonneg_l. exploit (two_p_gt_ZERO (N + l)). lia. lia. lia. + lia. Qed. (** This is theorem 5.1 from Granlund and Montgomery, PLDI 1994. *) @@ -159,13 +159,13 @@ Lemma Zquot_mul: Z.quot n d = Z.div (m * n) (two_p (N + l)) + (if zlt n 0 then 1 else 0). Proof. intros. destruct (zlt n 0). - exploit (Zdiv_mul_opp m l H H0 (-n)). omega. + exploit (Zdiv_mul_opp m l H H0 (-n)). lia. replace (- - n) with n by ring. replace (Z.quot n d) with (- Z.quot (-n) d). - rewrite Zquot_Zdiv_pos by omega. omega. - rewrite Z.quot_opp_l by omega. ring. - rewrite Z.add_0_r. rewrite Zquot_Zdiv_pos by omega. - apply Zdiv_mul_pos; omega. + rewrite Zquot_Zdiv_pos by lia. lia. + rewrite Z.quot_opp_l by lia. ring. + rewrite Z.add_0_r. rewrite Zquot_Zdiv_pos by lia. + apply Zdiv_mul_pos; lia. Qed. End Z_DIV_MUL. @@ -194,11 +194,11 @@ Proof with (try discriminate). destruct (zlt p1 32)... intros EQ; inv EQ. split. auto. split. auto. intros. - replace (32 + p') with (31 + (p' + 1)) by omega. - apply Zquot_mul; try omega. - replace (31 + (p' + 1)) with (32 + p') by omega. omega. + replace (32 + p') with (31 + (p' + 1)) by lia. + apply Zquot_mul; try lia. + replace (31 + (p' + 1)) with (32 + p') by lia. lia. change (Int.min_signed <= n < Int.half_modulus). - unfold Int.max_signed in H. omega. + unfold Int.max_signed in H. lia. Qed. Lemma divu_mul_params_sound: @@ -223,7 +223,7 @@ Proof with (try discriminate). destruct (zlt p1 32)... intros EQ; inv EQ. split. auto. split. auto. intros. - apply Zdiv_mul_pos; try omega. assumption. + apply Zdiv_mul_pos; try lia. assumption. Qed. Lemma divs_mul_shift_gen: @@ -237,25 +237,25 @@ Proof. exploit divs_mul_params_sound; eauto. intros (A & B & C). split. auto. split. auto. unfold Int.divs. fold n; fold d. rewrite C by (apply Int.signed_range). - rewrite two_p_is_exp by omega. rewrite <- Zdiv_Zdiv. + rewrite two_p_is_exp by lia. rewrite <- Zdiv_Zdiv. rewrite Int.shru_lt_zero. unfold Int.add. apply Int.eqm_samerepr. apply Int.eqm_add. rewrite Int.shr_div_two_p. apply Int.eqm_unsigned_repr_r. apply Int.eqm_refl2. rewrite Int.unsigned_repr. f_equal. rewrite Int.signed_repr. rewrite Int.modulus_power. f_equal. ring. cut (Int.min_signed <= n * m / Int.modulus < Int.half_modulus). - unfold Int.max_signed; omega. - apply Zdiv_interval_1. generalize Int.min_signed_neg; omega. apply Int.half_modulus_pos. + unfold Int.max_signed; lia. + apply Zdiv_interval_1. generalize Int.min_signed_neg; lia. apply Int.half_modulus_pos. apply Int.modulus_pos. split. apply Z.le_trans with (Int.min_signed * m). - apply Z.mul_le_mono_nonpos_l. generalize Int.min_signed_neg; omega. omega. - apply Z.mul_le_mono_nonneg_r. omega. unfold n; generalize (Int.signed_range x); tauto. + apply Z.mul_le_mono_nonpos_l. generalize Int.min_signed_neg; lia. lia. + apply Z.mul_le_mono_nonneg_r. lia. unfold n; generalize (Int.signed_range x); tauto. apply Z.le_lt_trans with (Int.half_modulus * m). - apply Z.mul_le_mono_nonneg_r. tauto. generalize (Int.signed_range x); unfold n, Int.max_signed; omega. - apply Zmult_lt_compat_l. generalize Int.half_modulus_pos; omega. tauto. - assert (32 < Int.max_unsigned) by (compute; auto). omega. + apply Z.mul_le_mono_nonneg_r. tauto. generalize (Int.signed_range x); unfold n, Int.max_signed; lia. + apply Zmult_lt_compat_l. generalize Int.half_modulus_pos; lia. tauto. + assert (32 < Int.max_unsigned) by (compute; auto). lia. unfold Int.lt; fold n. rewrite Int.signed_zero. destruct (zlt n 0); apply Int.eqm_unsigned_repr. - apply two_p_gt_ZERO. omega. - apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. + apply two_p_gt_ZERO. lia. Qed. Theorem divs_mul_shift_1: @@ -269,7 +269,7 @@ Proof. intros. exploit divs_mul_shift_gen; eauto. instantiate (1 := x). intros (A & B & C). split. auto. rewrite C. unfold Int.mulhs. rewrite Int.signed_repr. auto. - generalize Int.min_signed_neg; unfold Int.max_signed; omega. + generalize Int.min_signed_neg; unfold Int.max_signed; lia. Qed. Theorem divs_mul_shift_2: @@ -305,18 +305,18 @@ Proof. split. auto. rewrite Int.shru_div_two_p. rewrite Int.unsigned_repr. unfold Int.divu, Int.mulhu. f_equal. rewrite C by apply Int.unsigned_range. - rewrite two_p_is_exp by omega. rewrite <- Zdiv_Zdiv by (apply two_p_gt_ZERO; omega). + rewrite two_p_is_exp by lia. rewrite <- Zdiv_Zdiv by (apply two_p_gt_ZERO; lia). f_equal. rewrite (Int.unsigned_repr m). rewrite Int.unsigned_repr. f_equal. ring. cut (0 <= Int.unsigned x * m / Int.modulus < Int.modulus). - unfold Int.max_unsigned; omega. - apply Zdiv_interval_1. omega. compute; auto. compute; auto. - split. simpl. apply Z.mul_nonneg_nonneg. generalize (Int.unsigned_range x); omega. omega. + unfold Int.max_unsigned; lia. + apply Zdiv_interval_1. lia. compute; auto. compute; auto. + split. simpl. apply Z.mul_nonneg_nonneg. generalize (Int.unsigned_range x); lia. lia. apply Z.le_lt_trans with (Int.modulus * m). - apply Zmult_le_compat_r. generalize (Int.unsigned_range x); omega. omega. - apply Zmult_lt_compat_l. compute; auto. omega. - unfold Int.max_unsigned; omega. - assert (32 < Int.max_unsigned) by (compute; auto). omega. + apply Zmult_le_compat_r. generalize (Int.unsigned_range x); lia. lia. + apply Zmult_lt_compat_l. compute; auto. lia. + unfold Int.max_unsigned; lia. + assert (32 < Int.max_unsigned) by (compute; auto). lia. Qed. (** Same, for 64-bit integers *) @@ -343,11 +343,11 @@ Proof with (try discriminate). destruct (zlt p1 64)... intros EQ; inv EQ. split. auto. split. auto. intros. - replace (64 + p') with (63 + (p' + 1)) by omega. - apply Zquot_mul; try omega. - replace (63 + (p' + 1)) with (64 + p') by omega. omega. + replace (64 + p') with (63 + (p' + 1)) by lia. + apply Zquot_mul; try lia. + replace (63 + (p' + 1)) with (64 + p') by lia. lia. change (Int64.min_signed <= n < Int64.half_modulus). - unfold Int64.max_signed in H. omega. + unfold Int64.max_signed in H. lia. Qed. Lemma divlu_mul_params_sound: @@ -372,13 +372,13 @@ Proof with (try discriminate). destruct (zlt p1 64)... intros EQ; inv EQ. split. auto. split. auto. intros. - apply Zdiv_mul_pos; try omega. assumption. + apply Zdiv_mul_pos; try lia. assumption. Qed. Remark int64_shr'_div_two_p: forall x y, Int64.shr' x y = Int64.repr (Int64.signed x / two_p (Int.unsigned y)). Proof. - intros; unfold Int64.shr'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. + intros; unfold Int64.shr'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); lia. Qed. Lemma divls_mul_shift_gen: @@ -392,25 +392,25 @@ Proof. exploit divls_mul_params_sound; eauto. intros (A & B & C). split. auto. split. auto. unfold Int64.divs. fold n; fold d. rewrite C by (apply Int64.signed_range). - rewrite two_p_is_exp by omega. rewrite <- Zdiv_Zdiv. + rewrite two_p_is_exp by lia. rewrite <- Zdiv_Zdiv. rewrite Int64.shru_lt_zero. unfold Int64.add. apply Int64.eqm_samerepr. apply Int64.eqm_add. rewrite int64_shr'_div_two_p. apply Int64.eqm_unsigned_repr_r. apply Int64.eqm_refl2. rewrite Int.unsigned_repr. f_equal. rewrite Int64.signed_repr. rewrite Int64.modulus_power. f_equal. ring. cut (Int64.min_signed <= n * m / Int64.modulus < Int64.half_modulus). - unfold Int64.max_signed; omega. - apply Zdiv_interval_1. generalize Int64.min_signed_neg; omega. apply Int64.half_modulus_pos. + unfold Int64.max_signed; lia. + apply Zdiv_interval_1. generalize Int64.min_signed_neg; lia. apply Int64.half_modulus_pos. apply Int64.modulus_pos. split. apply Z.le_trans with (Int64.min_signed * m). - apply Z.mul_le_mono_nonpos_l. generalize Int64.min_signed_neg; omega. omega. + apply Z.mul_le_mono_nonpos_l. generalize Int64.min_signed_neg; lia. lia. apply Z.mul_le_mono_nonneg_r. tauto. unfold n; generalize (Int64.signed_range x); tauto. apply Z.le_lt_trans with (Int64.half_modulus * m). - apply Zmult_le_compat_r. generalize (Int64.signed_range x); unfold n, Int64.max_signed; omega. tauto. - apply Zmult_lt_compat_l. generalize Int64.half_modulus_pos; omega. tauto. - assert (64 < Int.max_unsigned) by (compute; auto). omega. + apply Zmult_le_compat_r. generalize (Int64.signed_range x); unfold n, Int64.max_signed; lia. tauto. + apply Zmult_lt_compat_l. generalize Int64.half_modulus_pos; lia. tauto. + assert (64 < Int.max_unsigned) by (compute; auto). lia. unfold Int64.lt; fold n. rewrite Int64.signed_zero. destruct (zlt n 0); apply Int64.eqm_unsigned_repr. - apply two_p_gt_ZERO. omega. - apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. + apply two_p_gt_ZERO. lia. Qed. Theorem divls_mul_shift_1: @@ -424,7 +424,7 @@ Proof. intros. exploit divls_mul_shift_gen; eauto. instantiate (1 := x). intros (A & B & C). split. auto. rewrite C. unfold Int64.mulhs. rewrite Int64.signed_repr. auto. - generalize Int64.min_signed_neg; unfold Int64.max_signed; omega. + generalize Int64.min_signed_neg; unfold Int64.max_signed; lia. Qed. Theorem divls_mul_shift_2: @@ -453,7 +453,7 @@ Qed. Remark int64_shru'_div_two_p: forall x y, Int64.shru' x y = Int64.repr (Int64.unsigned x / two_p (Int.unsigned y)). Proof. - intros; unfold Int64.shru'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. + intros; unfold Int64.shru'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); lia. Qed. Theorem divlu_mul_shift: @@ -466,18 +466,18 @@ Proof. split. auto. rewrite int64_shru'_div_two_p. rewrite Int.unsigned_repr. unfold Int64.divu, Int64.mulhu. f_equal. rewrite C by apply Int64.unsigned_range. - rewrite two_p_is_exp by omega. rewrite <- Zdiv_Zdiv by (apply two_p_gt_ZERO; omega). + rewrite two_p_is_exp by lia. rewrite <- Zdiv_Zdiv by (apply two_p_gt_ZERO; lia). f_equal. rewrite (Int64.unsigned_repr m). rewrite Int64.unsigned_repr. f_equal. ring. cut (0 <= Int64.unsigned x * m / Int64.modulus < Int64.modulus). - unfold Int64.max_unsigned; omega. - apply Zdiv_interval_1. omega. compute; auto. compute; auto. - split. simpl. apply Z.mul_nonneg_nonneg. generalize (Int64.unsigned_range x); omega. omega. + unfold Int64.max_unsigned; lia. + apply Zdiv_interval_1. lia. compute; auto. compute; auto. + split. simpl. apply Z.mul_nonneg_nonneg. generalize (Int64.unsigned_range x); lia. lia. apply Z.le_lt_trans with (Int64.modulus * m). - apply Zmult_le_compat_r. generalize (Int64.unsigned_range x); omega. omega. - apply Zmult_lt_compat_l. compute; auto. omega. - unfold Int64.max_unsigned; omega. - assert (64 < Int.max_unsigned) by (compute; auto). omega. + apply Zmult_le_compat_r. generalize (Int64.unsigned_range x); lia. lia. + apply Zmult_lt_compat_l. compute; auto. lia. + unfold Int64.max_unsigned; lia. + assert (64 < Int.max_unsigned) by (compute; auto). lia. Qed. (** * Correctness of the smart constructors for division and modulus *) @@ -515,7 +515,7 @@ Proof. replace (Int.ltu (Int.repr p) Int.iwordsize) with true in Q. inv Q. rewrite B. auto. unfold Int.ltu. rewrite Int.unsigned_repr. rewrite zlt_true; auto. tauto. - assert (32 < Int.max_unsigned) by (compute; auto). omega. + assert (32 < Int.max_unsigned) by (compute; auto). lia. Qed. Theorem eval_divuimm: @@ -630,7 +630,7 @@ Proof. simpl in LD. inv LD. assert (RANGE: 0 <= p < 32 -> Int.ltu (Int.repr p) Int.iwordsize = true). { intros. unfold Int.ltu. rewrite Int.unsigned_repr. rewrite zlt_true by tauto. auto. - assert (32 < Int.max_unsigned) by (compute; auto). omega. } + assert (32 < Int.max_unsigned) by (compute; auto). lia. } destruct (zlt M Int.half_modulus). - exploit (divs_mul_shift_1 x); eauto. intros [A B]. exploit eval_shrimm. eexact X. instantiate (1 := Int.repr p). intros [v1 [Z LD]]. @@ -768,7 +768,7 @@ Proof. simpl in B1; inv B1. simpl in B2. replace (Int.ltu (Int.repr p) Int64.iwordsize') with true in B2. inv B2. rewrite B. assumption. unfold Int.ltu. rewrite Int.unsigned_repr. rewrite zlt_true; auto. tauto. - assert (64 < Int.max_unsigned) by (compute; auto). omega. + assert (64 < Int.max_unsigned) by (compute; auto). lia. Qed. Theorem eval_divlu: @@ -847,10 +847,10 @@ Proof. exploit eval_addl. auto. eexact A5. eexact A3. intros (v6 & A6 & B6). assert (RANGE: forall x, 0 <= x < 64 -> Int.ltu (Int.repr x) Int64.iwordsize' = true). { intros. unfold Int.ltu. rewrite Int.unsigned_repr. rewrite zlt_true by tauto. auto. - assert (64 < Int.max_unsigned) by (compute; auto). omega. } + assert (64 < Int.max_unsigned) by (compute; auto). lia. } simpl in B1; inv B1. simpl in B2; inv B2. - simpl in B3; rewrite RANGE in B3 by omega; inv B3. + simpl in B3; rewrite RANGE in B3 by lia; inv B3. destruct (zlt M Int64.half_modulus). - exploit (divls_mul_shift_1 x); eauto. intros [A B]. simpl in B5; rewrite RANGE in B5 by auto; inv B5. diff --git a/backend/Selectionproof.v b/backend/Selectionproof.v index 987926aa..4755ab79 100644 --- a/backend/Selectionproof.v +++ b/backend/Selectionproof.v @@ -531,7 +531,7 @@ Lemma sel_switch_correct: (XElet arg (sel_switch make_cmp_eq make_cmp_ltu make_sub make_to_int O t)) (switch_target i dfl cases). Proof. - intros. exploit validate_switch_correct; eauto. omega. intros [A B]. + intros. exploit validate_switch_correct; eauto. lia. intros [A B]. econstructor. eauto. eapply sel_switch_correct_rec; eauto. Qed. @@ -564,7 +564,7 @@ Proof. inv R. unfold Val.cmp in B. simpl in B. revert B. predSpec Int.eq Int.eq_spec n0 (Int.repr n); intros B; inv B. rewrite Int.unsigned_repr. unfold proj_sumbool; rewrite zeq_true; auto. - unfold Int.max_unsigned; omega. + unfold Int.max_unsigned; lia. unfold proj_sumbool; rewrite zeq_false; auto. red; intros; elim H1. rewrite <- (Int.repr_unsigned n0). congruence. - intros until n; intros EVAL R RANGE. @@ -573,7 +573,7 @@ Proof. inv R. unfold Val.cmpu in B. simpl in B. unfold Int.ltu in B. rewrite Int.unsigned_repr in B. destruct (zlt (Int.unsigned n0) n); inv B; auto. - unfold Int.max_unsigned; omega. + unfold Int.max_unsigned; lia. - intros until n; intros EVAL R RANGE. exploit eval_sub. eexact EVAL. apply (INTCONST (Int.repr n)). intros (vb & A & B). inv R. simpl in B. inv B. econstructor; split; eauto. @@ -581,7 +581,7 @@ Proof. with (Int.unsigned (Int.sub n0 (Int.repr n))). constructor. unfold Int.sub. rewrite Int.unsigned_repr_eq. f_equal. f_equal. - apply Int.unsigned_repr. unfold Int.max_unsigned; omega. + apply Int.unsigned_repr. unfold Int.max_unsigned; lia. - intros until i0; intros EVAL R. exists v; split; auto. inv R. rewrite Z.mod_small by (apply Int.unsigned_range). constructor. - constructor. @@ -599,12 +599,12 @@ Proof. eapply eval_cmpl. eexact EVAL. apply eval_longconst with (n := Int64.repr n). inv R. unfold Val.cmpl. simpl. f_equal; f_equal. unfold Int64.eq. rewrite Int64.unsigned_repr. destruct (zeq (Int64.unsigned n0) n); auto. - unfold Int64.max_unsigned; omega. + unfold Int64.max_unsigned; lia. - intros until n; intros EVAL R RANGE. eapply eval_cmplu; auto. eexact EVAL. apply eval_longconst with (n := Int64.repr n). inv R. unfold Val.cmplu. simpl. f_equal; f_equal. unfold Int64.ltu. rewrite Int64.unsigned_repr. destruct (zlt (Int64.unsigned n0) n); auto. - unfold Int64.max_unsigned; omega. + unfold Int64.max_unsigned; lia. - intros until n; intros EVAL R RANGE. exploit eval_subl; auto; try apply HF'. eexact EVAL. apply eval_longconst with (n := Int64.repr n). intros (vb & A & B). @@ -613,7 +613,7 @@ Proof. with (Int64.unsigned (Int64.sub n0 (Int64.repr n))). constructor. unfold Int64.sub. rewrite Int64.unsigned_repr_eq. f_equal. f_equal. - apply Int64.unsigned_repr. unfold Int64.max_unsigned; omega. + apply Int64.unsigned_repr. unfold Int64.max_unsigned; lia. - intros until i0; intros EVAL R. exploit eval_lowlong. eexact EVAL. intros (vb & A & B). inv R. simpl in B. inv B. econstructor; split; eauto. @@ -1295,7 +1295,7 @@ Proof. eapply match_cont_call with (cunit := cunit) (hf := hf); eauto. + (* turned into Sbuiltin *) intros EQ. subst fd. - right; left; split. simpl; omega. split; auto. econstructor; eauto. + right; left; split. simpl; lia. split; auto. econstructor; eauto. - (* Stailcall *) exploit Mem.free_parallel_extends; eauto. intros [m2' [P Q]]. erewrite <- stackspace_function_translated in P by eauto. @@ -1413,7 +1413,7 @@ Proof. apply plus_one; econstructor. econstructor; eauto. destruct optid; simpl; auto. apply set_var_lessdef; auto. - (* return of an external call turned into a Sbuiltin *) - right; left; split. simpl; omega. split. auto. econstructor; eauto. + right; left; split. simpl; lia. split. auto. econstructor; eauto. Qed. Lemma sel_initial_states: diff --git a/backend/SplitLongproof.v b/backend/SplitLongproof.v index 18c1f18d..1e50b1c2 100644 --- a/backend/SplitLongproof.v +++ b/backend/SplitLongproof.v @@ -335,7 +335,7 @@ Proof. fold (Int.testbit i i0). destruct (zlt i0 Int.zwordsize). auto. - rewrite Int.bits_zero. rewrite Int.bits_above by omega. auto. + rewrite Int.bits_zero. rewrite Int.bits_above by lia. auto. Qed. Theorem eval_longofint: unary_constructor_sound longofint Val.longofint. @@ -352,13 +352,13 @@ Proof. apply Int64.same_bits_eq; intros. rewrite Int64.testbit_repr by auto. rewrite Int64.bits_ofwords by auto. - rewrite Int.bits_signed by omega. + rewrite Int.bits_signed by lia. destruct (zlt i0 Int.zwordsize). auto. assert (Int64.zwordsize = 2 * Int.zwordsize) by reflexivity. - rewrite Int.bits_shr by omega. + rewrite Int.bits_shr by lia. change (Int.unsigned (Int.repr 31)) with (Int.zwordsize - 1). - f_equal. destruct (zlt (i0 - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); omega. + f_equal. destruct (zlt (i0 - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); lia. Qed. Theorem eval_negl: unary_constructor_sound negl Val.negl. @@ -545,24 +545,24 @@ Proof. { red; intros. elim H. rewrite <- (Int.repr_unsigned n). rewrite H0. auto. } destruct (Int.ltu n Int.iwordsize) eqn:LT. exploit Int.ltu_iwordsize_inv; eauto. intros RANGE. - assert (0 <= Int.zwordsize - Int.unsigned n < Int.zwordsize) by omega. + assert (0 <= Int.zwordsize - Int.unsigned n < Int.zwordsize) by lia. apply A1. auto. auto. unfold Int.ltu, Int.sub. rewrite Int.unsigned_repr_wordsize. - rewrite Int.unsigned_repr. rewrite zlt_true; auto. omega. - generalize Int.wordsize_max_unsigned; omega. + rewrite Int.unsigned_repr. rewrite zlt_true; auto. lia. + generalize Int.wordsize_max_unsigned; lia. unfold Int.ltu. rewrite zlt_true; auto. change (Int.unsigned Int64.iwordsize') with 64. - change Int.zwordsize with 32 in RANGE. omega. + change Int.zwordsize with 32 in RANGE. lia. destruct (Int.ltu n Int64.iwordsize') eqn:LT'. exploit Int.ltu_inv; eauto. change (Int.unsigned Int64.iwordsize') with (Int.zwordsize * 2). intros RANGE. assert (Int.zwordsize <= Int.unsigned n). unfold Int.ltu in LT. rewrite Int.unsigned_repr_wordsize in LT. - destruct (zlt (Int.unsigned n) Int.zwordsize). discriminate. omega. + destruct (zlt (Int.unsigned n) Int.zwordsize). discriminate. lia. apply A2. tauto. unfold Int.ltu, Int.sub. rewrite Int.unsigned_repr_wordsize. - rewrite Int.unsigned_repr. rewrite zlt_true; auto. omega. - generalize Int.wordsize_max_unsigned; omega. + rewrite Int.unsigned_repr. rewrite zlt_true; auto. lia. + generalize Int.wordsize_max_unsigned; lia. auto. Qed. @@ -918,19 +918,19 @@ Proof. rewrite Int.bits_zero. rewrite Int.bits_or by auto. symmetry. apply orb_false_intro. transitivity (Int64.testbit (Int64.ofwords h l) (i + Int.zwordsize)). - rewrite Int64.bits_ofwords by omega. rewrite zlt_false by omega. f_equal; omega. + rewrite Int64.bits_ofwords by lia. rewrite zlt_false by lia. f_equal; lia. rewrite H0. apply Int64.bits_zero. transitivity (Int64.testbit (Int64.ofwords h l) i). - rewrite Int64.bits_ofwords by omega. rewrite zlt_true by omega. auto. + rewrite Int64.bits_ofwords by lia. rewrite zlt_true by lia. auto. rewrite H0. apply Int64.bits_zero. symmetry. apply Int.eq_false. red; intros; elim H0. apply Int64.same_bits_eq; intros. rewrite Int64.bits_zero. rewrite Int64.bits_ofwords by auto. destruct (zlt i Int.zwordsize). assert (Int.testbit (Int.or h l) i = false) by (rewrite H1; apply Int.bits_zero). - rewrite Int.bits_or in H3 by omega. exploit orb_false_elim; eauto. tauto. + rewrite Int.bits_or in H3 by lia. exploit orb_false_elim; eauto. tauto. assert (Int.testbit (Int.or h l) (i - Int.zwordsize) = false) by (rewrite H1; apply Int.bits_zero). - rewrite Int.bits_or in H3 by omega. exploit orb_false_elim; eauto. tauto. + rewrite Int.bits_or in H3 by lia. exploit orb_false_elim; eauto. tauto. Qed. Lemma eval_cmpl_eq_zero: diff --git a/backend/Stackingproof.v b/backend/Stackingproof.v index ffd9b227..7724c5d6 100644 --- a/backend/Stackingproof.v +++ b/backend/Stackingproof.v @@ -58,7 +58,7 @@ Lemma slot_outgoing_argument_valid: Proof. intros. exploit loc_arguments_acceptable_2; eauto. intros [A B]. unfold slot_valid. unfold proj_sumbool. - rewrite zle_true by omega. + rewrite zle_true by lia. rewrite pred_dec_true by auto. auto. Qed. @@ -126,7 +126,7 @@ Proof. destruct (wt_function f); simpl negb. destruct (zlt Ptrofs.max_unsigned (fe_size (make_env (function_bounds f)))). intros; discriminate. - intros. unfold fe. unfold b. omega. + intros. unfold fe. unfold b. lia. intros; discriminate. Qed. @@ -200,7 +200,7 @@ Next Obligation. - exploit H4; eauto. intros (v & A & B). exists v; split; auto. eapply Mem.load_unchanged_on; eauto. simpl; intros. rewrite size_type_chunk, typesize_typesize in H8. - split; auto. omega. + split; auto. lia. Qed. Next Obligation. eauto with mem. @@ -215,7 +215,7 @@ Remark valid_access_location: Proof. intros; split. - red; intros. apply Mem.perm_implies with Freeable; auto with mem. - apply H0. rewrite size_type_chunk, typesize_typesize in H4. omega. + apply H0. rewrite size_type_chunk, typesize_typesize in H4. lia. - rewrite align_type_chunk. apply Z.divide_add_r. apply Z.divide_trans with 8; auto. exists (8 / (4 * typealign ty)); destruct ty; reflexivity. @@ -233,7 +233,7 @@ Proof. intros. destruct H as (D & E & F & G & H). exploit H; eauto. intros (v & U & V). exists v; split; auto. unfold load_stack; simpl. rewrite Ptrofs.add_zero_l, Ptrofs.unsigned_repr; auto. - unfold Ptrofs.max_unsigned. generalize (typesize_pos ty). omega. + unfold Ptrofs.max_unsigned. generalize (typesize_pos ty). lia. Qed. Lemma set_location: @@ -252,19 +252,19 @@ Proof. { red; intros; eauto with mem. } exists m'; split. - unfold store_stack; simpl. rewrite Ptrofs.add_zero_l, Ptrofs.unsigned_repr; eauto. - unfold Ptrofs.max_unsigned. generalize (typesize_pos ty). omega. + unfold Ptrofs.max_unsigned. generalize (typesize_pos ty). lia. - simpl. intuition auto. + unfold Locmap.set. destruct (Loc.eq (S sl ofs ty) (S sl ofs0 ty0)); [|destruct (Loc.diff_dec (S sl ofs ty) (S sl ofs0 ty0))]. * (* same location *) inv e. rename ofs0 into ofs. rename ty0 into ty. exists (Val.load_result (chunk_of_type ty) v'); split. - eapply Mem.load_store_similar_2; eauto. omega. + eapply Mem.load_store_similar_2; eauto. lia. apply Val.load_result_inject; auto. * (* different locations *) exploit H; eauto. intros (v0 & X & Y). exists v0; split; auto. rewrite <- X; eapply Mem.load_store_other; eauto. - destruct d. congruence. right. rewrite ! size_type_chunk, ! typesize_typesize. omega. + destruct d. congruence. right. rewrite ! size_type_chunk, ! typesize_typesize. lia. * (* overlapping locations *) destruct (Mem.valid_access_load m' (chunk_of_type ty0) sp (pos + 4 * ofs0)) as [v'' LOAD]. apply Mem.valid_access_implies with Writable; auto with mem. @@ -273,7 +273,7 @@ Proof. + apply (m_invar P) with m; auto. eapply Mem.store_unchanged_on; eauto. intros i; rewrite size_type_chunk, typesize_typesize. intros; red; intros. - eelim C; eauto. simpl. split; auto. omega. + eelim C; eauto. simpl. split; auto. lia. Qed. Lemma initial_locations: @@ -933,8 +933,8 @@ Local Opaque mreg_type. { unfold pos1. apply Z.divide_trans with sz. unfold sz; rewrite <- size_type_chunk. apply align_size_chunk_divides. apply align_divides; auto. } - apply range_drop_left with (mid := pos1) in SEP; [ | omega ]. - apply range_split with (mid := pos1 + sz) in SEP; [ | omega ]. + apply range_drop_left with (mid := pos1) in SEP; [ | lia ]. + apply range_split with (mid := pos1 + sz) in SEP; [ | lia ]. unfold sz at 1 in SEP. rewrite <- size_type_chunk in SEP. apply range_contains in SEP; auto. exploit (contains_set_stack (fun v' => Val.inject j (ls (R r)) v') (rs r)). @@ -1073,7 +1073,7 @@ Local Opaque b fe. instantiate (1 := fe_stack_data fe). tauto. reflexivity. instantiate (1 := fe_stack_data fe + bound_stack_data b). rewrite Z.max_comm. reflexivity. - generalize (bound_stack_data_pos b) size_no_overflow; omega. + generalize (bound_stack_data_pos b) size_no_overflow; lia. tauto. tauto. clear SEP. intros (j' & SEP & INCR & SAME). @@ -1607,7 +1607,7 @@ Proof. + simpl in SEP. unfold parent_sp. assert (slot_valid f Outgoing pos ty = true). { destruct H0. unfold slot_valid, proj_sumbool. - rewrite zle_true by omega. rewrite pred_dec_true by auto. reflexivity. } + rewrite zle_true by lia. rewrite pred_dec_true by auto. reflexivity. } assert (slot_within_bounds (function_bounds f) Outgoing pos ty) by eauto. exploit frame_get_outgoing; eauto. intros (v & A & B). exists v; split. diff --git a/backend/Tailcallproof.v b/backend/Tailcallproof.v index 9ec89553..7a5be5ed 100644 --- a/backend/Tailcallproof.v +++ b/backend/Tailcallproof.v @@ -47,11 +47,11 @@ Proof. intro f. assert (forall n pc, (return_measure_rec n f pc <= n)%nat). induction n; intros; simpl. - omega. - destruct (f!pc); try omega. - destruct i; try omega. - generalize (IHn n0). omega. - generalize (IHn n0). omega. + lia. + destruct (f!pc); try lia. + destruct i; try lia. + generalize (IHn n0). lia. + generalize (IHn n0). lia. intros. unfold return_measure. apply H. Qed. @@ -61,11 +61,11 @@ Remark return_measure_rec_incr: (return_measure_rec n1 f pc <= return_measure_rec n2 f pc)%nat. Proof. induction n1; intros; simpl. - omega. - destruct n2. omegaContradiction. assert (n1 <= n2)%nat by omega. - simpl. destruct f!pc; try omega. destruct i; try omega. - generalize (IHn1 n2 n H0). omega. - generalize (IHn1 n2 n H0). omega. + lia. + destruct n2. extlia. assert (n1 <= n2)%nat by lia. + simpl. destruct f!pc; try lia. destruct i; try lia. + generalize (IHn1 n2 n H0). lia. + generalize (IHn1 n2 n H0). lia. Qed. Lemma is_return_measure_rec: @@ -75,13 +75,13 @@ Lemma is_return_measure_rec: Proof. induction n; simpl; intros. congruence. - destruct n'. omegaContradiction. simpl. + destruct n'. extlia. simpl. destruct (fn_code f)!pc; try congruence. destruct i; try congruence. - decEq. apply IHn with r. auto. omega. + decEq. apply IHn with r. auto. lia. destruct (is_move_operation o l); try congruence. destruct (Reg.eq r r1); try congruence. - decEq. apply IHn with r0. auto. omega. + decEq. apply IHn with r0. auto. lia. Qed. (** ** Relational characterization of the code transformation *) @@ -117,22 +117,22 @@ Proof. generalize H. simpl. caseEq ((fn_code f)!pc); try congruence. intro i. caseEq i; try congruence. - intros s; intros. eapply is_return_nop; eauto. eapply IHn; eauto. omega. + intros s; intros. eapply is_return_nop; eauto. eapply IHn; eauto. lia. unfold return_measure. rewrite <- (is_return_measure_rec f (S n) niter pc rret); auto. rewrite <- (is_return_measure_rec f n niter s rret); auto. - simpl. rewrite H2. omega. omega. + simpl. rewrite H2. lia. lia. intros op args dst s EQ1 EQ2. caseEq (is_move_operation op args); try congruence. intros src IMO. destruct (Reg.eq rret src); try congruence. subst rret. intro. exploit is_move_operation_correct; eauto. intros [A B]. subst. - eapply is_return_move; eauto. eapply IHn; eauto. omega. + eapply is_return_move; eauto. eapply IHn; eauto. lia. unfold return_measure. rewrite <- (is_return_measure_rec f (S n) niter pc src); auto. rewrite <- (is_return_measure_rec f n niter s dst); auto. - simpl. rewrite EQ2. omega. omega. + simpl. rewrite EQ2. lia. lia. intros or EQ1 EQ2. destruct or; intros. assert (r = rret). eapply proj_sumbool_true; eauto. subst r. @@ -407,7 +407,7 @@ Proof. eapply exec_Inop; eauto. constructor; auto. - (* eliminated nop *) assert (s0 = pc') by congruence. subst s0. - right. split. simpl. omega. split. auto. + right. split. simpl. lia. split. auto. econstructor; eauto. - (* op *) @@ -421,7 +421,7 @@ Proof. econstructor; eauto. apply set_reg_lessdef; auto. - (* eliminated move *) rewrite H1 in H. clear H1. inv H. - right. split. simpl. omega. split. auto. + right. split. simpl. lia. split. auto. econstructor; eauto. simpl in H0. rewrite PMap.gss. congruence. - (* load *) @@ -455,13 +455,13 @@ Proof. + (* call turned tailcall *) assert ({ m'' | Mem.free m' sp0 0 (fn_stacksize (transf_function f)) = Some m''}). apply Mem.range_perm_free. rewrite stacksize_preserved. rewrite H7. - red; intros; omegaContradiction. + red; intros; extlia. destruct X as [m'' FREE]. left. exists (Callstate s' (transf_fundef fd) (rs'##args) m''); split. eapply exec_Itailcall; eauto. apply sig_preserved. constructor. eapply match_stackframes_tail; eauto. apply regs_lessdef_regs; auto. eapply Mem.free_right_extends; eauto. - rewrite stacksize_preserved. rewrite H7. intros. omegaContradiction. + rewrite stacksize_preserved. rewrite H7. intros. extlia. + (* call that remains a call *) left. exists (Callstate (Stackframe res (transf_function f) (Vptr sp0 Ptrofs.zero) pc' rs' :: s') (transf_fundef fd) (rs'##args) m'); split. @@ -514,22 +514,22 @@ Proof. - (* eliminated return None *) assert (or = None) by congruence. subst or. - right. split. simpl. omega. split. auto. + right. split. simpl. lia. split. auto. constructor. auto. simpl. constructor. eapply Mem.free_left_extends; eauto. - (* eliminated return Some *) assert (or = Some r) by congruence. subst or. - right. split. simpl. omega. split. auto. + right. split. simpl. lia. split. auto. constructor. auto. simpl. auto. eapply Mem.free_left_extends; eauto. - (* internal call *) exploit Mem.alloc_extends; eauto. - instantiate (1 := 0). omega. - instantiate (1 := fn_stacksize f). omega. + instantiate (1 := 0). lia. + instantiate (1 := fn_stacksize f). lia. intros [m'1 [ALLOC EXT]]. assert (fn_stacksize (transf_function f) = fn_stacksize f /\ fn_entrypoint (transf_function f) = fn_entrypoint f /\ @@ -559,7 +559,7 @@ Proof. right. split. unfold measure. simpl length. change (S (length s) * (niter + 2))%nat with ((niter + 2) + (length s) * (niter + 2))%nat. - generalize (return_measure_bounds (fn_code f) pc). omega. + generalize (return_measure_bounds (fn_code f) pc). lia. split. auto. econstructor; eauto. rewrite Regmap.gss. auto. diff --git a/backend/Tunnelingproof.v b/backend/Tunnelingproof.v index 4f95ac9b..c612995b 100644 --- a/backend/Tunnelingproof.v +++ b/backend/Tunnelingproof.v @@ -87,7 +87,7 @@ Proof. * (* The new instruction *) rewrite (U.repr_union_2 u pc s); auto. rewrite U.repr_union_3. unfold measure_edge. destruct (peq (U.repr u s) pc). auto. right. split. auto. - rewrite PC. rewrite peq_true. omega. + rewrite PC. rewrite peq_true. lia. * (* An old instruction *) assert (U.repr u pc' = pc' -> U.repr (U.union u pc s) pc' = pc'). @@ -96,7 +96,7 @@ Proof. intros [P | [P Q]]. left; auto. right. split. apply U.sameclass_union_2. auto. unfold measure_edge. destruct (peq (U.repr u s) pc). auto. - rewrite P. destruct (peq (U.repr u s0) pc). omega. auto. + rewrite P. destruct (peq (U.repr u s0) pc). lia. auto. Qed. Definition record_gotos' (f: function) := @@ -420,7 +420,7 @@ Proof. generalize (record_gotos_correct f pc). rewrite H. destruct bb; auto. destruct i; auto. intros [A | [B C]]. auto. - right. split. simpl. omega. + right. split. simpl. lia. split. auto. rewrite B. econstructor; eauto. @@ -487,7 +487,7 @@ Proof. eapply exec_Lbranch; eauto. fold (branch_target f pc). econstructor; eauto. - (* Lbranch (eliminated) *) - right; split. simpl. omega. split. auto. constructor; auto. + right; split. simpl. lia. split. auto. constructor; auto. - (* Lcond *) simpl tunneled_block. diff --git a/backend/Unusedglobproof.v b/backend/Unusedglobproof.v index 680daba7..3216ec50 100644 --- a/backend/Unusedglobproof.v +++ b/backend/Unusedglobproof.v @@ -982,7 +982,7 @@ Proof. intros. exploit G; eauto. intros [U V]. assert (Mem.valid_block m sp0) by (eapply Mem.valid_block_inject_1; eauto). assert (Mem.valid_block tm tsp) by (eapply Mem.valid_block_inject_2; eauto). - unfold Mem.valid_block in *; xomega. + unfold Mem.valid_block in *; extlia. apply set_res_inject; auto. apply regset_inject_incr with j; auto. - (* cond *) @@ -1036,7 +1036,7 @@ Proof. apply match_stacks_bound with (Mem.nextblock m) (Mem.nextblock tm). apply match_stacks_incr with j; auto. intros. exploit G; eauto. intros [P Q]. - unfold Mem.valid_block in *; xomega. + unfold Mem.valid_block in *; extlia. eapply external_call_nextblock; eauto. eapply external_call_nextblock; eauto. @@ -1063,7 +1063,7 @@ Proof. - apply IHl. unfold Genv.add_global, P; simpl. intros LT. apply Plt_succ_inv in LT. destruct LT. + rewrite PTree.gso. apply H; auto. apply Plt_ne; auto. + rewrite H0. rewrite PTree.gss. exists g1; auto. } - apply H. red; simpl; intros. exfalso; xomega. + apply H. red; simpl; intros. exfalso; extlia. Qed. *) @@ -1123,10 +1123,10 @@ Lemma Mem_getN_forall2: P (ZMap.get i c1) (ZMap.get i c2). Proof. induction n; simpl Mem.getN; intros. -- simpl in H1. omegaContradiction. +- simpl in H1. extlia. - inv H. rewrite Nat2Z.inj_succ in H1. destruct (zeq i p0). + congruence. -+ apply IHn with (p0 + 1); auto. omega. omega. ++ apply IHn with (p0 + 1); auto. lia. lia. Qed. Lemma init_mem_inj_1: @@ -1143,7 +1143,7 @@ Proof. + intros (P2 & Q2 & R2 & S2) (P1 & Q1 & R1 & S1). apply Q1 in H0. destruct H0. subst. apply Mem.perm_cur. eapply Mem.perm_implies; eauto. - apply P2. omega. + apply P2. lia. - exploit init_meminj_invert; eauto. intros (A & id & B & C). subst delta. apply Z.divide_0_r. - exploit init_meminj_invert_strong; eauto. intros (A & id & gd & B & C & D & E & F). @@ -1162,8 +1162,8 @@ Local Transparent Mem.loadbytes. rewrite Z.add_0_r. apply Mem_getN_forall2 with (p := 0) (n := Z.to_nat (init_data_list_size (gvar_init v))). rewrite H3, H4. apply bytes_of_init_inject. auto. - omega. - rewrite Z2Nat.id by (apply Z.ge_le; apply init_data_list_size_pos). omega. + lia. + rewrite Z2Nat.id by (apply Z.ge_le; apply init_data_list_size_pos). lia. Qed. Lemma init_mem_inj_2: @@ -1181,18 +1181,18 @@ Proof. exploit init_meminj_invert. eexact H1. intros (A2 & id2 & B2 & C2). destruct (ident_eq id1 id2). congruence. left; eapply Genv.global_addresses_distinct; eauto. - exploit init_meminj_invert; eauto. intros (A & id & B & C). subst delta. - split. omega. generalize (Ptrofs.unsigned_range_2 ofs). omega. + split. lia. generalize (Ptrofs.unsigned_range_2 ofs). lia. - exploit init_meminj_invert_strong; eauto. intros (A & id & gd & B & C & D & E & F). exploit (Genv.init_mem_characterization_gen p); eauto. exploit (Genv.init_mem_characterization_gen tp); eauto. destruct gd as [f|v]. + intros (P2 & Q2) (P1 & Q1). - apply Q2 in H0. destruct H0. subst. replace ofs with 0 by omega. + apply Q2 in H0. destruct H0. subst. replace ofs with 0 by lia. left; apply Mem.perm_cur; auto. + intros (P2 & Q2 & R2 & S2) (P1 & Q1 & R1 & S1). apply Q2 in H0. destruct H0. subst. left. apply Mem.perm_cur. eapply Mem.perm_implies; eauto. - apply P1. omega. + apply P1. lia. Qed. End INIT_MEM. diff --git a/backend/ValueAnalysis.v b/backend/ValueAnalysis.v index b0ce019c..f7e4f0ed 100644 --- a/backend/ValueAnalysis.v +++ b/backend/ValueAnalysis.v @@ -544,8 +544,8 @@ Proof. eapply SM; auto. eapply mmatch_top; eauto. + (* below *) red; simpl; intros. rewrite NB. destruct (eq_block b sp). - subst b; rewrite SP; xomega. - exploit mmatch_below; eauto. xomega. + subst b; rewrite SP; extlia. + exploit mmatch_below; eauto. extlia. - (* unchanged *) simpl; intros. apply dec_eq_false. apply Plt_ne. auto. - (* values *) @@ -1147,11 +1147,11 @@ Proof. - constructor. - assert (Plt sp bound') by eauto with va. eapply sound_stack_public_call; eauto. apply IHsound_stack; intros. - apply INV. xomega. rewrite SAME; auto with ordered_type. xomega. auto. auto. + apply INV. extlia. rewrite SAME; auto with ordered_type. extlia. auto. auto. - assert (Plt sp bound') by eauto with va. eapply sound_stack_private_call; eauto. apply IHsound_stack; intros. - apply INV. xomega. rewrite SAME; auto with ordered_type. xomega. auto. auto. - apply bmatch_ext with m; auto. intros. apply INV. xomega. auto. auto. auto. + apply INV. extlia. rewrite SAME; auto with ordered_type. extlia. auto. auto. + apply bmatch_ext with m; auto. intros. apply INV. extlia. auto. auto. auto. Qed. Lemma sound_stack_inv: @@ -1210,8 +1210,8 @@ Lemma sound_stack_new_bound: Proof. intros. inv H. - constructor. -- eapply sound_stack_public_call with (bound' := bound'0); eauto. xomega. -- eapply sound_stack_private_call with (bound' := bound'0); eauto. xomega. +- eapply sound_stack_public_call with (bound' := bound'0); eauto. extlia. +- eapply sound_stack_private_call with (bound' := bound'0); eauto. extlia. Qed. Lemma sound_stack_exten: @@ -1224,12 +1224,12 @@ Proof. - constructor. - assert (Plt sp bound') by eauto with va. eapply sound_stack_public_call; eauto. - rewrite H0; auto. xomega. - intros. rewrite H0; auto. xomega. + rewrite H0; auto. extlia. + intros. rewrite H0; auto. extlia. - assert (Plt sp bound') by eauto with va. eapply sound_stack_private_call; eauto. - rewrite H0; auto. xomega. - intros. rewrite H0; auto. xomega. + rewrite H0; auto. extlia. + intros. rewrite H0; auto. extlia. Qed. (** ** Preservation of the semantic invariant by one step of execution *) diff --git a/backend/ValueDomain.v b/backend/ValueDomain.v index c132ce7c..45894bfc 100644 --- a/backend/ValueDomain.v +++ b/backend/ValueDomain.v @@ -45,10 +45,10 @@ Qed. Hint Extern 2 (_ = _) => congruence : va. Hint Extern 2 (_ <> _) => congruence : va. -Hint Extern 2 (_ < _) => xomega : va. -Hint Extern 2 (_ <= _) => xomega : va. -Hint Extern 2 (_ > _) => xomega : va. -Hint Extern 2 (_ >= _) => xomega : va. +Hint Extern 2 (_ < _) => extlia : va. +Hint Extern 2 (_ <= _) => extlia : va. +Hint Extern 2 (_ > _) => extlia : va. +Hint Extern 2 (_ >= _) => extlia : va. Section MATCH. @@ -595,17 +595,17 @@ Hint Extern 1 (vmatch _ _) => constructor : va. Lemma is_uns_mon: forall n1 n2 i, is_uns n1 i -> n1 <= n2 -> is_uns n2 i. Proof. - intros; red; intros. apply H; omega. + intros; red; intros. apply H; lia. Qed. Lemma is_sgn_mon: forall n1 n2 i, is_sgn n1 i -> n1 <= n2 -> is_sgn n2 i. Proof. - intros; red; intros. apply H; omega. + intros; red; intros. apply H; lia. Qed. Lemma is_uns_sgn: forall n1 n2 i, is_uns n1 i -> n1 < n2 -> is_sgn n2 i. Proof. - intros; red; intros. rewrite ! H by omega. auto. + intros; red; intros. rewrite ! H by lia. auto. Qed. Definition usize := Int.size. @@ -616,7 +616,7 @@ Lemma is_uns_usize: forall i, is_uns (usize i) i. Proof. unfold usize; intros; red; intros. - apply Int.bits_size_2. omega. + apply Int.bits_size_2. lia. Qed. Lemma is_sgn_ssize: @@ -628,10 +628,10 @@ Proof. rewrite <- (negb_involutive (Int.testbit i (Int.zwordsize - 1))). f_equal. generalize (Int.size_range (Int.not i)); intros RANGE. - rewrite <- ! Int.bits_not by omega. - rewrite ! Int.bits_size_2 by omega. + rewrite <- ! Int.bits_not by lia. + rewrite ! Int.bits_size_2 by lia. auto. -- rewrite ! Int.bits_size_2 by omega. +- rewrite ! Int.bits_size_2 by lia. auto. Qed. @@ -639,8 +639,8 @@ Lemma is_uns_zero_ext: forall n i, is_uns n i <-> Int.zero_ext n i = i. Proof. intros; split; intros. - Int.bit_solve. destruct (zlt i0 n); auto. symmetry; apply H; auto. omega. - rewrite <- H. red; intros. rewrite Int.bits_zero_ext by omega. rewrite zlt_false by omega. auto. + Int.bit_solve. destruct (zlt i0 n); auto. symmetry; apply H; auto. lia. + rewrite <- H. red; intros. rewrite Int.bits_zero_ext by lia. rewrite zlt_false by lia. auto. Qed. Lemma is_sgn_sign_ext: @@ -649,18 +649,18 @@ Proof. intros; split; intros. Int.bit_solve. destruct (zlt i0 n); auto. transitivity (Int.testbit i (Int.zwordsize - 1)). - apply H0; omega. symmetry; apply H0; omega. - rewrite <- H0. red; intros. rewrite ! Int.bits_sign_ext by omega. - f_equal. transitivity (n-1). destruct (zlt m n); omega. - destruct (zlt (Int.zwordsize - 1) n); omega. + apply H0; lia. symmetry; apply H0; lia. + rewrite <- H0. red; intros. rewrite ! Int.bits_sign_ext by lia. + f_equal. transitivity (n-1). destruct (zlt m n); lia. + destruct (zlt (Int.zwordsize - 1) n); lia. Qed. Lemma is_zero_ext_uns: forall i n m, is_uns m i \/ n <= m -> is_uns m (Int.zero_ext n i). Proof. - intros. red; intros. rewrite Int.bits_zero_ext by omega. - destruct (zlt m0 n); auto. destruct H. apply H; omega. omegaContradiction. + intros. red; intros. rewrite Int.bits_zero_ext by lia. + destruct (zlt m0 n); auto. destruct H. apply H; lia. extlia. Qed. Lemma is_zero_ext_sgn: @@ -668,9 +668,9 @@ Lemma is_zero_ext_sgn: n < m -> is_sgn m (Int.zero_ext n i). Proof. - intros. red; intros. rewrite ! Int.bits_zero_ext by omega. - transitivity false. apply zlt_false; omega. - symmetry; apply zlt_false; omega. + intros. red; intros. rewrite ! Int.bits_zero_ext by lia. + transitivity false. apply zlt_false; lia. + symmetry; apply zlt_false; lia. Qed. Lemma is_sign_ext_uns: @@ -679,8 +679,8 @@ Lemma is_sign_ext_uns: is_uns m i -> is_uns m (Int.sign_ext n i). Proof. - intros; red; intros. rewrite Int.bits_sign_ext by omega. - apply H0. destruct (zlt m0 n); omega. destruct (zlt m0 n); omega. + intros; red; intros. rewrite Int.bits_sign_ext by lia. + apply H0. destruct (zlt m0 n); lia. destruct (zlt m0 n); lia. Qed. Lemma is_sign_ext_sgn: @@ -690,9 +690,9 @@ Lemma is_sign_ext_sgn: Proof. intros. apply is_sgn_sign_ext; auto. destruct (zlt m n). destruct H1. apply is_sgn_sign_ext in H1; auto. - rewrite <- H1. rewrite (Int.sign_ext_widen i) by omega. apply Int.sign_ext_idem; auto. - omegaContradiction. - apply Int.sign_ext_widen; omega. + rewrite <- H1. rewrite (Int.sign_ext_widen i) by lia. apply Int.sign_ext_idem; auto. + extlia. + apply Int.sign_ext_widen; lia. Qed. Hint Resolve is_uns_mon is_sgn_mon is_uns_sgn is_uns_usize is_sgn_ssize : va. @@ -701,8 +701,8 @@ Lemma is_uns_1: forall n, is_uns 1 n -> n = Int.zero \/ n = Int.one. Proof. intros. destruct (Int.testbit n 0) eqn:B0; [right|left]; apply Int.same_bits_eq; intros. - rewrite Int.bits_one. destruct (zeq i 0). subst i; auto. apply H; omega. - rewrite Int.bits_zero. destruct (zeq i 0). subst i; auto. apply H; omega. + rewrite Int.bits_one. destruct (zeq i 0). subst i; auto. apply H; lia. + rewrite Int.bits_zero. destruct (zeq i 0). subst i; auto. apply H; lia. Qed. (** Tracking leakage of pointers through arithmetic operations. @@ -958,13 +958,13 @@ Hint Resolve vge_uns_uns' vge_uns_i' vge_sgn_sgn' vge_sgn_i' : va. Lemma usize_pos: forall n, 0 <= usize n. Proof. - unfold usize; intros. generalize (Int.size_range n); omega. + unfold usize; intros. generalize (Int.size_range n); lia. Qed. Lemma ssize_pos: forall n, 0 < ssize n. Proof. unfold ssize; intros. - generalize (Int.size_range (if Int.lt n Int.zero then Int.not n else n)); omega. + generalize (Int.size_range (if Int.lt n Int.zero then Int.not n else n)); lia. Qed. Lemma vge_lub_l: @@ -975,12 +975,12 @@ Proof. unfold vlub; destruct x, y; eauto using pge_lub_l with va. - predSpec Int.eq Int.eq_spec n n0. auto with va. destruct (Int.lt n Int.zero || Int.lt n0 Int.zero). - apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. - apply vge_uns_i'. generalize (usize_pos n); xomega. eauto with va. + apply vge_sgn_i'. generalize (ssize_pos n); extlia. eauto with va. + apply vge_uns_i'. generalize (usize_pos n); extlia. eauto with va. - destruct (Int.lt n Int.zero). - apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. - apply vge_uns_i'. generalize (usize_pos n); xomega. eauto with va. -- apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. + apply vge_sgn_i'. generalize (ssize_pos n); extlia. eauto with va. + apply vge_uns_i'. generalize (usize_pos n); extlia. eauto with va. +- apply vge_sgn_i'. generalize (ssize_pos n); extlia. eauto with va. - destruct (Int.lt n0 Int.zero). eapply vge_trans. apply vge_sgn_sgn'. apply vge_trans with (Sgn p (n + 1)); eauto with va. @@ -1269,12 +1269,12 @@ Proof. destruct (Int.ltu n Int.iwordsize) eqn:LTU; auto. exploit Int.ltu_inv; eauto. intros RANGE. inv H; auto with va. -- apply vmatch_uns'. red; intros. rewrite Int.bits_shl by omega. - destruct (zlt m (Int.unsigned n)). auto. apply H1; xomega. +- apply vmatch_uns'. red; intros. rewrite Int.bits_shl by lia. + destruct (zlt m (Int.unsigned n)). auto. apply H1; extlia. - apply vmatch_sgn'. red; intros. zify. - rewrite ! Int.bits_shl by omega. - rewrite ! zlt_false by omega. - rewrite H1 by omega. symmetry. rewrite H1 by omega. auto. + rewrite ! Int.bits_shl by lia. + rewrite ! zlt_false by lia. + rewrite H1 by lia. symmetry. rewrite H1 by lia. auto. - destruct v; constructor. Qed. @@ -1306,13 +1306,13 @@ Proof. assert (DEFAULT2: forall i, vmatch (Vint (Int.shru i n)) (uns (provenance x) (Int.zwordsize - Int.unsigned n))). { intros. apply vmatch_uns. red; intros. - rewrite Int.bits_shru by omega. apply zlt_false. omega. + rewrite Int.bits_shru by lia. apply zlt_false. lia. } inv H; auto with va. - apply vmatch_uns'. red; intros. zify. - rewrite Int.bits_shru by omega. + rewrite Int.bits_shru by lia. destruct (zlt (m + Int.unsigned n) Int.zwordsize); auto. - apply H1; omega. + apply H1; lia. - destruct v; constructor. Qed. @@ -1345,22 +1345,22 @@ Proof. assert (DEFAULT2: forall i, vmatch (Vint (Int.shr i n)) (sgn (provenance x) (Int.zwordsize - Int.unsigned n))). { intros. apply vmatch_sgn. red; intros. - rewrite ! Int.bits_shr by omega. f_equal. + rewrite ! Int.bits_shr by lia. f_equal. destruct (zlt (m + Int.unsigned n) Int.zwordsize); destruct (zlt (Int.zwordsize - 1 + Int.unsigned n) Int.zwordsize); - omega. + lia. } assert (SGN: forall q i p, is_sgn p i -> 0 < p -> vmatch (Vint (Int.shr i n)) (sgn q (p - Int.unsigned n))). { intros. apply vmatch_sgn'. red; intros. zify. - rewrite ! Int.bits_shr by omega. + rewrite ! Int.bits_shr by lia. transitivity (Int.testbit i (Int.zwordsize - 1)). destruct (zlt (m + Int.unsigned n) Int.zwordsize). - apply H0; omega. + apply H0; lia. auto. symmetry. destruct (zlt (Int.zwordsize - 1 + Int.unsigned n) Int.zwordsize). - apply H0; omega. + apply H0; lia. auto. } inv H; eauto with va. @@ -1418,12 +1418,12 @@ Proof. assert (UNS: forall i j n m, is_uns n i -> is_uns m j -> is_uns (Z.max n m) (Int.or i j)). { intros; red; intros. rewrite Int.bits_or by auto. - rewrite H by xomega. rewrite H0 by xomega. auto. + rewrite H by extlia. rewrite H0 by extlia. auto. } assert (SGN: forall i j n m, is_sgn n i -> is_sgn m j -> is_sgn (Z.max n m) (Int.or i j)). { - intros; red; intros. rewrite ! Int.bits_or by xomega. - rewrite H by xomega. rewrite H0 by xomega. auto. + intros; red; intros. rewrite ! Int.bits_or by extlia. + rewrite H by extlia. rewrite H0 by extlia. auto. } intros. unfold or, Val.or; inv H; eauto with va; inv H0; eauto with va. Qed. @@ -1443,12 +1443,12 @@ Proof. assert (UNS: forall i j n m, is_uns n i -> is_uns m j -> is_uns (Z.max n m) (Int.xor i j)). { intros; red; intros. rewrite Int.bits_xor by auto. - rewrite H by xomega. rewrite H0 by xomega. auto. + rewrite H by extlia. rewrite H0 by extlia. auto. } assert (SGN: forall i j n m, is_sgn n i -> is_sgn m j -> is_sgn (Z.max n m) (Int.xor i j)). { - intros; red; intros. rewrite ! Int.bits_xor by xomega. - rewrite H by xomega. rewrite H0 by xomega. auto. + intros; red; intros. rewrite ! Int.bits_xor by extlia. + rewrite H by extlia. rewrite H0 by extlia. auto. } intros. unfold xor, Val.xor; inv H; eauto with va; inv H0; eauto with va. Qed. @@ -1466,7 +1466,7 @@ Lemma notint_sound: Proof. assert (SGN: forall n i, is_sgn n i -> is_sgn n (Int.not i)). { - intros; red; intros. rewrite ! Int.bits_not by omega. + intros; red; intros. rewrite ! Int.bits_not by lia. f_equal. apply H; auto. } intros. unfold Val.notint, notint; inv H; eauto with va. @@ -1492,13 +1492,13 @@ Proof. inv H; auto with va. - apply vmatch_uns. red; intros. rewrite Int.bits_rol by auto. generalize (Int.unsigned_range n); intros. - rewrite Z.mod_small by omega. - apply H1. omega. omega. + rewrite Z.mod_small by lia. + apply H1. lia. lia. - destruct (zlt n0 Int.zwordsize); auto with va. - apply vmatch_sgn. red; intros. rewrite ! Int.bits_rol by omega. + apply vmatch_sgn. red; intros. rewrite ! Int.bits_rol by lia. generalize (Int.unsigned_range n); intros. - rewrite ! Z.mod_small by omega. - rewrite H1 by omega. symmetry. rewrite H1 by omega. auto. + rewrite ! Z.mod_small by lia. + rewrite H1 by lia. symmetry. rewrite H1 by lia. auto. - destruct (zlt n0 Int.zwordsize); auto with va. Qed. @@ -1674,8 +1674,8 @@ Proof. generalize (Int.unsigned_range_2 j); intros RANGE. assert (Int.unsigned j <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned j). rewrite H0. auto. } - exploit (Z_mod_lt (Int.unsigned i) (Int.unsigned j)). omega. intros MOD. - unfold Int.modu. rewrite Int.unsigned_repr. omega. omega. + exploit (Z_mod_lt (Int.unsigned i) (Int.unsigned j)). lia. intros MOD. + unfold Int.modu. rewrite Int.unsigned_repr. lia. lia. } intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.eq i0 Int.zero) eqn:Z; inv H1. @@ -2084,12 +2084,12 @@ Lemma zero_ext_sound: Proof. assert (DFL: forall nbits i, is_uns nbits (Int.zero_ext nbits i)). { - intros; red; intros. rewrite Int.bits_zero_ext by omega. apply zlt_false; auto. + intros; red; intros. rewrite Int.bits_zero_ext by lia. apply zlt_false; auto. } intros. inv H; simpl; auto with va. apply vmatch_uns. red; intros. zify. - rewrite Int.bits_zero_ext by omega. - destruct (zlt m nbits); auto. apply H1; omega. + rewrite Int.bits_zero_ext by lia. + destruct (zlt m nbits); auto. apply H1; lia. Qed. Definition sign_ext (nbits: Z) (v: aval) := @@ -2109,7 +2109,7 @@ Proof. intros. apply vmatch_sgn. apply is_sign_ext_sgn; auto with va. } intros. unfold sign_ext. destruct (zle nbits 0). -- destruct v; simpl; auto with va. constructor. omega. +- destruct v; simpl; auto with va. constructor. lia. rewrite Int.sign_ext_below by auto. red; intros; apply Int.bits_zero. - inv H; simpl; auto with va. + destruct (zlt n nbits); eauto with va. @@ -2514,26 +2514,26 @@ Proof. intros c [lo hi] x n; simpl; intros R. destruct c; unfold zcmp, proj_sumbool. - (* eq *) - destruct (zlt n lo). rewrite zeq_false by omega. constructor. - destruct (zlt hi n). rewrite zeq_false by omega. constructor. + destruct (zlt n lo). rewrite zeq_false by lia. constructor. + destruct (zlt hi n). rewrite zeq_false by lia. constructor. constructor. - (* ne *) constructor. - (* lt *) - destruct (zlt hi n). rewrite zlt_true by omega. constructor. - destruct (zle n lo). rewrite zlt_false by omega. constructor. + destruct (zlt hi n). rewrite zlt_true by lia. constructor. + destruct (zle n lo). rewrite zlt_false by lia. constructor. constructor. - (* le *) - destruct (zle hi n). rewrite zle_true by omega. constructor. - destruct (zlt n lo). rewrite zle_false by omega. constructor. + destruct (zle hi n). rewrite zle_true by lia. constructor. + destruct (zlt n lo). rewrite zle_false by lia. constructor. constructor. - (* gt *) - destruct (zlt n lo). rewrite zlt_true by omega. constructor. - destruct (zle hi n). rewrite zlt_false by omega. constructor. + destruct (zlt n lo). rewrite zlt_true by lia. constructor. + destruct (zle hi n). rewrite zlt_false by lia. constructor. constructor. - (* ge *) - destruct (zle n lo). rewrite zle_true by omega. constructor. - destruct (zlt hi n). rewrite zle_false by omega. constructor. + destruct (zle n lo). rewrite zle_true by lia. constructor. + destruct (zlt hi n). rewrite zle_false by lia. constructor. constructor. Qed. @@ -2567,10 +2567,10 @@ Lemma uintv_sound: forall n v, vmatch (Vint n) v -> fst (uintv v) <= Int.unsigned n <= snd (uintv v). Proof. intros. inv H; simpl; try (apply Int.unsigned_range_2). -- omega. +- lia. - destruct (zlt n0 Int.zwordsize); simpl. -+ rewrite is_uns_zero_ext in H2. rewrite <- H2. rewrite Int.zero_ext_mod by omega. - exploit (Z_mod_lt (Int.unsigned n) (two_p n0)). apply two_p_gt_ZERO; auto. omega. ++ rewrite is_uns_zero_ext in H2. rewrite <- H2. rewrite Int.zero_ext_mod by lia. + exploit (Z_mod_lt (Int.unsigned n) (two_p n0)). apply two_p_gt_ZERO; auto. lia. + apply Int.unsigned_range_2. Qed. @@ -2582,8 +2582,8 @@ Proof. intros. simpl. replace (Int.cmpu c n1 n2) with (zcmp c (Int.unsigned n1) (Int.unsigned n2)). apply zcmp_intv_sound; apply uintv_sound; auto. destruct c; simpl; auto. - unfold Int.ltu. destruct (zle (Int.unsigned n1) (Int.unsigned n2)); [rewrite zlt_false|rewrite zlt_true]; auto; omega. - unfold Int.ltu. destruct (zle (Int.unsigned n2) (Int.unsigned n1)); [rewrite zlt_false|rewrite zlt_true]; auto; omega. + unfold Int.ltu. destruct (zle (Int.unsigned n1) (Int.unsigned n2)); [rewrite zlt_false|rewrite zlt_true]; auto; lia. + unfold Int.ltu. destruct (zle (Int.unsigned n2) (Int.unsigned n1)); [rewrite zlt_false|rewrite zlt_true]; auto; lia. Qed. Lemma cmpu_intv_sound_2: @@ -2610,22 +2610,22 @@ Lemma sintv_sound: forall n v, vmatch (Vint n) v -> fst (sintv v) <= Int.signed n <= snd (sintv v). Proof. intros. inv H; simpl; try (apply Int.signed_range). -- omega. +- lia. - destruct (zlt n0 Int.zwordsize); simpl. + rewrite is_uns_zero_ext in H2. rewrite <- H2. - assert (Int.unsigned (Int.zero_ext n0 n) = Int.unsigned n mod two_p n0) by (apply Int.zero_ext_mod; omega). + assert (Int.unsigned (Int.zero_ext n0 n) = Int.unsigned n mod two_p n0) by (apply Int.zero_ext_mod; lia). exploit (Z_mod_lt (Int.unsigned n) (two_p n0)). apply two_p_gt_ZERO; auto. intros. replace (Int.signed (Int.zero_ext n0 n)) with (Int.unsigned (Int.zero_ext n0 n)). - rewrite H. omega. + rewrite H. lia. unfold Int.signed. rewrite zlt_true. auto. assert (two_p n0 <= Int.half_modulus). { change Int.half_modulus with (two_p (Int.zwordsize - 1)). - apply two_p_monotone. omega. } - omega. + apply two_p_monotone. lia. } + lia. + apply Int.signed_range. - destruct (zlt n0 (Int.zwordsize)); simpl. + rewrite is_sgn_sign_ext in H2 by auto. rewrite <- H2. - exploit (Int.sign_ext_range n0 n). omega. omega. + exploit (Int.sign_ext_range n0 n). lia. lia. + apply Int.signed_range. Qed. @@ -2637,8 +2637,8 @@ Proof. intros. simpl. replace (Int.cmp c n1 n2) with (zcmp c (Int.signed n1) (Int.signed n2)). apply zcmp_intv_sound; apply sintv_sound; auto. destruct c; simpl; rewrite ? Int.eq_signed; auto. - unfold Int.lt. destruct (zle (Int.signed n1) (Int.signed n2)); [rewrite zlt_false|rewrite zlt_true]; auto; omega. - unfold Int.lt. destruct (zle (Int.signed n2) (Int.signed n1)); [rewrite zlt_false|rewrite zlt_true]; auto; omega. + unfold Int.lt. destruct (zle (Int.signed n1) (Int.signed n2)); [rewrite zlt_false|rewrite zlt_true]; auto; lia. + unfold Int.lt. destruct (zle (Int.signed n2) (Int.signed n1)); [rewrite zlt_false|rewrite zlt_true]; auto; lia. Qed. Lemma cmp_intv_sound_2: @@ -2823,7 +2823,7 @@ Proof. assert (DEFAULT: vmatch (Val.of_optbool ob) (Uns Pbot 1)). { destruct ob; simpl; auto with va. - destruct b; constructor; try omega. + destruct b; constructor; try lia. change 1 with (usize Int.one). apply is_uns_usize. red; intros. apply Int.bits_zero. } @@ -2942,27 +2942,27 @@ Proof. - destruct (zlt n 8); constructor; auto with va. apply is_sign_ext_uns; auto. apply is_sign_ext_sgn; auto with va. -- constructor. xomega. apply is_zero_ext_uns. apply Z.min_case; auto with va. +- constructor. extlia. apply is_zero_ext_uns. apply Z.min_case; auto with va. - destruct (zlt n 16); constructor; auto with va. apply is_sign_ext_uns; auto. apply is_sign_ext_sgn; auto with va. -- constructor. xomega. apply is_zero_ext_uns. apply Z.min_case; auto with va. +- constructor. extlia. apply is_zero_ext_uns. apply Z.min_case; auto with va. - destruct (zlt n 8); auto with va. - destruct (zlt n 16); auto with va. -- constructor. xomega. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. -- constructor. omega. apply is_zero_ext_uns; auto with va. -- constructor. xomega. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. -- constructor. omega. apply is_zero_ext_uns; auto with va. +- constructor. extlia. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. +- constructor. lia. apply is_zero_ext_uns; auto with va. +- constructor. extlia. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. +- constructor. lia. apply is_zero_ext_uns; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. -- constructor. omega. apply is_sign_ext_sgn; auto with va. -- constructor. omega. apply is_zero_ext_uns; auto with va. -- constructor. omega. apply is_sign_ext_sgn; auto with va. -- constructor. omega. apply is_zero_ext_uns; auto with va. +- constructor. lia. apply is_sign_ext_sgn; auto with va. +- constructor. lia. apply is_zero_ext_uns; auto with va. +- constructor. lia. apply is_sign_ext_sgn; auto with va. +- constructor. lia. apply is_zero_ext_uns; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. @@ -2977,13 +2977,13 @@ Proof. intros. exploit Mem.load_cast; eauto. exploit Mem.load_type; eauto. destruct chunk; simpl; intros. - (* int8signed *) - rewrite H2. destruct v; simpl; constructor. omega. apply is_sign_ext_sgn; auto with va. + rewrite H2. destruct v; simpl; constructor. lia. apply is_sign_ext_sgn; auto with va. - (* int8unsigned *) - rewrite H2. destruct v; simpl; constructor. omega. apply is_zero_ext_uns; auto with va. + rewrite H2. destruct v; simpl; constructor. lia. apply is_zero_ext_uns; auto with va. - (* int16signed *) - rewrite H2. destruct v; simpl; constructor. omega. apply is_sign_ext_sgn; auto with va. + rewrite H2. destruct v; simpl; constructor. lia. apply is_sign_ext_sgn; auto with va. - (* int16unsigned *) - rewrite H2. destruct v; simpl; constructor. omega. apply is_zero_ext_uns; auto with va. + rewrite H2. destruct v; simpl; constructor. lia. apply is_zero_ext_uns; auto with va. - (* int32 *) auto. - (* int64 *) @@ -3025,9 +3025,9 @@ Proof with (auto using provenance_monotone with va). apply is_sign_ext_sgn... - constructor... apply is_zero_ext_uns... apply Z.min_case... - unfold provenance; destruct (va_strict tt)... -- destruct (zlt n1 8). rewrite zlt_true by omega... +- destruct (zlt n1 8). rewrite zlt_true by lia... destruct (zlt n2 8)... -- destruct (zlt n1 16). rewrite zlt_true by omega... +- destruct (zlt n1 16). rewrite zlt_true by lia... destruct (zlt n2 16)... - constructor... apply is_sign_ext_sgn... apply Z.min_case... - constructor... apply is_zero_ext_uns... @@ -3148,7 +3148,7 @@ Function inval_after (lo: Z) (hi: Z) (c: ZTree.t acontent) { wf (Zwf lo) hi } : then inval_after lo (hi - 1) (ZTree.remove hi c) else c. Proof. - intros; red; omega. + intros; red; lia. apply Zwf_well_founded. Qed. @@ -3163,7 +3163,7 @@ Function inval_before (hi: Z) (lo: Z) (c: ZTree.t acontent) { wf (Zwf_up hi) lo then inval_before hi (lo + 1) (inval_if hi lo c) else c. Proof. - intros; red; omega. + intros; red; lia. apply Zwf_up_well_founded. Qed. @@ -3201,7 +3201,7 @@ Remark loadbytes_load_ext: Proof. intros. exploit Mem.load_loadbytes; eauto. intros [bytes [A B]]. exploit Mem.load_valid_access; eauto. intros [C D]. - subst v. apply Mem.loadbytes_load; auto. apply H; auto. generalize (size_chunk_pos chunk); omega. + subst v. apply Mem.loadbytes_load; auto. apply H; auto. generalize (size_chunk_pos chunk); lia. Qed. Lemma smatch_ext: @@ -3212,7 +3212,7 @@ Lemma smatch_ext: Proof. intros. destruct H. split; intros. eapply H; eauto. eapply loadbytes_load_ext; eauto. - eapply H1; eauto. apply H0; eauto. omega. + eapply H1; eauto. apply H0; eauto. lia. Qed. Lemma smatch_inv: @@ -3247,19 +3247,19 @@ Proof. + rewrite (Mem.loadbytes_empty m b ofs sz) in LOAD by auto. inv LOAD. contradiction. + exploit (Mem.loadbytes_split m b ofs 1 (sz - 1) bytes). - replace (1 + (sz - 1)) with sz by omega. auto. - omega. - omega. + replace (1 + (sz - 1)) with sz by lia. auto. + lia. + lia. intros (bytes1 & bytes2 & LOAD1 & LOAD2 & CONCAT). subst bytes. exploit Mem.loadbytes_length. eexact LOAD1. change (Z.to_nat 1) with 1%nat. intros LENGTH1. rewrite in_app_iff in IN. destruct IN. * destruct bytes1; try discriminate. destruct bytes1; try discriminate. simpl in H. destruct H; try contradiction. subst m0. - exists ofs; split. omega. auto. - * exploit (REC (sz - 1)). red; omega. eexact LOAD2. auto. + exists ofs; split. lia. auto. + * exploit (REC (sz - 1)). red; lia. eexact LOAD2. auto. intros (ofs' & A & B). - exists ofs'; split. omega. auto. + exists ofs'; split. lia. auto. Qed. Lemma smatch_loadbytes: @@ -3285,13 +3285,13 @@ Proof. - apply Zwf_well_founded. - intros sz REC ofs bytes LOAD LOAD1 IN. exploit (Mem.loadbytes_split m b ofs 1 (sz - 1) bytes). - replace (1 + (sz - 1)) with sz by omega. auto. - omega. - omega. + replace (1 + (sz - 1)) with sz by lia. auto. + lia. + lia. intros (bytes1 & bytes2 & LOAD3 & LOAD4 & CONCAT). subst bytes. rewrite in_app_iff. destruct (zeq ofs ofs'). + subst ofs'. rewrite LOAD1 in LOAD3; inv LOAD3. left; simpl; auto. -+ right. eapply (REC (sz - 1)). red; omega. eexact LOAD4. auto. omega. ++ right. eapply (REC (sz - 1)). red; lia. eexact LOAD4. auto. lia. Qed. Lemma storebytes_provenance: @@ -3309,10 +3309,10 @@ Proof. destruct (eq_block b' b); auto. destruct (zle (ofs' + 1) ofs); auto. destruct (zle (ofs + Z.of_nat (length bytes)) ofs'); auto. - right. split. auto. omega. + right. split. auto. lia. } destruct EITHER as [A | (A & B)]. -- right. rewrite <- H0. symmetry. eapply Mem.loadbytes_storebytes_other; eauto. omega. +- right. rewrite <- H0. symmetry. eapply Mem.loadbytes_storebytes_other; eauto. lia. - subst b'. left. eapply loadbytes_provenance; eauto. eapply Mem.loadbytes_storebytes_same; eauto. @@ -3457,7 +3457,7 @@ Remark inval_after_outside: forall i lo hi c, i < lo \/ i > hi -> (inval_after lo hi c)##i = c##i. Proof. intros until c. functional induction (inval_after lo hi c); intros. - rewrite IHt by omega. apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; omega. + rewrite IHt by lia. apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; lia. auto. Qed. @@ -3468,18 +3468,18 @@ Remark inval_after_contents: Proof. intros until c. functional induction (inval_after lo hi c); intros. destruct (zeq i hi). - subst i. rewrite inval_after_outside in H by omega. rewrite ZTree.grs in H. discriminate. - exploit IHt; eauto. intros [A B]. rewrite ZTree.gro in A by auto. split. auto. omega. - split. auto. omega. + subst i. rewrite inval_after_outside in H by lia. rewrite ZTree.grs in H. discriminate. + exploit IHt; eauto. intros [A B]. rewrite ZTree.gro in A by auto. split. auto. lia. + split. auto. lia. Qed. Remark inval_before_outside: forall i hi lo c, i < lo \/ i >= hi -> (inval_before hi lo c)##i = c##i. Proof. intros until c. functional induction (inval_before hi lo c); intros. - rewrite IHt by omega. unfold inval_if. destruct (c##lo) as [[chunk av]|]; auto. + rewrite IHt by lia. unfold inval_if. destruct (c##lo) as [[chunk av]|]; auto. destruct (zle (lo + size_chunk chunk) hi); auto. - apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; omega. + apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; lia. auto. Qed. @@ -3490,21 +3490,21 @@ Remark inval_before_contents_1: Proof. intros until c. functional induction (inval_before hi lo c); intros. - destruct (zeq lo i). -+ subst i. rewrite inval_before_outside in H0 by omega. ++ subst i. rewrite inval_before_outside in H0 by lia. unfold inval_if in H0. destruct (c##lo) as [[chunk0 v0]|] eqn:C; try congruence. destruct (zle (lo + size_chunk chunk0) hi). rewrite C in H0; inv H0. auto. rewrite ZTree.grs in H0. congruence. -+ exploit IHt. omega. auto. intros [A B]; split; auto. ++ exploit IHt. lia. auto. intros [A B]; split; auto. unfold inval_if in A. destruct (c##lo) as [[chunk0 v0]|] eqn:C; auto. destruct (zle (lo + size_chunk chunk0) hi); auto. rewrite ZTree.gro in A; auto. -- omegaContradiction. +- extlia. Qed. Lemma max_size_chunk: forall chunk, size_chunk chunk <= 8. Proof. - destruct chunk; simpl; omega. + destruct chunk; simpl; lia. Qed. Remark inval_before_contents: @@ -3513,12 +3513,12 @@ Remark inval_before_contents: c##j = Some (ACval chunk' av') /\ (j + size_chunk chunk' <= i \/ i <= j). Proof. intros. destruct (zlt j (i - 7)). - rewrite inval_before_outside in H by omega. - split. auto. left. generalize (max_size_chunk chunk'); omega. + rewrite inval_before_outside in H by lia. + split. auto. left. generalize (max_size_chunk chunk'); lia. destruct (zlt j i). - exploit inval_before_contents_1; eauto. omega. tauto. - rewrite inval_before_outside in H by omega. - split. auto. omega. + exploit inval_before_contents_1; eauto. lia. tauto. + rewrite inval_before_outside in H by lia. + split. auto. lia. Qed. Lemma ablock_store_contents: @@ -3534,7 +3534,7 @@ Proof. right. rewrite ZTree.gso in H by auto. exploit inval_before_contents; eauto. intros [A B]. exploit inval_after_contents; eauto. intros [C D]. - split. auto. omega. + split. auto. lia. Qed. Lemma chunk_compat_true: @@ -3604,7 +3604,7 @@ Proof. unfold ablock_storebytes; simpl; intros. exploit inval_before_contents; eauto. clear H. intros [A B]. exploit inval_after_contents; eauto. clear A. intros [C D]. - split. auto. xomega. + split. auto. extlia. Qed. Lemma ablock_storebytes_sound: @@ -3627,7 +3627,7 @@ Proof. exploit ablock_storebytes_contents; eauto. intros [A B]. assert (Mem.load chunk' m b ofs' = Some v'). { rewrite <- LOAD'; symmetry. eapply Mem.load_storebytes_other; eauto. - rewrite U. rewrite LENGTH. rewrite Z_to_nat_max. right; omega. } + rewrite U. rewrite LENGTH. rewrite Z_to_nat_max. right; lia. } exploit BM2; eauto. unfold ablock_load. rewrite A. rewrite COMPAT. auto. Qed. @@ -3755,7 +3755,7 @@ Proof. apply bmatch_inv with m; auto. + intros. eapply Mem.loadbytes_store_other; eauto. left. red; intros; subst b0. elim (C ofs). apply Mem.perm_cur_max. - apply P. generalize (size_chunk_pos chunk); omega. + apply P. generalize (size_chunk_pos chunk); lia. - intros; red; intros; elim (C ofs0). eauto with mem. Qed. @@ -4184,7 +4184,7 @@ Proof. - apply bmatch_ext with m; eauto with va. - apply smatch_ext with m; auto with va. - apply smatch_ext with m; auto with va. -- red; intros. exploit mmatch_below0; eauto. xomega. +- red; intros. exploit mmatch_below0; eauto. extlia. Qed. Lemma mmatch_free: @@ -4195,7 +4195,7 @@ Lemma mmatch_free: Proof. intros. apply mmatch_ext with m; auto. intros. eapply Mem.loadbytes_free_2; eauto. - erewrite <- Mem.nextblock_free by eauto. xomega. + erewrite <- Mem.nextblock_free by eauto. extlia. Qed. Lemma mmatch_top': @@ -4419,7 +4419,7 @@ Proof. { Local Transparent Mem.loadbytes. unfold Mem.loadbytes. rewrite pred_dec_true. reflexivity. - red; intros. replace ofs0 with ofs by omega. auto. + red; intros. replace ofs0 with ofs by lia. auto. } destruct mv; econstructor. destruct v; econstructor. apply inj_of_bc_valid. @@ -4440,7 +4440,7 @@ Proof. auto. - (* overflow *) intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. - rewrite Z.add_0_r. split. omega. apply Ptrofs.unsigned_range_2. + rewrite Z.add_0_r. split. lia. apply Ptrofs.unsigned_range_2. - (* perm inv *) intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. rewrite Z.add_0_r in H2. auto. diff --git a/cfrontend/Cexec.v b/cfrontend/Cexec.v index b08c3ad7..d763c98c 100644 --- a/cfrontend/Cexec.v +++ b/cfrontend/Cexec.v @@ -290,7 +290,7 @@ Definition assign_copy_ok (ty: type) (b: block) (ofs: ptrofs) (b': block) (ofs': Remark check_assign_copy: forall (ty: type) (b: block) (ofs: ptrofs) (b': block) (ofs': ptrofs), { assign_copy_ok ty b ofs b' ofs' } + {~ assign_copy_ok ty b ofs b' ofs' }. -Proof with try (right; intuition omega). +Proof with try (right; intuition lia). intros. unfold assign_copy_ok. destruct (Zdivide_dec (alignof_blockcopy ge ty) (Ptrofs.unsigned ofs')); auto... destruct (Zdivide_dec (alignof_blockcopy ge ty) (Ptrofs.unsigned ofs)); auto... @@ -306,8 +306,8 @@ Proof with try (right; intuition omega). destruct (zeq (Ptrofs.unsigned ofs') (Ptrofs.unsigned ofs)); auto. destruct (zle (Ptrofs.unsigned ofs' + sizeof ge ty) (Ptrofs.unsigned ofs)); auto. destruct (zle (Ptrofs.unsigned ofs + sizeof ge ty) (Ptrofs.unsigned ofs')); auto. - right; intuition omega. - destruct Y... left; intuition omega. + right; intuition lia. + destruct Y... left; intuition lia. Defined. Definition do_assign_loc (w: world) (ty: type) (m: mem) (b: block) (ofs: ptrofs) (v: val): option (world * trace * mem) := @@ -579,7 +579,7 @@ Proof with try congruence. replace (Vlong Int64.zero) with Vnullptr. split; constructor. unfold Vnullptr; rewrite H0; auto. + destruct vargs... mydestr. - split. apply SIZE in Heqo0. econstructor; eauto. congruence. omega. + split. apply SIZE in Heqo0. econstructor; eauto. congruence. lia. constructor. - (* EF_memcpy *) unfold do_ef_memcpy. destruct vargs... destruct v... destruct vargs... @@ -636,7 +636,7 @@ Proof. inv H0. erewrite SIZE by eauto. rewrite H1, H2. auto. - (* EF_free *) inv H; unfold do_ef_free. -+ inv H0. rewrite H1. erewrite SIZE by eauto. rewrite zlt_true. rewrite H3. auto. omega. ++ inv H0. rewrite H1. erewrite SIZE by eauto. rewrite zlt_true. rewrite H3. auto. lia. + inv H0. unfold Vnullptr; destruct Archi.ptr64; auto. - (* EF_memcpy *) inv H; unfold do_ef_memcpy. diff --git a/cfrontend/Clight.v b/cfrontend/Clight.v index 8ab29fe9..239ca370 100644 --- a/cfrontend/Clight.v +++ b/cfrontend/Clight.v @@ -739,7 +739,7 @@ Proof. exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]]. exists (Returnstate vres2 k m2). econstructor; eauto. (* trace length *) - red; simpl; intros. inv H; simpl; try omega. + red; simpl; intros. inv H; simpl; try lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. Qed. diff --git a/cfrontend/Cminorgen.v b/cfrontend/Cminorgen.v index 45c21f96..1b031866 100644 --- a/cfrontend/Cminorgen.v +++ b/cfrontend/Cminorgen.v @@ -240,7 +240,7 @@ Module VarOrder <: TotalLeBool. Theorem leb_total: forall v1 v2, leb v1 v2 = true \/ leb v2 v1 = true. Proof. unfold leb; intros. - assert (snd v1 <= snd v2 \/ snd v2 <= snd v1) by omega. + assert (snd v1 <= snd v2 \/ snd v2 <= snd v1) by lia. unfold proj_sumbool. destruct H; [left|right]; apply zle_true; auto. Qed. End VarOrder. diff --git a/cfrontend/Cminorgenproof.v b/cfrontend/Cminorgenproof.v index 5acb996d..bc1c92ca 100644 --- a/cfrontend/Cminorgenproof.v +++ b/cfrontend/Cminorgenproof.v @@ -287,7 +287,7 @@ Lemma match_env_external_call: Proof. intros. apply match_env_invariant with f1; auto. intros. eapply inject_incr_separated_same'; eauto. - intros. eapply inject_incr_separated_same; eauto. red. destruct H. xomega. + intros. eapply inject_incr_separated_same; eauto. red. destruct H. extlia. Qed. (** [match_env] and allocations *) @@ -317,18 +317,18 @@ Proof. constructor; eauto. constructor. (* low-high *) - rewrite NEXTBLOCK; xomega. + rewrite NEXTBLOCK; extlia. (* bounded *) intros. rewrite PTree.gsspec in H. destruct (peq id0 id). - inv H. rewrite NEXTBLOCK; xomega. - exploit me_bounded0; eauto. rewrite NEXTBLOCK; xomega. + inv H. rewrite NEXTBLOCK; extlia. + exploit me_bounded0; eauto. rewrite NEXTBLOCK; extlia. (* inv *) intros. destruct (eq_block b (Mem.nextblock m1)). subst b. rewrite SAME in H; inv H. exists id; exists sz. apply PTree.gss. rewrite OTHER in H; auto. exploit me_inv0; eauto. intros [id1 [sz1 EQ]]. exists id1; exists sz1. rewrite PTree.gso; auto. congruence. (* incr *) - intros. rewrite OTHER in H. eauto. unfold block in *; xomega. + intros. rewrite OTHER in H. eauto. unfold block in *; extlia. Qed. (** The sizes of blocks appearing in [e] are respected. *) @@ -512,23 +512,23 @@ Proof. (* base case *) econstructor; eauto. inv H. constructor; intros; eauto. - eapply IMAGE; eauto. eapply H6; eauto. xomega. + eapply IMAGE; eauto. eapply H6; eauto. extlia. (* inductive case *) assert (Ple lo hi) by (eapply me_low_high; eauto). econstructor; eauto. eapply match_temps_invariant; eauto. eapply match_env_invariant; eauto. - intros. apply H3. xomega. + intros. apply H3. extlia. eapply match_bounds_invariant; eauto. intros. eapply H1; eauto. - exploit me_bounded; eauto. xomega. + exploit me_bounded; eauto. extlia. eapply padding_freeable_invariant; eauto. - intros. apply H3. xomega. + intros. apply H3. extlia. eapply IHmatch_callstack; eauto. - intros. eapply H1; eauto. xomega. - intros. eapply H2; eauto. xomega. - intros. eapply H3; eauto. xomega. - intros. eapply H4; eauto. xomega. + intros. eapply H1; eauto. extlia. + intros. eapply H2; eauto. extlia. + intros. eapply H3; eauto. extlia. + intros. eapply H4; eauto. extlia. Qed. Lemma match_callstack_incr_bound: @@ -538,8 +538,8 @@ Lemma match_callstack_incr_bound: match_callstack f m tm cs bound' tbound'. Proof. intros. inv H. - econstructor; eauto. xomega. xomega. - constructor; auto. xomega. xomega. + econstructor; eauto. extlia. extlia. + constructor; auto. extlia. extlia. Qed. (** Assigning a temporary variable. *) @@ -596,17 +596,17 @@ Proof. auto. inv A. assert (Mem.range_perm m b 0 sz Cur Freeable). eapply free_list_freeable; eauto. eapply in_blocks_of_env; eauto. - replace ofs with ((ofs - delta) + delta) by omega. - eapply Mem.perm_inject; eauto. apply H3. omega. + replace ofs with ((ofs - delta) + delta) by lia. + eapply Mem.perm_inject; eauto. apply H3. lia. destruct X as [tm' FREE]. exploit nextblock_freelist; eauto. intro NEXT. exploit Mem.nextblock_free; eauto. intro NEXT'. exists tm'. split. auto. split. rewrite NEXT; rewrite NEXT'. - apply match_callstack_incr_bound with lo sp; try omega. + apply match_callstack_incr_bound with lo sp; try lia. apply match_callstack_invariant with f m tm; auto. intros. eapply perm_freelist; eauto. - intros. eapply Mem.perm_free_1; eauto. left; unfold block; xomega. xomega. xomega. + intros. eapply Mem.perm_free_1; eauto. left; unfold block; extlia. extlia. extlia. eapply Mem.free_inject; eauto. intros. exploit me_inv0; eauto. intros [id [sz A]]. exists 0; exists sz; split. @@ -636,21 +636,21 @@ Proof. inv H. constructor; auto. intros. case_eq (f1 b1). intros [b2' delta'] EQ. rewrite (INCR _ _ _ EQ) in H. inv H. eauto. - intro EQ. exploit SEPARATED; eauto. intros [A B]. elim B. red. xomega. + intro EQ. exploit SEPARATED; eauto. intros [A B]. elim B. red. extlia. (* inductive case *) constructor. auto. auto. eapply match_temps_invariant; eauto. eapply match_env_invariant; eauto. red in SEPARATED. intros. destruct (f1 b) as [[b' delta']|] eqn:?. exploit INCR; eauto. congruence. - exploit SEPARATED; eauto. intros [A B]. elim B. red. xomega. + exploit SEPARATED; eauto. intros [A B]. elim B. red. extlia. intros. assert (Ple lo hi) by (eapply me_low_high; eauto). destruct (f1 b) as [[b' delta']|] eqn:?. apply INCR; auto. destruct (f2 b) as [[b' delta']|] eqn:?; auto. - exploit SEPARATED; eauto. intros [A B]. elim A. red. xomega. + exploit SEPARATED; eauto. intros [A B]. elim A. red. extlia. eapply match_bounds_invariant; eauto. - intros. eapply MAXPERMS; eauto. red. exploit me_bounded; eauto. xomega. + intros. eapply MAXPERMS; eauto. red. exploit me_bounded; eauto. extlia. (* padding-freeable *) red; intros. destruct (is_reachable_from_env_dec f1 e sp ofs). @@ -660,10 +660,10 @@ Proof. red; intros; red; intros. elim H3. exploit me_inv; eauto. intros [id [lv B]]. exploit BOUND0; eauto. intros C. - apply is_reachable_intro with id b0 lv delta; auto; omega. + apply is_reachable_intro with id b0 lv delta; auto; lia. eauto with mem. (* induction *) - eapply IHmatch_callstack; eauto. inv MENV; xomega. xomega. + eapply IHmatch_callstack; eauto. inv MENV; extlia. extlia. Qed. (** [match_callstack] and allocations *) @@ -683,12 +683,12 @@ Proof. exploit Mem.nextblock_alloc; eauto. intros NEXTBLOCK. exploit Mem.alloc_result; eauto. intros RES. constructor. - xomega. - unfold block in *; xomega. + extlia. + unfold block in *; extlia. auto. constructor; intros. rewrite H3. rewrite PTree.gempty. constructor. - xomega. + extlia. rewrite PTree.gempty in H4; discriminate. eelim Mem.fresh_block_alloc; eauto. eapply Mem.valid_block_inject_2; eauto. rewrite RES. change (Mem.valid_block tm tb). eapply Mem.valid_block_inject_2; eauto. @@ -719,23 +719,23 @@ Proof. exploit Mem.alloc_result; eauto. intros RES. assert (LO: Ple lo (Mem.nextblock m1)) by (eapply me_low_high; eauto). constructor. - xomega. + extlia. auto. eapply match_temps_invariant; eauto. eapply match_env_alloc; eauto. red; intros. rewrite PTree.gsspec in H. destruct (peq id0 id). inversion H. subst b0 sz0 id0. eapply Mem.perm_alloc_3; eauto. eapply BOUND0; eauto. eapply Mem.perm_alloc_4; eauto. - exploit me_bounded; eauto. unfold block in *; xomega. + exploit me_bounded; eauto. unfold block in *; extlia. red; intros. exploit PERM; eauto. intros [A|A]. auto. right. inv A. apply is_reachable_intro with id0 b0 sz0 delta; auto. rewrite PTree.gso. auto. congruence. eapply match_callstack_invariant with (m1 := m1); eauto. intros. eapply Mem.perm_alloc_4; eauto. - unfold block in *; xomega. - intros. apply H4. unfold block in *; xomega. + unfold block in *; extlia. + intros. apply H4. unfold block in *; extlia. intros. destruct (eq_block b0 b). - subst b0. rewrite H3 in H. inv H. xomegaContradiction. + subst b0. rewrite H3 in H. inv H. extlia. rewrite H4 in H; auto. Qed. @@ -828,11 +828,11 @@ Proof. eexact MINJ. eexact H. eexact VALID. - instantiate (1 := ofs). zify. omega. - intros. exploit STKSIZE; eauto. omega. - intros. apply STKPERMS. zify. omega. - replace (sz - 0) with sz by omega. auto. - intros. eapply SEP2. eauto with coqlib. eexact CENV. eauto. eauto. omega. + instantiate (1 := ofs). zify. lia. + intros. exploit STKSIZE; eauto. lia. + intros. apply STKPERMS. zify. lia. + replace (sz - 0) with sz by lia. auto. + intros. eapply SEP2. eauto with coqlib. eexact CENV. eauto. eauto. lia. intros [f2 [A [B [C D]]]]. exploit (IHalloc_variables f2); eauto. red; intros. eapply COMPAT. auto with coqlib. @@ -841,7 +841,7 @@ Proof. subst b. rewrite C in H5; inv H5. exploit SEP1. eapply in_eq. eapply in_cons; eauto. eauto. eauto. red; intros; subst id0. elim H3. change id with (fst (id, sz0)). apply in_map; auto. - omega. + lia. eapply SEP2. apply in_cons; eauto. eauto. rewrite D in H5; eauto. eauto. auto. intros. rewrite PTree.gso. eapply UNBOUND; eauto with coqlib. @@ -890,9 +890,9 @@ Remark block_alignment_pos: forall sz, block_alignment sz > 0. Proof. unfold block_alignment; intros. - destruct (zlt sz 2). omega. - destruct (zlt sz 4). omega. - destruct (zlt sz 8); omega. + destruct (zlt sz 2). lia. + destruct (zlt sz 4). lia. + destruct (zlt sz 8); lia. Qed. Remark assign_variable_incr: @@ -901,8 +901,8 @@ Remark assign_variable_incr: Proof. simpl; intros. inv H. generalize (align_le stksz (block_alignment sz) (block_alignment_pos sz)). - assert (0 <= Z.max 0 sz). apply Zmax_bound_l. omega. - omega. + assert (0 <= Z.max 0 sz). apply Zmax_bound_l. lia. + lia. Qed. Remark assign_variables_incr: @@ -910,7 +910,7 @@ Remark assign_variables_incr: assign_variables (cenv, sz) vars = (cenv', sz') -> sz <= sz'. Proof. induction vars; intros until sz'. - simpl; intros. inv H. omega. + simpl; intros. inv H. lia. Opaque assign_variable. destruct a as [id s]. simpl. intros. destruct (assign_variable (cenv, sz) (id, s)) as [cenv1 sz1] eqn:?. @@ -931,11 +931,11 @@ Proof. assert (2 | 8). exists 4; auto. assert (4 | 8). exists 2; auto. destruct (zlt sz 2). - destruct chunk; simpl in *; auto; omegaContradiction. + destruct chunk; simpl in *; auto; extlia. destruct (zlt sz 4). - destruct chunk; simpl in *; auto; omegaContradiction. + destruct chunk; simpl in *; auto; extlia. destruct (zlt sz 8). - destruct chunk; simpl in *; auto; omegaContradiction. + destruct chunk; simpl in *; auto; extlia. destruct chunk; simpl; auto. apply align_divides. apply block_alignment_pos. Qed. @@ -948,7 +948,7 @@ Proof. replace (block_alignment sz) with (block_alignment (Z.max 0 sz)). apply inj_offset_aligned_block. rewrite Zmax_spec. destruct (zlt sz 0); auto. - transitivity 1. reflexivity. unfold block_alignment. rewrite zlt_true. auto. omega. + transitivity 1. reflexivity. unfold block_alignment. rewrite zlt_true. auto. lia. Qed. Lemma assign_variable_sound: @@ -976,23 +976,23 @@ Proof. exploit COMPAT; eauto. intros [ofs [A [B [C D]]]]. exists ofs. split. rewrite PTree.gso; auto. - split. auto. split. auto. zify; omega. + split. auto. split. auto. zify; lia. inv P. exists (align sz1 (block_alignment sz)). split. apply PTree.gss. split. apply inj_offset_aligned_block. - split. omega. - omega. + split. lia. + lia. apply EITHER in H; apply EITHER in H0. destruct H as [[P Q] | P]; destruct H0 as [[R S] | R]. rewrite PTree.gso in *; auto. eapply SEP; eauto. inv R. rewrite PTree.gso in H1; auto. rewrite PTree.gss in H2; inv H2. exploit COMPAT; eauto. intros [ofs [A [B [C D]]]]. assert (ofs = ofs1) by congruence. subst ofs. - left. zify; omega. + left. zify; lia. inv P. rewrite PTree.gso in H2; auto. rewrite PTree.gss in H1; inv H1. exploit COMPAT; eauto. intros [ofs [A [B [C D]]]]. assert (ofs = ofs2) by congruence. subst ofs. - right. zify; omega. + right. zify; lia. congruence. Qed. @@ -1023,7 +1023,7 @@ Proof. split. rewrite map_app. apply list_norepet_append_commut. simpl. constructor; auto. rewrite map_app. simpl. red; intros. rewrite in_app in H4. destruct H4. eauto. simpl in H4. destruct H4. subst y. red; intros; subst x. tauto. tauto. - generalize (assign_variable_incr _ _ _ _ _ _ Heqp). omega. + generalize (assign_variable_incr _ _ _ _ _ _ Heqp). lia. auto. auto. rewrite app_ass. auto. Qed. @@ -1054,7 +1054,7 @@ Proof. eexact H. simpl. rewrite app_nil_r. apply permutation_norepet with (map fst vars1); auto. apply Permutation_map. auto. - omega. + lia. red; intros. contradiction. red; intros. contradiction. destruct H1 as [A B]. split. @@ -1680,11 +1680,11 @@ Lemma switch_table_default: /\ snd (switch_table sl base) = (n + base)%nat. Proof. induction sl; simpl; intros. -- exists O; split. constructor. omega. +- exists O; split. constructor. lia. - destruct o. + destruct (IHsl (S base)) as (n & P & Q). exists (S n); split. constructor; auto. - destruct (switch_table sl (S base)) as [tbl dfl]; simpl in *. omega. + destruct (switch_table sl (S base)) as [tbl dfl]; simpl in *. lia. + exists O; split. constructor. destruct (switch_table sl (S base)) as [tbl dfl]; simpl in *. auto. Qed. @@ -1708,11 +1708,11 @@ Proof. exists O; split; auto. constructor. specialize (IHsl (S base) dfl). rewrite ST in IHsl. simpl in *. destruct (select_switch_case i sl). - destruct IHsl as (x & P & Q). exists (S x); split. constructor; auto. omega. + destruct IHsl as (x & P & Q). exists (S x); split. constructor; auto. lia. auto. specialize (IHsl (S base) dfl). rewrite ST in IHsl. simpl in *. destruct (select_switch_case i sl). - destruct IHsl as (x & P & Q). exists (S x); split. constructor; auto. omega. + destruct IHsl as (x & P & Q). exists (S x); split. constructor; auto. lia. auto. Qed. @@ -1725,10 +1725,10 @@ Proof. unfold select_switch; intros. generalize (switch_table_case i sl O (snd (switch_table sl O))). destruct (select_switch_case i sl) as [sl'|]. - intros (n & P & Q). replace (n + O)%nat with n in Q by omega. congruence. + intros (n & P & Q). replace (n + O)%nat with n in Q by lia. congruence. intros E; rewrite E. destruct (switch_table_default sl O) as (n & P & Q). - replace (n + O)%nat with n in Q by omega. congruence. + replace (n + O)%nat with n in Q by lia. congruence. Qed. Inductive transl_lblstmt_cont(cenv: compilenv) (xenv: exit_env): lbl_stmt -> cont -> cont -> Prop := @@ -2039,7 +2039,7 @@ Proof. apply match_callstack_incr_bound with (Mem.nextblock m) (Mem.nextblock tm). eapply match_callstack_external_call; eauto. intros. eapply external_call_max_perm; eauto. - xomega. xomega. + extlia. extlia. eapply external_call_nextblock; eauto. eapply external_call_nextblock; eauto. econstructor; eauto. @@ -2191,7 +2191,7 @@ Opaque PTree.set. apply match_callstack_incr_bound with (Mem.nextblock m) (Mem.nextblock tm). eapply match_callstack_external_call; eauto. intros. eapply external_call_max_perm; eauto. - xomega. xomega. + extlia. extlia. eapply external_call_nextblock; eauto. eapply external_call_nextblock; eauto. @@ -2235,7 +2235,7 @@ Proof. eapply match_callstate with (f := Mem.flat_inj (Mem.nextblock m0)) (cs := @nil frame) (cenv := PTree.empty Z). auto. eapply Genv.initmem_inject; eauto. - apply mcs_nil with (Mem.nextblock m0). apply match_globalenvs_init; auto. xomega. xomega. + apply mcs_nil with (Mem.nextblock m0). apply match_globalenvs_init; auto. extlia. extlia. constructor. red; auto. constructor. Qed. diff --git a/cfrontend/Csem.v b/cfrontend/Csem.v index 6d2b470f..4fa70ae2 100644 --- a/cfrontend/Csem.v +++ b/cfrontend/Csem.v @@ -839,11 +839,11 @@ Proof. unfold semantics; intros; red; simpl; intros. set (ge := globalenv p) in *. assert (DEREF: forall chunk m b ofs t v, deref_loc ge chunk m b ofs t v -> (length t <= 1)%nat). - intros. inv H0; simpl; try omega. inv H3; simpl; try omega. + intros. inv H0; simpl; try lia. inv H3; simpl; try lia. assert (ASSIGN: forall chunk m b ofs t v m', assign_loc ge chunk m b ofs v t m' -> (length t <= 1)%nat). - intros. inv H0; simpl; try omega. inv H3; simpl; try omega. + intros. inv H0; simpl; try lia. inv H3; simpl; try lia. destruct H. - inv H; simpl; try omega. inv H0; eauto; simpl; try omega. + inv H; simpl; try lia. inv H0; eauto; simpl; try lia. eapply external_call_trace_length; eauto. - inv H; simpl; try omega. eapply external_call_trace_length; eauto. + inv H; simpl; try lia. eapply external_call_trace_length; eauto. Qed. diff --git a/cfrontend/Cshmgenproof.v b/cfrontend/Cshmgenproof.v index 1ceb8e4d..6e653e01 100644 --- a/cfrontend/Cshmgenproof.v +++ b/cfrontend/Cshmgenproof.v @@ -689,32 +689,32 @@ Proof. destruct (zlt 0 sz); try discriminate. destruct (zle sz Ptrofs.max_signed); simpl in SEM; inv SEM. assert (E1: Ptrofs.signed (Ptrofs.repr sz) = sz). - { apply Ptrofs.signed_repr. generalize Ptrofs.min_signed_neg; omega. } + { apply Ptrofs.signed_repr. generalize Ptrofs.min_signed_neg; lia. } destruct Archi.ptr64 eqn:SF; inversion EQ0; clear EQ0; subst c. + assert (E: Int64.signed (Int64.repr sz) = sz). { apply Int64.signed_repr. replace Int64.max_signed with Ptrofs.max_signed. - generalize Int64.min_signed_neg; omega. + generalize Int64.min_signed_neg; lia. unfold Ptrofs.max_signed, Ptrofs.half_modulus; rewrite Ptrofs.modulus_eq64 by auto. reflexivity. } econstructor; eauto with cshm. rewrite SF, dec_eq_true. simpl. predSpec Int64.eq Int64.eq_spec (Int64.repr sz) Int64.zero. - rewrite H in E; rewrite Int64.signed_zero in E; omegaContradiction. + rewrite H in E; rewrite Int64.signed_zero in E; extlia. predSpec Int64.eq Int64.eq_spec (Int64.repr sz) Int64.mone. - rewrite H0 in E; rewrite Int64.signed_mone in E; omegaContradiction. + rewrite H0 in E; rewrite Int64.signed_mone in E; extlia. rewrite andb_false_r; simpl. unfold Vptrofs; rewrite SF. apply f_equal. apply f_equal. symmetry. auto with ptrofs. + assert (E: Int.signed (Int.repr sz) = sz). { apply Int.signed_repr. replace Int.max_signed with Ptrofs.max_signed. - generalize Int.min_signed_neg; omega. + generalize Int.min_signed_neg; lia. unfold Ptrofs.max_signed, Ptrofs.half_modulus, Ptrofs.modulus, Ptrofs.wordsize, Wordsize_Ptrofs.wordsize. rewrite SF. reflexivity. } econstructor; eauto with cshm. rewrite SF, dec_eq_true. simpl. predSpec Int.eq Int.eq_spec (Int.repr sz) Int.zero. - rewrite H in E; rewrite Int.signed_zero in E; omegaContradiction. + rewrite H in E; rewrite Int.signed_zero in E; extlia. predSpec Int.eq Int.eq_spec (Int.repr sz) Int.mone. - rewrite H0 in E; rewrite Int.signed_mone in E; omegaContradiction. + rewrite H0 in E; rewrite Int.signed_mone in E; extlia. rewrite andb_false_r; simpl. unfold Vptrofs; rewrite SF. apply f_equal. apply f_equal. symmetry. auto with ptrofs. - destruct Archi.ptr64 eqn:SF; inv EQ0; rewrite (transl_sizeof _ _ _ _ LINK EQ). @@ -772,7 +772,7 @@ Proof. assert (Int64.unsigned i = Int.unsigned (Int64.loword i)). { unfold Int64.loword. rewrite Int.unsigned_repr; auto. - comput Int.max_unsigned; omega. + comput Int.max_unsigned; lia. } split; auto. unfold Int.ltu. apply zlt_true. rewrite <- H0. tauto. Qed. @@ -786,7 +786,7 @@ Proof. assert (Int64.unsigned i = Int.unsigned (Int64.loword i)). { unfold Int64.loword. rewrite Int.unsigned_repr; auto. - comput Int.max_unsigned; omega. + comput Int.max_unsigned; lia. } unfold Int.ltu. apply zlt_true. rewrite <- H0. tauto. Qed. @@ -797,7 +797,7 @@ Lemma small_shift_amount_3: Int64.unsigned (Int64.repr (Int.unsigned i)) = Int.unsigned i. Proof. intros. apply Int.ltu_inv in H. comput (Int.unsigned Int64.iwordsize'). - apply Int64.unsigned_repr. comput Int64.max_unsigned; omega. + apply Int64.unsigned_repr. comput Int64.max_unsigned; lia. Qed. Lemma make_shl_correct: shift_constructor_correct make_shl sem_shl. diff --git a/cfrontend/Cstrategy.v b/cfrontend/Cstrategy.v index c235031f..30e5c2ae 100644 --- a/cfrontend/Cstrategy.v +++ b/cfrontend/Cstrategy.v @@ -1553,13 +1553,13 @@ Proof. exploit external_call_trace_length; eauto. destruct t1; simpl; intros. exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]]. econstructor; econstructor. left; eapply step_builtin; eauto. - omegaContradiction. + extlia. (* external calls *) inv H1. exploit external_call_trace_length; eauto. destruct t1; simpl; intros. exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]]. exists (Returnstate vres2 k m2); exists E0; right; econstructor; eauto. - omegaContradiction. + extlia. (* well-behaved traces *) red; intros. inv H; inv H0; simpl; auto. (* valof volatile *) @@ -1582,10 +1582,10 @@ Proof. exploit deref_loc_trace; eauto. destruct t; auto. destruct t; tauto. (* builtins *) exploit external_call_trace_length; eauto. - destruct t; simpl; auto. destruct t; simpl; auto. intros; omegaContradiction. + destruct t; simpl; auto. destruct t; simpl; auto. intros; extlia. (* external calls *) exploit external_call_trace_length; eauto. - destruct t; simpl; auto. destruct t; simpl; auto. intros; omegaContradiction. + destruct t; simpl; auto. destruct t; simpl; auto. intros; extlia. Qed. (** The main simulation result. *) @@ -2734,7 +2734,7 @@ Proof. cofix COEL. intros. inv H. (* cons left *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ecall a1 (exprlist_app al (Econs x al0)) ty)). eauto. eapply leftcontext_compose; eauto. constructor. auto. apply exprlist_app_leftcontext; auto. traceEq. @@ -2745,7 +2745,7 @@ Proof. eapply leftcontext_compose; eauto. repeat constructor. auto. apply exprlist_app_leftcontext; auto. eapply forever_N_star with (a2 := (esizelist al0)). - eexact R. simpl; omega. + eexact R. simpl; lia. change (Econs a1' al0) with (exprlist_app (Econs a1' Enil) al0). rewrite <- exprlist_app_assoc. eapply COEL. eauto. auto. auto. @@ -2754,42 +2754,42 @@ Proof. intros. inv H. (* field *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Efield x f0 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* valof *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Evalof x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* deref *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ederef x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* addrof *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eaddrof x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* unop *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eunop op x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* binop left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ebinop op x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* binop right *) destruct (eval_expr_to_steps _ _ _ _ _ _ _ H1 (fun x => C(Ebinop op x a2 ty)) f k) as [P [Q R]]. eapply leftcontext_compose; eauto. repeat constructor. - eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; omega. + eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; lia. eapply COE with (C := fun x => C(Ebinop op a1' x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. auto. traceEq. (* cast *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ecast x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* seqand left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eseqand x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* seqand 2 *) @@ -2802,7 +2802,7 @@ Proof. eapply COE with (C := fun x => (C (Eparen x type_bool ty))). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* seqor left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eseqor x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* seqor 2 *) @@ -2815,7 +2815,7 @@ Proof. eapply COE with (C := fun x => (C (Eparen x type_bool ty))). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* condition top *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Econdition x a2 a3 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* condition *) @@ -2828,33 +2828,33 @@ Proof. eapply COE with (C := fun x => (C (Eparen x ty ty))). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* assign left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eassign x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* assign right *) destruct (eval_expr_to_steps _ _ _ _ _ _ _ H1 (fun x => C(Eassign x a2 ty)) f k) as [P [Q R]]. eapply leftcontext_compose; eauto. repeat constructor. - eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; omega. + eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; lia. eapply COE with (C := fun x => C(Eassign a1' x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. auto. traceEq. (* assignop left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Eassignop op x a2 tyres ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* assignop right *) destruct (eval_expr_to_steps _ _ _ _ _ _ _ H1 (fun x => C(Eassignop op x a2 tyres ty)) f k) as [P [Q R]]. eapply leftcontext_compose; eauto. repeat constructor. - eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; omega. + eapply forever_N_star with (a2 := (esize a2)). eexact R. simpl; lia. eapply COE with (C := fun x => C(Eassignop op a1' x tyres ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. auto. traceEq. (* postincr *) - eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a0)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Epostincr id x ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* comma left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ecomma x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* comma right *) @@ -2865,14 +2865,14 @@ Proof. left; eapply step_comma; eauto. reflexivity. eapply COE with (C := C); eauto. traceEq. (* call left *) - eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; omega. + eapply forever_N_star with (a2 := (esize a1)). apply star_refl. simpl; lia. eapply COE with (C := fun x => C(Ecall x a2 ty)). eauto. eapply leftcontext_compose; eauto. repeat constructor. traceEq. (* call right *) destruct (eval_expr_to_steps _ _ _ _ _ _ _ H1 (fun x => C(Ecall x a2 ty)) f k) as [P [Q R]]. eapply leftcontext_compose; eauto. repeat constructor. - eapply forever_N_star with (a2 := (esizelist a2)). eexact R. simpl; omega. + eapply forever_N_star with (a2 := (esizelist a2)). eexact R. simpl; lia. eapply COEL with (al := Enil). eauto. auto. auto. auto. traceEq. (* call *) destruct (eval_expr_to_steps _ _ _ _ _ _ _ H1 (fun x => C(Ecall x rargs ty)) f k) diff --git a/cfrontend/Ctypes.v b/cfrontend/Ctypes.v index 7ce50525..0de5075c 100644 --- a/cfrontend/Ctypes.v +++ b/cfrontend/Ctypes.v @@ -350,13 +350,16 @@ Fixpoint sizeof (env: composite_env) (t: type) : Z := Lemma sizeof_pos: forall env t, sizeof env t >= 0. Proof. - induction t; simpl; try omega. - destruct i; omega. - destruct f; omega. - destruct Archi.ptr64; omega. - change 0 with (0 * Z.max 0 z) at 2. apply Zmult_ge_compat_r. auto. xomega. - destruct (env!i). apply co_sizeof_pos. omega. - destruct (env!i). apply co_sizeof_pos. omega. + induction t; simpl. +- lia. +- destruct i; lia. +- lia. +- destruct f; lia. +- destruct Archi.ptr64; lia. +- change 0 with (0 * Z.max 0 z) at 2. apply Zmult_ge_compat_r. auto. lia. +- lia. +- destruct (env!i). apply co_sizeof_pos. lia. +- destruct (env!i). apply co_sizeof_pos. lia. Qed. (** The size of a type is an integral multiple of its alignment, @@ -435,18 +438,18 @@ Lemma sizeof_struct_incr: forall env m cur, cur <= sizeof_struct env cur m. Proof. induction m as [|[id t]]; simpl; intros. -- omega. +- lia. - apply Z.le_trans with (align cur (alignof env t)). apply align_le. apply alignof_pos. apply Z.le_trans with (align cur (alignof env t) + sizeof env t). - generalize (sizeof_pos env t); omega. + generalize (sizeof_pos env t); lia. apply IHm. Qed. Lemma sizeof_union_pos: forall env m, 0 <= sizeof_union env m. Proof. - induction m as [|[id t]]; simpl; xomega. + induction m as [|[id t]]; simpl; extlia. Qed. (** ** Byte offset for a field of a structure *) @@ -490,7 +493,7 @@ Proof. apply align_le. apply alignof_pos. apply sizeof_struct_incr. exploit IHfld; eauto. intros [A B]. split; auto. eapply Z.le_trans; eauto. apply Z.le_trans with (align pos (alignof env t)). - apply align_le. apply alignof_pos. generalize (sizeof_pos env t). omega. + apply align_le. apply alignof_pos. generalize (sizeof_pos env t). lia. Qed. Lemma field_offset_in_range: @@ -637,7 +640,7 @@ Proof. destruct n; auto. right; right; right. apply Z.min_l. rewrite two_power_nat_two_p. rewrite ! Nat2Z.inj_succ. - change 8 with (two_p 3). apply two_p_monotone. omega. + change 8 with (two_p 3). apply two_p_monotone. lia. } induction ty; simpl. auto. @@ -654,7 +657,7 @@ Qed. Lemma alignof_blockcopy_pos: forall env ty, alignof_blockcopy env ty > 0. Proof. - intros. generalize (alignof_blockcopy_1248 env ty). simpl. intuition omega. + intros. generalize (alignof_blockcopy_1248 env ty). simpl. intuition lia. Qed. Lemma sizeof_alignof_blockcopy_compat: @@ -670,8 +673,8 @@ Proof. apply Z.min_case. exists (two_p (Z.of_nat n)). change 8 with (two_p 3). - rewrite <- two_p_is_exp by omega. - rewrite two_power_nat_two_p. rewrite !Nat2Z.inj_succ. f_equal. omega. + rewrite <- two_p_is_exp by lia. + rewrite two_power_nat_two_p. rewrite !Nat2Z.inj_succ. f_equal. lia. apply Z.divide_refl. } induction ty; simpl. @@ -1090,8 +1093,8 @@ Remark rank_type_members: forall ce id t m, In (id, t) m -> (rank_type ce t <= rank_members ce m)%nat. Proof. induction m; simpl; intros; intuition auto. - subst a. xomega. - xomega. + subst a. extlia. + extlia. Qed. Lemma rank_struct_member: @@ -1104,7 +1107,7 @@ Proof. intros; simpl. rewrite H0. erewrite co_consistent_rank by eauto. exploit (rank_type_members ce); eauto. - omega. + lia. Qed. Lemma rank_union_member: @@ -1117,7 +1120,7 @@ Proof. intros; simpl. rewrite H0. erewrite co_consistent_rank by eauto. exploit (rank_type_members ce); eauto. - omega. + lia. Qed. (** * Programs and compilation units *) diff --git a/cfrontend/Ctyping.v b/cfrontend/Ctyping.v index d9637b6a..83f3cfe0 100644 --- a/cfrontend/Ctyping.v +++ b/cfrontend/Ctyping.v @@ -1428,8 +1428,8 @@ Lemma pres_cast_int_int: forall sz sg n, wt_int (cast_int_int sz sg n) sz sg. Proof. intros. unfold cast_int_int. destruct sz; simpl. -- destruct sg. apply Int.sign_ext_idem; omega. apply Int.zero_ext_idem; omega. -- destruct sg. apply Int.sign_ext_idem; omega. apply Int.zero_ext_idem; omega. +- destruct sg. apply Int.sign_ext_idem; lia. apply Int.zero_ext_idem; lia. +- destruct sg. apply Int.sign_ext_idem; lia. apply Int.zero_ext_idem; lia. - auto. - destruct (Int.eq n Int.zero); auto. Qed. @@ -1616,12 +1616,12 @@ Proof. unfold access_mode, Val.load_result. remember Archi.ptr64 as ptr64. intros until v; intros AC. destruct ty; simpl in AC; try discriminate AC. - destruct i; [destruct s|destruct s|idtac|idtac]; inv AC; simpl. - destruct v; auto with ty. constructor; red. apply Int.sign_ext_idem; omega. - destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; omega. - destruct v; auto with ty. constructor; red. apply Int.sign_ext_idem; omega. - destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; omega. + destruct v; auto with ty. constructor; red. apply Int.sign_ext_idem; lia. + destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; lia. + destruct v; auto with ty. constructor; red. apply Int.sign_ext_idem; lia. + destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; lia. destruct Archi.ptr64 eqn:SF; destruct v; auto with ty. - destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; omega. + destruct v; auto with ty. constructor; red. apply Int.zero_ext_idem; lia. - inv AC. destruct Archi.ptr64 eqn:SF; destruct v; auto with ty. - destruct f; inv AC; destruct v; auto with ty. - inv AC. unfold Mptr. destruct Archi.ptr64 eqn:SF; destruct v; auto with ty. @@ -1637,16 +1637,16 @@ Proof. destruct ty; simpl in ACC; try discriminate. - destruct i; [destruct s|destruct s|idtac|idtac]; inv ACC; unfold decode_val. destruct (proj_bytes vl); auto with ty. - constructor; red. apply Int.sign_ext_idem; omega. + constructor; red. apply Int.sign_ext_idem; lia. destruct (proj_bytes vl); auto with ty. - constructor; red. apply Int.zero_ext_idem; omega. + constructor; red. apply Int.zero_ext_idem; lia. destruct (proj_bytes vl); auto with ty. - constructor; red. apply Int.sign_ext_idem; omega. + constructor; red. apply Int.sign_ext_idem; lia. destruct (proj_bytes vl); auto with ty. - constructor; red. apply Int.zero_ext_idem; omega. + constructor; red. apply Int.zero_ext_idem; lia. destruct (proj_bytes vl). auto with ty. destruct Archi.ptr64 eqn:SF; auto with ty. destruct (proj_bytes vl); auto with ty. - constructor; red. apply Int.zero_ext_idem; omega. + constructor; red. apply Int.zero_ext_idem; lia. - inv ACC. unfold decode_val. destruct (proj_bytes vl). auto with ty. destruct Archi.ptr64 eqn:SF; auto with ty. - destruct f; inv ACC; unfold decode_val; destruct (proj_bytes vl); auto with ty. diff --git a/cfrontend/Initializersproof.v b/cfrontend/Initializersproof.v index 272b929f..10ccbeff 100644 --- a/cfrontend/Initializersproof.v +++ b/cfrontend/Initializersproof.v @@ -561,7 +561,7 @@ Local Opaque sizeof. + destruct (zeq sz 0). inv TR. exists (@nil init_data); split; auto. constructor. destruct (zle 0 sz). - inv TR. econstructor; split. constructor. omega. auto. + inv TR. econstructor; split. constructor. lia. auto. discriminate. + monadInv TR. destruct (transl_init_rec_spec _ _ _ _ EQ) as (d1 & A1 & B1). @@ -672,8 +672,8 @@ Remark padding_size: forall frm to, frm <= to -> idlsize (tr_padding frm to) = to - frm. Proof. unfold tr_padding; intros. destruct (zlt frm to). - simpl. xomega. - simpl. omega. + simpl. extlia. + simpl. lia. Qed. Remark idlsize_app: @@ -681,7 +681,7 @@ Remark idlsize_app: Proof. induction d1; simpl; intros. auto. - rewrite IHd1. omega. + rewrite IHd1. lia. Qed. Remark union_field_size: @@ -690,8 +690,8 @@ Proof. induction fl as [|[i t]]; simpl; intros. - inv H. - destruct (ident_eq f i). - + inv H. xomega. - + specialize (IHfl H). xomega. + + inv H. extlia. + + specialize (IHfl H). extlia. Qed. Hypothesis ce_consistent: composite_env_consistent ge. @@ -712,16 +712,16 @@ with tr_init_struct_size: Proof. Local Opaque sizeof. - destruct 1; simpl. -+ erewrite transl_init_single_size by eauto. omega. ++ erewrite transl_init_single_size by eauto. lia. + Local Transparent sizeof. simpl. eapply tr_init_array_size; eauto. -+ replace (idlsize d) with (idlsize d + 0) by omega. ++ replace (idlsize d) with (idlsize d + 0) by lia. eapply tr_init_struct_size; eauto. simpl. unfold lookup_composite in H. destruct (ge.(genv_cenv)!id) as [co'|] eqn:?; inv H. erewrite co_consistent_sizeof by (eapply ce_consistent; eauto). unfold sizeof_composite. rewrite H0. apply align_le. destruct (co_alignof_two_p co) as [n EQ]. rewrite EQ. apply two_power_nat_pos. + rewrite idlsize_app, padding_size. - exploit tr_init_size; eauto. intros EQ; rewrite EQ. omega. + exploit tr_init_size; eauto. intros EQ; rewrite EQ. lia. simpl. unfold lookup_composite in H. destruct (ge.(genv_cenv)!id) as [co'|] eqn:?; inv H. apply Z.le_trans with (sizeof_union ge (co_members co)). eapply union_field_size; eauto. @@ -730,21 +730,21 @@ Local Opaque sizeof. destruct (co_alignof_two_p co) as [n EQ]. rewrite EQ. apply two_power_nat_pos. - destruct 1; simpl. -+ omega. ++ lia. + rewrite Z.mul_comm. assert (0 <= sizeof ge ty * sz). - { apply Zmult_gt_0_le_0_compat. omega. generalize (sizeof_pos ge ty); omega. } - xomega. + { apply Zmult_gt_0_le_0_compat. lia. generalize (sizeof_pos ge ty); lia. } + extlia. + rewrite idlsize_app. erewrite tr_init_size by eauto. erewrite tr_init_array_size by eauto. ring. - destruct 1; simpl; intros. -+ rewrite padding_size by auto. omega. ++ rewrite padding_size by auto. lia. + rewrite ! idlsize_app, padding_size. erewrite tr_init_size by eauto. - rewrite <- (tr_init_struct_size _ _ _ _ _ H0 H1). omega. + rewrite <- (tr_init_struct_size _ _ _ _ _ H0 H1). lia. unfold pos1. apply align_le. apply alignof_pos. Qed. @@ -806,7 +806,7 @@ Remark exec_init_array_length: forall m b ofs ty sz il m', exec_init_array m b ofs ty sz il m' -> sz >= 0. Proof. - induction 1; omega. + induction 1; lia. Qed. Lemma store_init_data_list_app: @@ -847,10 +847,10 @@ Local Opaque sizeof. inv H3. simpl. erewrite transl_init_single_steps by eauto. auto. - (* array *) inv H1. replace (Z.max 0 sz) with sz in H7. eauto. - assert (sz >= 0) by (eapply exec_init_array_length; eauto). xomega. + assert (sz >= 0) by (eapply exec_init_array_length; eauto). extlia. - (* struct *) inv H3. unfold lookup_composite in H7. rewrite H in H7. inv H7. - replace ofs with (ofs + 0) by omega. eauto. + replace ofs with (ofs + 0) by lia. eauto. - (* union *) inv H4. unfold lookup_composite in H9. rewrite H in H9. inv H9. rewrite H1 in H12; inv H12. eapply store_init_data_list_app. eauto. @@ -870,7 +870,7 @@ Local Opaque sizeof. inv H4. simpl in H3; inv H3. eapply store_init_data_list_app. apply store_init_data_list_padding. rewrite padding_size. - replace (ofs + pos0 + (pos2 - pos0)) with (ofs + pos2) by omega. + replace (ofs + pos0 + (pos2 - pos0)) with (ofs + pos2) by lia. eapply store_init_data_list_app. eauto. rewrite (tr_init_size _ _ _ H9). diff --git a/cfrontend/SimplExprproof.v b/cfrontend/SimplExprproof.v index 9a3f32ec..2d059ddd 100644 --- a/cfrontend/SimplExprproof.v +++ b/cfrontend/SimplExprproof.v @@ -1449,13 +1449,13 @@ Proof. (* for val *) intros [SL1 [TY1 EV1]]. subst sl. econstructor; split. - right; split. apply star_refl. destruct r; simpl; (contradiction || omega). + right; split. apply star_refl. destruct r; simpl; (contradiction || lia). econstructor; eauto. instantiate (1 := tmps). apply tr_top_val_val; auto. (* for effects *) intros SL1. subst sl. econstructor; split. - right; split. apply star_refl. destruct r; simpl; (contradiction || omega). + right; split. apply star_refl. destruct r; simpl; (contradiction || lia). econstructor; eauto. instantiate (1 := tmps). apply tr_top_base. constructor. (* for set *) @@ -1779,7 +1779,7 @@ Proof. subst; simpl Kseqlist. econstructor; split. right; split. rewrite app_ass. rewrite Kseqlist_app. eexact EXEC. - simpl. omega. + simpl. lia. constructor. (* for value *) exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]]. @@ -1788,7 +1788,7 @@ Proof. subst; simpl Kseqlist. econstructor; split. right; split. rewrite app_ass. rewrite Kseqlist_app. eexact EXEC. - simpl. omega. + simpl. lia. constructor. (* postincr *) exploit tr_top_leftcontext; eauto. clear H14. @@ -1846,7 +1846,7 @@ Proof. subst. simpl Kseqlist. econstructor; split. right; split. rewrite app_ass; rewrite Kseqlist_app. eexact EXEC. - simpl; omega. + simpl; lia. constructor. (* for value *) exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]]. @@ -1863,7 +1863,7 @@ Proof. subst sl0; simpl Kseqlist. econstructor; split. right; split. apply star_refl. simpl. apply plus_lt_compat_r. - apply (leftcontext_size _ _ _ H). simpl. omega. + apply (leftcontext_size _ _ _ H). simpl. lia. econstructor; eauto. apply S. eapply tr_expr_monotone; eauto. auto. auto. @@ -1885,7 +1885,7 @@ Proof. (* for effects *) econstructor; split. right; split. apply star_refl. simpl. apply plus_lt_compat_r. - apply (leftcontext_size _ _ _ H). simpl. omega. + apply (leftcontext_size _ _ _ H). simpl. lia. econstructor; eauto. exploit tr_simple_rvalue; eauto. simpl. intros A. subst sl1. apply S. constructor; auto. auto. auto. @@ -2015,12 +2015,12 @@ Proof. inv H6. inv H0. econstructor; split. right; split. apply push_seq. - simpl. omega. + simpl. lia. econstructor; eauto. constructor. auto. (* do 2 *) inv H7. inv H6. inv H. econstructor; split. - right; split. apply star_refl. simpl. omega. + right; split. apply star_refl. simpl. lia. econstructor; eauto. constructor. (* seq *) diff --git a/cfrontend/SimplLocalsproof.v b/cfrontend/SimplLocalsproof.v index 2dd34389..8246a748 100644 --- a/cfrontend/SimplLocalsproof.v +++ b/cfrontend/SimplLocalsproof.v @@ -173,10 +173,10 @@ Proof. eapply H1; eauto. destruct (f' b) as [[b' delta]|] eqn:?; auto. exploit H2; eauto. unfold Mem.valid_block. intros [A B]. - xomegaContradiction. + extlia. intros. destruct (f b) as [[b'' delta']|] eqn:?. eauto. exploit H2; eauto. unfold Mem.valid_block. intros [A B]. - xomegaContradiction. + extlia. Qed. (** Properties of values resulting from a cast *) @@ -606,7 +606,7 @@ Proof. generalize (alloc_variables_nextblock _ _ _ _ _ _ H0). intros A B C. subst b. split. apply Ple_refl. eapply Pos.lt_le_trans; eauto. rewrite B. apply Plt_succ. auto. - right. exploit Mem.nextblock_alloc; eauto. intros B. rewrite B in A. xomega. + right. exploit Mem.nextblock_alloc; eauto. intros B. rewrite B in A. extlia. Qed. Lemma alloc_variables_injective: @@ -622,12 +622,12 @@ Proof. repeat rewrite PTree.gsspec; intros. destruct (peq id1 id); destruct (peq id2 id). congruence. - inv H6. exploit Mem.alloc_result; eauto. exploit H2; eauto. unfold block; xomega. - inv H7. exploit Mem.alloc_result; eauto. exploit H2; eauto. unfold block; xomega. + inv H6. exploit Mem.alloc_result; eauto. exploit H2; eauto. unfold block; extlia. + inv H7. exploit Mem.alloc_result; eauto. exploit H2; eauto. unfold block; extlia. eauto. intros. rewrite PTree.gsspec in H6. destruct (peq id0 id). inv H6. - exploit Mem.alloc_result; eauto. exploit Mem.nextblock_alloc; eauto. unfold block; xomega. - exploit H2; eauto. exploit Mem.nextblock_alloc; eauto. unfold block; xomega. + exploit Mem.alloc_result; eauto. exploit Mem.nextblock_alloc; eauto. unfold block; extlia. + exploit H2; eauto. exploit Mem.nextblock_alloc; eauto. unfold block; extlia. Qed. Lemma match_alloc_variables: @@ -719,7 +719,7 @@ Proof. eapply Mem.valid_new_block; eauto. eapply Q; eauto. unfold Mem.valid_block in *. exploit Mem.nextblock_alloc. eexact A. exploit Mem.alloc_result. eexact A. - unfold block; xomega. + unfold block; extlia. split. intros. destruct (ident_eq id0 id). (* same var *) subst id0. @@ -760,7 +760,7 @@ Proof. destruct ty; try destruct i; try destruct s; try destruct f; inv H; auto; unfold Mptr; simpl; destruct Archi.ptr64; auto. } - omega. + lia. Qed. Definition env_initial_value (e: env) (m: mem) := @@ -778,7 +778,7 @@ Proof. apply IHalloc_variables. red; intros. rewrite PTree.gsspec in H2. destruct (peq id0 id). inv H2. eapply Mem.load_alloc_same'; eauto. - omega. rewrite Z.add_0_l. eapply sizeof_by_value; eauto. + lia. rewrite Z.add_0_l. eapply sizeof_by_value; eauto. apply Z.divide_0_r. eapply Mem.load_alloc_other; eauto. Qed. @@ -985,7 +985,7 @@ Proof. (* flat *) exploit alloc_variables_range. eexact A. eauto. rewrite PTree.gempty. intros [P|P]. congruence. - exploit K; eauto. unfold Mem.valid_block. xomega. + exploit K; eauto. unfold Mem.valid_block. extlia. intros [id0 [ty0 [U [V W]]]]. split; auto. destruct (ident_eq id id0). congruence. assert (b' <> b'). @@ -1032,34 +1032,34 @@ Proof. + (* special case size = 0 *) assert (bytes = nil). { exploit (Mem.loadbytes_empty m bsrc (Ptrofs.unsigned osrc) (sizeof tge ty)). - omega. congruence. } + lia. congruence. } subst. destruct (Mem.range_perm_storebytes tm bdst' (Ptrofs.unsigned (Ptrofs.add odst (Ptrofs.repr delta))) nil) as [tm' SB]. - simpl. red; intros; omegaContradiction. + simpl. red; intros; extlia. exists tm'. split. eapply assign_loc_copy; eauto. - intros; omegaContradiction. - intros; omegaContradiction. - rewrite e; right; omega. - apply Mem.loadbytes_empty. omega. + intros; extlia. + intros; extlia. + rewrite e; right; lia. + apply Mem.loadbytes_empty. lia. split. eapply Mem.storebytes_empty_inject; eauto. intros. rewrite <- H0. eapply Mem.load_storebytes_other; eauto. left. congruence. + (* general case size > 0 *) exploit Mem.loadbytes_length; eauto. intros LEN. assert (SZPOS: sizeof tge ty > 0). - { generalize (sizeof_pos tge ty); omega. } + { generalize (sizeof_pos tge ty); lia. } assert (RPSRC: Mem.range_perm m bsrc (Ptrofs.unsigned osrc) (Ptrofs.unsigned osrc + sizeof tge ty) Cur Nonempty). eapply Mem.range_perm_implies. eapply Mem.loadbytes_range_perm; eauto. auto with mem. assert (RPDST: Mem.range_perm m bdst (Ptrofs.unsigned odst) (Ptrofs.unsigned odst + sizeof tge ty) Cur Nonempty). replace (sizeof tge ty) with (Z.of_nat (List.length bytes)). eapply Mem.range_perm_implies. eapply Mem.storebytes_range_perm; eauto. auto with mem. - rewrite LEN. apply Z2Nat.id. omega. + rewrite LEN. apply Z2Nat.id. lia. assert (PSRC: Mem.perm m bsrc (Ptrofs.unsigned osrc) Cur Nonempty). - apply RPSRC. omega. + apply RPSRC. lia. assert (PDST: Mem.perm m bdst (Ptrofs.unsigned odst) Cur Nonempty). - apply RPDST. omega. + apply RPDST. lia. exploit Mem.address_inject. eauto. eexact PSRC. eauto. intros EQ1. exploit Mem.address_inject. eauto. eexact PDST. eauto. intros EQ2. exploit Mem.loadbytes_inject; eauto. intros [bytes2 [A B]]. @@ -1244,7 +1244,7 @@ Proof. destruct (Mem.range_perm_free m b lo hi) as [m1 A]; auto. rewrite A. apply IHl; auto. intros. red; intros. eapply Mem.perm_free_1; eauto. - exploit H1; eauto. intros [B|B]. auto. right; omega. + exploit H1; eauto. intros [B|B]. auto. right; lia. eapply H; eauto. Qed. @@ -1276,11 +1276,11 @@ Proof. change id' with (fst (id', (b', ty'))). apply List.in_map; auto. } assert (Mem.perm m b0 0 Max Nonempty). { apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable. - eapply PERMS; eauto. omega. auto with mem. } + eapply PERMS; eauto. lia. auto with mem. } assert (Mem.perm m b0' 0 Max Nonempty). { apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable. - eapply PERMS; eauto. omega. auto with mem. } - exploit Mem.mi_no_overlap; eauto. intros [A|A]. auto. omegaContradiction. + eapply PERMS; eauto. lia. auto with mem. } + exploit Mem.mi_no_overlap; eauto. intros [A|A]. auto. extlia. Qed. Lemma free_list_right_inject: @@ -1326,7 +1326,7 @@ Local Opaque ge tge. unfold block_of_binding in EQ; inv EQ. exploit me_mapped; eauto. eapply PTree.elements_complete; eauto. intros [b [A B]]. - change 0 with (0 + 0). replace (sizeof ge ty) with (sizeof ge ty + 0) by omega. + change 0 with (0 + 0). replace (sizeof ge ty) with (sizeof ge ty + 0) by lia. eapply Mem.range_perm_inject; eauto. eapply free_blocks_of_env_perm_2; eauto. - (* no overlap *) @@ -1343,7 +1343,7 @@ Local Opaque ge tge. intros [[id [b' ty]] [EQ IN]]. unfold block_of_binding in EQ. inv EQ. exploit me_flat; eauto. apply PTree.elements_complete; eauto. intros [P Q]. subst delta. eapply free_blocks_of_env_perm_1 with (m := m); eauto. - rewrite <- comp_env_preserved. omega. + rewrite <- comp_env_preserved. lia. Qed. (** Matching global environments *) @@ -1577,17 +1577,17 @@ Proof. induction 1; intros LOAD INCR INJ1 INJ2; econstructor; eauto. (* globalenvs *) inv H. constructor; intros; eauto. - assert (f b1 = Some (b2, delta)). rewrite <- H; symmetry; eapply INJ2; eauto. xomega. + assert (f b1 = Some (b2, delta)). rewrite <- H; symmetry; eapply INJ2; eauto. extlia. eapply IMAGE; eauto. (* call *) eapply match_envs_invariant; eauto. - intros. apply LOAD; auto. xomega. - intros. apply INJ1; auto; xomega. - intros. eapply INJ2; eauto; xomega. + intros. apply LOAD; auto. extlia. + intros. apply INJ1; auto; extlia. + intros. eapply INJ2; eauto; extlia. eapply IHmatch_cont; eauto. - intros; apply LOAD; auto. inv H0; xomega. - intros; apply INJ1. inv H0; xomega. - intros; eapply INJ2; eauto. inv H0; xomega. + intros; apply LOAD; auto. inv H0; extlia. + intros; apply INJ1. inv H0; extlia. + intros; eapply INJ2; eauto. inv H0; extlia. Qed. (** Invariance by assignment to location "above" *) @@ -1602,9 +1602,9 @@ Proof. intros. eapply match_cont_invariant; eauto. intros. rewrite <- H4. inv H0. (* scalar *) - simpl in H6. eapply Mem.load_store_other; eauto. left. unfold block; xomega. + simpl in H6. eapply Mem.load_store_other; eauto. left. unfold block; extlia. (* block copy *) - eapply Mem.load_storebytes_other; eauto. left. unfold block; xomega. + eapply Mem.load_storebytes_other; eauto. left. unfold block; extlia. Qed. (** Invariance by external calls *) @@ -1622,9 +1622,9 @@ Proof. intros. eapply Mem.load_unchanged_on; eauto. red in H2. intros. destruct (f b) as [[b' delta] | ] eqn:?. auto. destruct (f' b) as [[b' delta] | ] eqn:?; auto. - exploit H2; eauto. unfold Mem.valid_block. intros [A B]. xomegaContradiction. + exploit H2; eauto. unfold Mem.valid_block. intros [A B]. extlia. red in H2. intros. destruct (f b) as [[b'' delta''] | ] eqn:?. auto. - exploit H2; eauto. unfold Mem.valid_block. intros [A B]. xomegaContradiction. + exploit H2; eauto. unfold Mem.valid_block. intros [A B]. extlia. Qed. (** Invariance by change of bounds *) @@ -1636,7 +1636,7 @@ Lemma match_cont_incr_bounds: Ple bound bound' -> Ple tbound tbound' -> match_cont f cenv k tk m bound' tbound'. Proof. - induction 1; intros; econstructor; eauto; xomega. + induction 1; intros; econstructor; eauto; extlia. Qed. (** [match_cont] and call continuations. *) @@ -1690,7 +1690,7 @@ Proof. inv H; auto. destruct a. destruct p. destruct (Mem.free m b z0 z) as [m1|] eqn:?; try discriminate. transitivity (Mem.load chunk m1 b' 0). eauto. - eapply Mem.load_free. eauto. left. assert (Plt b' b) by eauto. unfold block; xomega. + eapply Mem.load_free. eauto. left. assert (Plt b' b) by eauto. unfold block; extlia. Qed. Lemma match_cont_free_env: @@ -1708,9 +1708,9 @@ Proof. intros. rewrite <- H7. eapply free_list_load; eauto. unfold blocks_of_env; intros. exploit list_in_map_inv; eauto. intros [[id [b1 ty]] [P Q]]. simpl in P. inv P. - exploit me_range; eauto. eapply PTree.elements_complete; eauto. xomega. - rewrite (free_list_nextblock _ _ _ H3). inv H; xomega. - rewrite (free_list_nextblock _ _ _ H4). inv H; xomega. + exploit me_range; eauto. eapply PTree.elements_complete; eauto. extlia. + rewrite (free_list_nextblock _ _ _ H3). inv H; extlia. + rewrite (free_list_nextblock _ _ _ H4). inv H; extlia. Qed. (** Matching of global environments *) @@ -2018,7 +2018,7 @@ Proof. eapply step_Sset_debug. eauto. rewrite typeof_simpl_expr. eauto. econstructor; eauto with compat. eapply match_envs_assign_lifted; eauto. eapply cast_val_is_casted; eauto. - eapply match_cont_assign_loc; eauto. exploit me_range; eauto. xomega. + eapply match_cont_assign_loc; eauto. exploit me_range; eauto. extlia. inv MV; try congruence. inv H2; try congruence. unfold Mem.storev in H3. eapply Mem.store_unmapped_inject; eauto. congruence. erewrite assign_loc_nextblock; eauto. @@ -2068,7 +2068,7 @@ Proof. eapply match_envs_set_opttemp; eauto. eapply match_envs_extcall; eauto. eapply match_cont_extcall; eauto. - inv MENV; xomega. inv MENV; xomega. + inv MENV; extlia. inv MENV; extlia. eapply Ple_trans; eauto. eapply external_call_nextblock; eauto. eapply Ple_trans; eauto. eapply external_call_nextblock; eauto. @@ -2212,11 +2212,11 @@ Proof. eapply bind_parameters_load; eauto. intros. exploit alloc_variables_range. eexact H1. eauto. unfold empty_env. rewrite PTree.gempty. intros [?|?]. congruence. - red; intros; subst b'. xomega. + red; intros; subst b'. extlia. eapply alloc_variables_load; eauto. apply compat_cenv_for. - rewrite (bind_parameters_nextblock _ _ _ _ _ _ H2). xomega. - rewrite T; xomega. + rewrite (bind_parameters_nextblock _ _ _ _ _ _ H2). extlia. + rewrite T; extlia. (* external function *) monadInv TRFD. inv FUNTY. @@ -2227,7 +2227,7 @@ Proof. apply plus_one. econstructor; eauto. eapply external_call_symbols_preserved; eauto. apply senv_preserved. econstructor; eauto. intros. apply match_cont_incr_bounds with (Mem.nextblock m) (Mem.nextblock tm). - eapply match_cont_extcall; eauto. xomega. xomega. + eapply match_cont_extcall; eauto. extlia. extlia. eapply external_call_nextblock; eauto. eapply external_call_nextblock; eauto. @@ -2262,7 +2262,7 @@ Proof. eapply Genv.find_symbol_not_fresh; eauto. eapply Genv.find_funct_ptr_not_fresh; eauto. eapply Genv.find_var_info_not_fresh; eauto. - xomega. xomega. + extlia. extlia. eapply Genv.initmem_inject; eauto. constructor. Qed. diff --git a/common/AST.v b/common/AST.v index 4f954c5c..7ccbb6cc 100644 --- a/common/AST.v +++ b/common/AST.v @@ -61,7 +61,7 @@ Definition typesize (ty: typ) : Z := end. Lemma typesize_pos: forall ty, typesize ty > 0. -Proof. destruct ty; simpl; omega. Qed. +Proof. destruct ty; simpl; lia. Qed. Lemma typesize_Tptr: typesize Tptr = if Archi.ptr64 then 8 else 4. Proof. unfold Tptr; destruct Archi.ptr64; auto. Qed. @@ -265,13 +265,13 @@ Fixpoint init_data_list_size (il: list init_data) {struct il} : Z := Lemma init_data_size_pos: forall i, init_data_size i >= 0. Proof. - destruct i; simpl; try xomega. destruct Archi.ptr64; omega. + destruct i; simpl; try extlia. destruct Archi.ptr64; lia. Qed. Lemma init_data_list_size_pos: forall il, init_data_list_size il >= 0. Proof. - induction il; simpl. omega. generalize (init_data_size_pos a); omega. + induction il; simpl. lia. generalize (init_data_size_pos a); lia. Qed. (** Information attached to global variables. *) diff --git a/common/Events.v b/common/Events.v index 28bb992a..ee2d529d 100644 --- a/common/Events.v +++ b/common/Events.v @@ -798,7 +798,7 @@ Proof. exists f; exists v'; exists m1'; intuition. constructor; auto. red; intros. congruence. (* trace length *) -- inv H; inv H0; simpl; omega. +- inv H; inv H0; simpl; lia. (* receptive *) - inv H. exploit volatile_load_receptive; eauto. intros [v2 A]. exists v2; exists m1; constructor; auto. @@ -925,7 +925,7 @@ Proof. eelim H3; eauto. exploit Mem.store_valid_access_3. eexact H0. intros [X Y]. apply Mem.perm_cur_max. apply Mem.perm_implies with Writable; auto with mem. - apply X. omega. + apply X. lia. Qed. Lemma volatile_store_receptive: @@ -960,7 +960,7 @@ Proof. exploit volatile_store_inject; eauto. intros [m2' [A [B [C D]]]]. exists f; exists Vundef; exists m2'; intuition. constructor; auto. red; intros; congruence. (* trace length *) -- inv H; inv H0; simpl; omega. +- inv H; inv H0; simpl; lia. (* receptive *) - assert (t1 = t2). inv H. eapply volatile_store_receptive; eauto. subst t2; exists vres1; exists m1; auto. @@ -1042,7 +1042,7 @@ Proof. subst b1. rewrite C in H2. inv H2. eauto with mem. rewrite D in H2 by auto. congruence. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - assert (t1 = t2). inv H; inv H0; auto. subst t2. exists vres1; exists m1; auto. @@ -1122,21 +1122,21 @@ Proof. exploit Mem.address_inject; eauto. apply Mem.perm_implies with Freeable; auto with mem. apply P. instantiate (1 := lo). - generalize (size_chunk_pos Mptr); omega. + generalize (size_chunk_pos Mptr); lia. intro EQ. exploit Mem.free_parallel_inject; eauto. intros (m2' & C & D). exists f, Vundef, m2'; split. apply extcall_free_sem_ptr with (sz := sz) (m' := m2'). - rewrite EQ. rewrite <- A. f_equal. omega. + rewrite EQ. rewrite <- A. f_equal. lia. auto. auto. - rewrite ! EQ. rewrite <- C. f_equal; omega. + rewrite ! EQ. rewrite <- C. f_equal; lia. split. auto. split. auto. split. eapply Mem.free_unchanged_on; eauto. unfold loc_unmapped. intros; congruence. split. eapply Mem.free_unchanged_on; eauto. unfold loc_out_of_reach. intros. red; intros. eelim H2; eauto. apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable; auto with mem. - apply P. omega. + apply P. lia. split. auto. red; intros. congruence. + inv H2. inv H6. replace v' with Vnullptr. @@ -1145,7 +1145,7 @@ Proof. red; intros; congruence. unfold Vnullptr in *; destruct Archi.ptr64; inv H4; auto. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - assert (t1 = t2) by (inv H; inv H0; auto). subst t2. exists vres1; exists m1; auto. @@ -1217,23 +1217,23 @@ Proof. destruct (zeq sz 0). + (* special case sz = 0 *) assert (bytes = nil). - { exploit (Mem.loadbytes_empty m1 bsrc (Ptrofs.unsigned osrc) sz). omega. congruence. } + { exploit (Mem.loadbytes_empty m1 bsrc (Ptrofs.unsigned osrc) sz). lia. congruence. } subst. destruct (Mem.range_perm_storebytes m1' b0 (Ptrofs.unsigned (Ptrofs.add odst (Ptrofs.repr delta0))) nil) as [m2' SB]. - simpl. red; intros; omegaContradiction. + simpl. red; intros; extlia. exists f, Vundef, m2'. split. econstructor; eauto. - intros; omegaContradiction. - intros; omegaContradiction. - right; omega. - apply Mem.loadbytes_empty. omega. + intros; extlia. + intros; extlia. + right; lia. + apply Mem.loadbytes_empty. lia. split. auto. split. eapply Mem.storebytes_empty_inject; eauto. split. eapply Mem.storebytes_unchanged_on; eauto. unfold loc_unmapped; intros. congruence. split. eapply Mem.storebytes_unchanged_on; eauto. - simpl; intros; omegaContradiction. + simpl; intros; extlia. split. apply inject_incr_refl. red; intros; congruence. + (* general case sz > 0 *) @@ -1243,11 +1243,11 @@ Proof. assert (RPDST: Mem.range_perm m1 bdst (Ptrofs.unsigned odst) (Ptrofs.unsigned odst + sz) Cur Nonempty). replace sz with (Z.of_nat (length bytes)). eapply Mem.range_perm_implies. eapply Mem.storebytes_range_perm; eauto. auto with mem. - rewrite LEN. apply Z2Nat.id. omega. + rewrite LEN. apply Z2Nat.id. lia. assert (PSRC: Mem.perm m1 bsrc (Ptrofs.unsigned osrc) Cur Nonempty). - apply RPSRC. omega. + apply RPSRC. lia. assert (PDST: Mem.perm m1 bdst (Ptrofs.unsigned odst) Cur Nonempty). - apply RPDST. omega. + apply RPDST. lia. exploit Mem.address_inject. eauto. eexact PSRC. eauto. intros EQ1. exploit Mem.address_inject. eauto. eexact PDST. eauto. intros EQ2. exploit Mem.loadbytes_inject; eauto. intros [bytes2 [A B]]. @@ -1258,7 +1258,7 @@ Proof. intros; eapply Mem.aligned_area_inject with (m := m1); eauto. eapply Mem.disjoint_or_equal_inject with (m := m1); eauto. apply Mem.range_perm_max with Cur; auto. - apply Mem.range_perm_max with Cur; auto. omega. + apply Mem.range_perm_max with Cur; auto. lia. split. constructor. split. auto. split. eapply Mem.storebytes_unchanged_on; eauto. unfold loc_unmapped; intros. @@ -1268,11 +1268,11 @@ Proof. apply Mem.perm_cur_max. apply Mem.perm_implies with Writable; auto with mem. eapply Mem.storebytes_range_perm; eauto. erewrite list_forall2_length; eauto. - omega. + lia. split. apply inject_incr_refl. red; intros; congruence. - (* trace length *) - intros; inv H. simpl; omega. + intros; inv H. simpl; lia. - (* receptive *) intros. assert (t1 = t2). inv H; inv H0; auto. subst t2. @@ -1318,7 +1318,7 @@ Proof. eapply eventval_list_match_inject; eauto. red; intros; congruence. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - assert (t1 = t2). inv H; inv H0; auto. exists vres1; exists m1; congruence. @@ -1363,7 +1363,7 @@ Proof. eapply eventval_match_inject; eauto. red; intros; congruence. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - assert (t1 = t2). inv H; inv H0; auto. subst t2. exists vres1; exists m1; auto. @@ -1404,7 +1404,7 @@ Proof. econstructor; eauto. red; intros; congruence. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - inv H; inv H0. exists Vundef, m1; constructor. (* determ *) @@ -1458,7 +1458,7 @@ Proof. constructor; auto. red; intros; congruence. (* trace length *) -- inv H; simpl; omega. +- inv H; simpl; lia. (* receptive *) - inv H; inv H0. exists vres1, m1; constructor; auto. (* determ *) @@ -1582,7 +1582,7 @@ Proof. intros. destruct (plt (Mem.nextblock m2) (Mem.nextblock m1)). exploit external_call_valid_block; eauto. intros. eelim Plt_strict; eauto. - unfold Plt, Ple in *; zify; omega. + unfold Plt, Ple in *; zify; lia. Qed. (** Special case of [external_call_mem_inject_gen] (for backward compatibility) *) diff --git a/common/Globalenvs.v b/common/Globalenvs.v index d37fbd46..40496044 100644 --- a/common/Globalenvs.v +++ b/common/Globalenvs.v @@ -55,7 +55,7 @@ Function store_zeros (m: mem) (b: block) (p: Z) (n: Z) {wf (Zwf 0) n}: option me | None => None end. Proof. - intros. red. omega. + intros. red. lia. apply Zwf_well_founded. Qed. @@ -849,8 +849,8 @@ Proof. intros until n. functional induction (store_zeros m b p n); intros. - inv H; apply Mem.unchanged_on_refl. - apply Mem.unchanged_on_trans with m'. -+ eapply Mem.store_unchanged_on; eauto. simpl. intros. apply H0. omega. -+ apply IHo; auto. intros; apply H0; omega. ++ eapply Mem.store_unchanged_on; eauto. simpl. intros. apply H0. lia. ++ apply IHo; auto. intros; apply H0; lia. - discriminate. Qed. @@ -879,7 +879,7 @@ Proof. - destruct (store_init_data m b p a) as [m1|] eqn:?; try congruence. apply Mem.unchanged_on_trans with m1. eapply store_init_data_unchanged; eauto. intros; apply H0; tauto. - eapply IHil; eauto. intros; apply H0. generalize (init_data_size_pos a); omega. + eapply IHil; eauto. intros; apply H0. generalize (init_data_size_pos a); lia. Qed. (** Properties related to [loadbytes] *) @@ -895,24 +895,24 @@ Lemma store_zeros_loadbytes: readbytes_as_zero m' b p n. Proof. intros until n; functional induction (store_zeros m b p n); red; intros. -- destruct n0. simpl. apply Mem.loadbytes_empty. omega. - rewrite Nat2Z.inj_succ in H1. omegaContradiction. +- destruct n0. simpl. apply Mem.loadbytes_empty. lia. + rewrite Nat2Z.inj_succ in H1. extlia. - destruct (zeq p0 p). - + subst p0. destruct n0. simpl. apply Mem.loadbytes_empty. omega. + + subst p0. destruct n0. simpl. apply Mem.loadbytes_empty. lia. rewrite Nat2Z.inj_succ in H1. rewrite Nat2Z.inj_succ. - replace (Z.succ (Z.of_nat n0)) with (1 + Z.of_nat n0) by omega. + replace (Z.succ (Z.of_nat n0)) with (1 + Z.of_nat n0) by lia. change (list_repeat (S n0) (Byte Byte.zero)) with ((Byte Byte.zero :: nil) ++ list_repeat n0 (Byte Byte.zero)). apply Mem.loadbytes_concat. eapply Mem.loadbytes_unchanged_on with (P := fun b1 ofs1 => ofs1 = p). - eapply store_zeros_unchanged; eauto. intros; omega. - intros; omega. + eapply store_zeros_unchanged; eauto. intros; lia. + intros; lia. replace (Byte Byte.zero :: nil) with (encode_val Mint8unsigned Vzero). change 1 with (size_chunk Mint8unsigned). eapply Mem.loadbytes_store_same; eauto. unfold encode_val; unfold encode_int; unfold rev_if_be; destruct Archi.big_endian; reflexivity. - eapply IHo; eauto. omega. omega. omega. omega. - + eapply IHo; eauto. omega. omega. + eapply IHo; eauto. lia. lia. lia. lia. + + eapply IHo; eauto. lia. lia. - discriminate. Qed. @@ -947,8 +947,8 @@ Proof. intros; destruct i; simpl in H; try apply (Mem.loadbytes_store_same _ _ _ _ _ _ H). - inv H. simpl. assert (EQ: Z.of_nat (Z.to_nat z) = Z.max z 0). - { destruct (zle 0 z). rewrite Z2Nat.id; xomega. destruct z; try discriminate. simpl. xomega. } - rewrite <- EQ. apply H0. omega. simpl. omega. + { destruct (zle 0 z). rewrite Z2Nat.id; extlia. destruct z; try discriminate. simpl. extlia. } + rewrite <- EQ. apply H0. lia. simpl. lia. - rewrite init_data_size_addrof. simpl. destruct (find_symbol ge i) as [b'|]; try discriminate. rewrite (Mem.loadbytes_store_same _ _ _ _ _ _ H). @@ -968,23 +968,23 @@ Lemma store_init_data_list_loadbytes: Mem.loadbytes m' b p (init_data_list_size il) = Some (bytes_of_init_data_list il). Proof. induction il as [ | i1 il]; simpl; intros. -- apply Mem.loadbytes_empty. omega. +- apply Mem.loadbytes_empty. lia. - generalize (init_data_size_pos i1) (init_data_list_size_pos il); intros P1 PL. destruct (store_init_data m b p i1) as [m1|] eqn:S; try discriminate. apply Mem.loadbytes_concat. eapply Mem.loadbytes_unchanged_on with (P := fun b1 ofs1 => ofs1 < p + init_data_size i1). eapply store_init_data_list_unchanged; eauto. - intros; omega. - intros; omega. + intros; lia. + intros; lia. eapply store_init_data_loadbytes; eauto. - red; intros; apply H0. omega. omega. + red; intros; apply H0. lia. lia. apply IHil with m1; auto. red; intros. eapply Mem.loadbytes_unchanged_on with (P := fun b1 ofs1 => p + init_data_size i1 <= ofs1). eapply store_init_data_unchanged; eauto. - intros; omega. - intros; omega. - apply H0. omega. omega. + intros; lia. + intros; lia. + apply H0. lia. lia. auto. auto. Qed. @@ -1011,7 +1011,7 @@ Remark read_as_zero_unchanged: read_as_zero m' b ofs len. Proof. intros; red; intros. eapply Mem.load_unchanged_on; eauto. - intros; apply H1. omega. + intros; apply H1. lia. Qed. Lemma store_zeros_read_as_zero: @@ -1068,7 +1068,7 @@ Proof. { intros. eapply Mem.load_unchanged_on with (P := fun b' ofs' => ofs' < p + size_chunk chunk). - eapply store_init_data_list_unchanged; eauto. intros; omega. + eapply store_init_data_list_unchanged; eauto. intros; lia. intros; tauto. eapply Mem.load_store_same; eauto. } @@ -1078,10 +1078,10 @@ Proof. exploit IHil; eauto. set (P := fun (b': block) ofs' => p + init_data_size a <= ofs'). apply read_as_zero_unchanged with (m := m) (P := P). - red; intros; apply H0; auto. generalize (init_data_size_pos a); omega. omega. + red; intros; apply H0; auto. generalize (init_data_size_pos a); lia. lia. eapply store_init_data_unchanged with (P := P); eauto. - intros; unfold P. omega. - intros; unfold P. omega. + intros; unfold P. lia. + intros; unfold P. lia. intro D. destruct a; simpl in Heqo. + split; auto. eapply (A Mint8unsigned (Vint i)); eauto. @@ -1093,10 +1093,10 @@ Proof. + split; auto. set (P := fun (b': block) ofs' => ofs' < p + init_data_size (Init_space z)). inv Heqo. apply read_as_zero_unchanged with (m := m1) (P := P). - red; intros. apply H0; auto. simpl. generalize (init_data_list_size_pos il); xomega. + red; intros. apply H0; auto. simpl. generalize (init_data_list_size_pos il); extlia. eapply store_init_data_list_unchanged; eauto. - intros; unfold P. omega. - intros; unfold P. simpl; xomega. + intros; unfold P. lia. + intros; unfold P. simpl; extlia. + rewrite init_data_size_addrof in *. split; auto. destruct (find_symbol ge i); try congruence. @@ -1195,11 +1195,11 @@ Proof. * destruct (Mem.alloc m 0 1) as [m1 b] eqn:ALLOC. exploit Mem.alloc_result; eauto. intros RES. rewrite H, <- RES. split. - eapply Mem.perm_drop_1; eauto. omega. + eapply Mem.perm_drop_1; eauto. lia. intros. assert (0 <= ofs < 1). { eapply Mem.perm_alloc_3; eauto. eapply Mem.perm_drop_4; eauto. } exploit Mem.perm_drop_2; eauto. intros ORD. - split. omega. inv ORD; auto. + split. lia. inv ORD; auto. * set (init := gvar_init v) in *. set (sz := init_data_list_size init) in *. destruct (Mem.alloc m 0 sz) as [m1 b] eqn:?. @@ -1442,7 +1442,7 @@ Proof. exploit alloc_global_neutral; eauto. assert (Ple (Pos.succ (Mem.nextblock m)) (Mem.nextblock m')). { rewrite EQ. apply advance_next_le. } - unfold Plt, Ple in *; zify; omega. + unfold Plt, Ple in *; zify; lia. Qed. End INITMEM_INJ. @@ -1563,9 +1563,9 @@ Lemma store_zeros_exists: Proof. intros until n. functional induction (store_zeros m b p n); intros PERM. - exists m; auto. -- apply IHo. red; intros. eapply Mem.perm_store_1; eauto. apply PERM. omega. +- apply IHo. red; intros. eapply Mem.perm_store_1; eauto. apply PERM. lia. - destruct (Mem.valid_access_store m Mint8unsigned b p Vzero) as (m' & STORE). - split. red; intros. apply Mem.perm_cur. apply PERM. simpl in H. omega. + split. red; intros. apply Mem.perm_cur. apply PERM. simpl in H. lia. simpl. apply Z.divide_1_l. congruence. Qed. @@ -1603,10 +1603,10 @@ Proof. - exists m; auto. - destruct H0. destruct (@store_init_data_exists m b p i1) as (m1 & S1); eauto. - red; intros. apply H. generalize (init_data_list_size_pos il); omega. + red; intros. apply H. generalize (init_data_list_size_pos il); lia. rewrite S1. apply IHil; eauto. - red; intros. erewrite <- store_init_data_perm by eauto. apply H. generalize (init_data_size_pos i1); omega. + red; intros. erewrite <- store_init_data_perm by eauto. apply H. generalize (init_data_size_pos i1); lia. Qed. Lemma alloc_global_exists: diff --git a/common/Linking.v b/common/Linking.v index ec828ea4..a5cf0a4a 100644 --- a/common/Linking.v +++ b/common/Linking.v @@ -123,7 +123,7 @@ Defined. Next Obligation. inv H; inv H0; constructor; auto. congruence. - simpl. generalize (init_data_list_size_pos z). xomega. + simpl. generalize (init_data_list_size_pos z). extlia. Defined. Next Obligation. revert H; unfold link_varinit. diff --git a/common/Memdata.v b/common/Memdata.v index f3016efe..05a3d4ed 100644 --- a/common/Memdata.v +++ b/common/Memdata.v @@ -47,7 +47,7 @@ Definition size_chunk (chunk: memory_chunk) : Z := Lemma size_chunk_pos: forall chunk, size_chunk chunk > 0. Proof. - intros. destruct chunk; simpl; omega. + intros. destruct chunk; simpl; lia. Qed. Definition size_chunk_nat (chunk: memory_chunk) : nat := @@ -65,7 +65,7 @@ Proof. intros. generalize (size_chunk_pos chunk). rewrite size_chunk_conv. destruct (size_chunk_nat chunk). - simpl; intros; omegaContradiction. + simpl; intros; extlia. intros; exists n; auto. Qed. @@ -101,7 +101,7 @@ Definition align_chunk (chunk: memory_chunk) : Z := Lemma align_chunk_pos: forall chunk, align_chunk chunk > 0. Proof. - intro. destruct chunk; simpl; omega. + intro. destruct chunk; simpl; lia. Qed. Lemma align_chunk_Mptr: align_chunk Mptr = if Archi.ptr64 then 8 else 4. @@ -120,7 +120,7 @@ Lemma align_le_divides: align_chunk chunk1 <= align_chunk chunk2 -> (align_chunk chunk1 | align_chunk chunk2). Proof. intros. destruct chunk1; destruct chunk2; simpl in *; - solve [ omegaContradiction + solve [ extlia | apply Z.divide_refl | exists 2; reflexivity | exists 4; reflexivity @@ -216,12 +216,12 @@ Proof. simpl. rewrite Zmod_1_r. auto. Opaque Byte.wordsize. rewrite Nat2Z.inj_succ. simpl. - replace (Z.succ (Z.of_nat n) * 8) with (Z.of_nat n * 8 + 8) by omega. - rewrite two_p_is_exp; try omega. + replace (Z.succ (Z.of_nat n) * 8) with (Z.of_nat n * 8 + 8) by lia. + rewrite two_p_is_exp; try lia. rewrite Zmod_recombine. rewrite IHn. rewrite Z.add_comm. change (Byte.unsigned (Byte.repr x)) with (Byte.Z_mod_modulus x). rewrite Byte.Z_mod_modulus_eq. reflexivity. - apply two_p_gt_ZERO. omega. apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. apply two_p_gt_ZERO. lia. Qed. Lemma rev_if_be_involutive: @@ -280,15 +280,15 @@ Proof. intros; simpl; auto. intros until y. rewrite Nat2Z.inj_succ. - replace (Z.succ (Z.of_nat n) * 8) with (Z.of_nat n * 8 + 8) by omega. - rewrite two_p_is_exp; try omega. + replace (Z.succ (Z.of_nat n) * 8) with (Z.of_nat n * 8 + 8) by lia. + rewrite two_p_is_exp; try lia. intro EQM. simpl; decEq. apply Byte.eqm_samerepr. red. eapply eqmod_divides; eauto. apply Z.divide_factor_r. apply IHn. destruct EQM as [k EQ]. exists k. rewrite EQ. - rewrite <- Z_div_plus_full_l. decEq. change (two_p 8) with 256. ring. omega. + rewrite <- Z_div_plus_full_l. decEq. change (two_p 8) with 256. ring. lia. Qed. Lemma encode_int_8_mod: @@ -517,9 +517,9 @@ Ltac solve_decode_encode_val_general := | |- context [ Int.repr(decode_int (encode_int 2 (Int.unsigned _))) ] => rewrite decode_encode_int_2 | |- context [ Int.repr(decode_int (encode_int 4 (Int.unsigned _))) ] => rewrite decode_encode_int_4 | |- context [ Int64.repr(decode_int (encode_int 8 (Int64.unsigned _))) ] => rewrite decode_encode_int_8 - | |- Vint (Int.sign_ext _ (Int.sign_ext _ _)) = Vint _ => f_equal; apply Int.sign_ext_idem; omega - | |- Vint (Int.zero_ext _ (Int.zero_ext _ _)) = Vint _ => f_equal; apply Int.zero_ext_idem; omega - | |- Vint (Int.sign_ext _ (Int.zero_ext _ _)) = Vint _ => f_equal; apply Int.sign_ext_zero_ext; omega + | |- Vint (Int.sign_ext _ (Int.sign_ext _ _)) = Vint _ => f_equal; apply Int.sign_ext_idem; lia + | |- Vint (Int.zero_ext _ (Int.zero_ext _ _)) = Vint _ => f_equal; apply Int.zero_ext_idem; lia + | |- Vint (Int.sign_ext _ (Int.zero_ext _ _)) = Vint _ => f_equal; apply Int.sign_ext_zero_ext; lia end. Lemma decode_encode_val_general: @@ -543,7 +543,7 @@ Lemma decode_encode_val_similar: v2 = Val.load_result chunk2 v1. Proof. intros until v2; intros TY SZ DE. - destruct chunk1; destruct chunk2; simpl in TY; try discriminate; simpl in SZ; try omegaContradiction; + destruct chunk1; destruct chunk2; simpl in TY; try discriminate; simpl in SZ; try extlia; destruct v1; auto. Qed. @@ -553,7 +553,7 @@ Lemma decode_val_rettype: Proof. intros. unfold decode_val. destruct (proj_bytes cl). -- destruct chunk; simpl; rewrite ? Int.sign_ext_idem, ? Int.zero_ext_idem by omega; auto. +- destruct chunk; simpl; rewrite ? Int.sign_ext_idem, ? Int.zero_ext_idem by lia; auto. - Local Opaque Val.load_result. destruct chunk; simpl; (exact I || apply Val.load_result_type || destruct Archi.ptr64; (exact I || apply Val.load_result_type)). @@ -653,7 +653,7 @@ Proof. exists j, mv = Fragment v q j /\ S j <> size_quantity_nat q). { induction n; simpl; intros. contradiction. destruct H0. - exists n; split; auto. omega. apply IHn; auto. omega. + exists n; split; auto. lia. apply IHn; auto. lia. } assert (B: forall q, q = quantity_chunk chunk -> @@ -663,7 +663,7 @@ Proof. Local Transparent inj_value. intros. unfold inj_value. destruct (size_quantity_nat_pos q) as [sz' EQ']. rewrite EQ'. simpl. constructor; auto. - intros; eapply A; eauto. omega. + intros; eapply A; eauto. lia. } assert (C: forall bl, match v with Vint _ => True | Vlong _ => True | Vfloat _ => True | Vsingle _ => True | _ => False end -> @@ -719,8 +719,8 @@ Proof. induction n; destruct mvs; simpl; intros; try discriminate. contradiction. destruct m; try discriminate. InvBooleans. apply beq_nat_true in H4. subst. - destruct H0. subst mv. exists n0; split; auto. omega. - eapply IHn; eauto. omega. + destruct H0. subst mv. exists n0; split; auto. lia. + eapply IHn; eauto. lia. } assert (U: forall mvs, shape_decoding chunk mvs (Val.load_result chunk Vundef)). { @@ -740,7 +740,7 @@ Proof. simpl. apply beq_nat_true in EQN. subst n q0. constructor. auto. destruct H0 as [E|[E|[E|E]]]; subst chunk; destruct q; auto || discriminate. congruence. - intros. eapply B; eauto. omega. + intros. eapply B; eauto. lia. } unfold decode_val. destruct (proj_bytes (mv1 :: mvl)) as [bl|] eqn:PB. @@ -955,22 +955,22 @@ Proof. induction l1; simpl int_of_bytes; intros. simpl. ring. simpl length. rewrite Nat2Z.inj_succ. - replace (Z.succ (Z.of_nat (length l1)) * 8) with (Z.of_nat (length l1) * 8 + 8) by omega. + replace (Z.succ (Z.of_nat (length l1)) * 8) with (Z.of_nat (length l1) * 8 + 8) by lia. rewrite two_p_is_exp. change (two_p 8) with 256. rewrite IHl1. ring. - omega. omega. + lia. lia. Qed. Lemma int_of_bytes_range: forall l, 0 <= int_of_bytes l < two_p (Z.of_nat (length l) * 8). Proof. induction l; intros. - simpl. omega. + simpl. lia. simpl length. rewrite Nat2Z.inj_succ. - replace (Z.succ (Z.of_nat (length l)) * 8) with (Z.of_nat (length l) * 8 + 8) by omega. + replace (Z.succ (Z.of_nat (length l)) * 8) with (Z.of_nat (length l) * 8 + 8) by lia. rewrite two_p_is_exp. change (two_p 8) with 256. simpl int_of_bytes. generalize (Byte.unsigned_range a). - change Byte.modulus with 256. omega. - omega. omega. + change Byte.modulus with 256. lia. + lia. lia. Qed. Lemma length_proj_bytes: @@ -1014,7 +1014,7 @@ Proof. intros. apply Int.unsigned_repr. generalize (int_of_bytes_range l). rewrite H2. change (two_p (Z.of_nat 4 * 8)) with (Int.max_unsigned + 1). - omega. + lia. apply Val.lessdef_same. unfold decode_int, rev_if_be. destruct Archi.big_endian; rewrite B1; rewrite B2. + rewrite <- (rev_length b1) in L1. @@ -1036,18 +1036,18 @@ Lemma bytes_of_int_append: bytes_of_int n1 x1 ++ bytes_of_int n2 x2. Proof. induction n1; intros. -- simpl in *. f_equal. omega. +- simpl in *. f_equal. lia. - assert (E: two_p (Z.of_nat (S n1) * 8) = two_p (Z.of_nat n1 * 8) * 256). { rewrite Nat2Z.inj_succ. change 256 with (two_p 8). rewrite <- two_p_is_exp. - f_equal. omega. omega. omega. + f_equal. lia. lia. lia. } rewrite E in *. simpl. f_equal. apply Byte.eqm_samerepr. exists (x2 * two_p (Z.of_nat n1 * 8)). change Byte.modulus with 256. ring. rewrite Z.mul_assoc. rewrite Z_div_plus. apply IHn1. - apply Zdiv_interval_1. omega. apply two_p_gt_ZERO; omega. omega. - assumption. omega. + apply Zdiv_interval_1. lia. apply two_p_gt_ZERO; lia. lia. + assumption. lia. Qed. Lemma bytes_of_int64: diff --git a/common/Memory.v b/common/Memory.v index 9f9934c2..641a9243 100644 --- a/common/Memory.v +++ b/common/Memory.v @@ -208,11 +208,11 @@ Proof. induction lo using (well_founded_induction_type (Zwf_up_well_founded hi)). destruct (zlt lo hi). destruct (perm_dec m b lo k p). - destruct (H (lo + 1)). red. omega. - left; red; intros. destruct (zeq lo ofs). congruence. apply r. omega. - right; red; intros. elim n. red; intros; apply H0; omega. - right; red; intros. elim n. apply H0. omega. - left; red; intros. omegaContradiction. + destruct (H (lo + 1)). red. lia. + left; red; intros. destruct (zeq lo ofs). congruence. apply r. lia. + right; red; intros. elim n. red; intros; apply H0; lia. + right; red; intros. elim n. apply H0. lia. + left; red; intros. extlia. Defined. (** [valid_access m chunk b ofs p] holds if a memory access @@ -253,7 +253,7 @@ Theorem valid_access_valid_block: Proof. intros. destruct H. assert (perm m b ofs Cur Nonempty). - apply H. generalize (size_chunk_pos chunk). omega. + apply H. generalize (size_chunk_pos chunk). lia. eauto with mem. Qed. @@ -264,7 +264,7 @@ Lemma valid_access_perm: valid_access m chunk b ofs p -> perm m b ofs k p. Proof. - intros. destruct H. apply perm_cur. apply H. generalize (size_chunk_pos chunk). omega. + intros. destruct H. apply perm_cur. apply H. generalize (size_chunk_pos chunk). lia. Qed. Lemma valid_access_compat: @@ -310,9 +310,9 @@ Theorem valid_pointer_valid_access: Proof. intros. rewrite valid_pointer_nonempty_perm. split; intros. - split. simpl; red; intros. replace ofs0 with ofs by omega. auto. + split. simpl; red; intros. replace ofs0 with ofs by lia. auto. simpl. apply Z.divide_1_l. - destruct H. apply H. simpl. omega. + destruct H. apply H. simpl. lia. Qed. (** C allows pointers one past the last element of an array. These are not @@ -482,8 +482,8 @@ Proof. auto. simpl length in H. rewrite Nat2Z.inj_succ in H. transitivity (ZMap.get q (ZMap.set p a c)). - apply IHvl. intros. apply H. omega. - apply ZMap.gso. apply not_eq_sym. apply H. omega. + apply IHvl. intros. apply H. lia. + apply ZMap.gso. apply not_eq_sym. apply H. lia. Qed. Remark setN_outside: @@ -492,7 +492,7 @@ Remark setN_outside: ZMap.get q (setN vl p c) = ZMap.get q c. Proof. intros. apply setN_other. - intros. omega. + intros. lia. Qed. Remark getN_setN_same: @@ -502,7 +502,7 @@ Proof. induction vl; intros; simpl. auto. decEq. - rewrite setN_outside. apply ZMap.gss. omega. + rewrite setN_outside. apply ZMap.gss. lia. apply IHvl. Qed. @@ -512,7 +512,7 @@ Remark getN_exten: getN n p c1 = getN n p c2. Proof. induction n; intros. auto. rewrite Nat2Z.inj_succ in H. simpl. decEq. - apply H. omega. apply IHn. intros. apply H. omega. + apply H. lia. apply IHn. intros. apply H. lia. Qed. Remark getN_setN_disjoint: @@ -636,7 +636,7 @@ Qed. Theorem valid_access_empty: forall chunk b ofs p, ~valid_access empty chunk b ofs p. Proof. intros. red; intros. elim (perm_empty b ofs Cur p). apply H. - generalize (size_chunk_pos chunk); omega. + generalize (size_chunk_pos chunk); lia. Qed. (** ** Properties related to [load] *) @@ -801,7 +801,7 @@ Theorem loadbytes_empty: n <= 0 -> loadbytes m b ofs n = Some nil. Proof. intros. unfold loadbytes. rewrite pred_dec_true. rewrite Z_to_nat_neg; auto. - red; intros. omegaContradiction. + red; intros. extlia. Qed. Lemma getN_concat: @@ -809,9 +809,9 @@ Lemma getN_concat: getN (n1 + n2)%nat p c = getN n1 p c ++ getN n2 (p + Z.of_nat n1) c. Proof. induction n1; intros. - simpl. decEq. omega. + simpl. decEq. lia. rewrite Nat2Z.inj_succ. simpl. decEq. - replace (p + Z.succ (Z.of_nat n1)) with ((p + 1) + Z.of_nat n1) by omega. + replace (p + Z.succ (Z.of_nat n1)) with ((p + 1) + Z.of_nat n1) by lia. auto. Qed. @@ -825,12 +825,12 @@ Proof. unfold loadbytes; intros. destruct (range_perm_dec m b ofs (ofs + n1) Cur Readable); try congruence. destruct (range_perm_dec m b (ofs + n1) (ofs + n1 + n2) Cur Readable); try congruence. - rewrite pred_dec_true. rewrite Z2Nat.inj_add by omega. - rewrite getN_concat. rewrite Z2Nat.id by omega. + rewrite pred_dec_true. rewrite Z2Nat.inj_add by lia. + rewrite getN_concat. rewrite Z2Nat.id by lia. congruence. red; intros. - assert (ofs0 < ofs + n1 \/ ofs0 >= ofs + n1) by omega. - destruct H4. apply r; omega. apply r0; omega. + assert (ofs0 < ofs + n1 \/ ofs0 >= ofs + n1) by lia. + destruct H4. apply r; lia. apply r0; lia. Qed. Theorem loadbytes_split: @@ -845,13 +845,13 @@ Proof. unfold loadbytes; intros. destruct (range_perm_dec m b ofs (ofs + (n1 + n2)) Cur Readable); try congruence. - rewrite Z2Nat.inj_add in H by omega. rewrite getN_concat in H. - rewrite Z2Nat.id in H by omega. + rewrite Z2Nat.inj_add in H by lia. rewrite getN_concat in H. + rewrite Z2Nat.id in H by lia. repeat rewrite pred_dec_true. econstructor; econstructor. split. reflexivity. split. reflexivity. congruence. - red; intros; apply r; omega. - red; intros; apply r; omega. + red; intros; apply r; lia. + red; intros; apply r; lia. Qed. Theorem load_rep: @@ -871,13 +871,13 @@ Proof. revert ofs H; induction n; intros; simpl; auto. f_equal. rewrite Nat2Z.inj_succ in H. - replace ofs with (ofs+0) by omega. - apply H; omega. + replace ofs with (ofs+0) by lia. + apply H; lia. apply IHn. intros. rewrite <- Z.add_assoc. apply H. - rewrite Nat2Z.inj_succ. omega. + rewrite Nat2Z.inj_succ. lia. Qed. Theorem load_int64_split: @@ -892,7 +892,7 @@ Proof. exploit load_valid_access; eauto. intros [A B]. simpl in *. exploit load_loadbytes. eexact H. simpl. intros [bytes [LB EQ]]. change 8 with (4 + 4) in LB. - exploit loadbytes_split. eexact LB. omega. omega. + exploit loadbytes_split. eexact LB. lia. lia. intros (bytes1 & bytes2 & LB1 & LB2 & APP). change 4 with (size_chunk Mint32) in LB1. exploit loadbytes_load. eexact LB1. @@ -924,11 +924,11 @@ Proof. change (Int.unsigned (Int.repr 4)) with 4. apply Ptrofs.unsigned_repr. exploit (Zdivide_interval (Ptrofs.unsigned i) Ptrofs.modulus 8). - omega. apply Ptrofs.unsigned_range. auto. + lia. apply Ptrofs.unsigned_range. auto. exists (two_p (Ptrofs.zwordsize - 3)). unfold Ptrofs.modulus, Ptrofs.zwordsize, Ptrofs.wordsize. unfold Wordsize_Ptrofs.wordsize. destruct Archi.ptr64; reflexivity. - unfold Ptrofs.max_unsigned. omega. + unfold Ptrofs.max_unsigned. lia. Qed. Theorem loadv_int64_split: @@ -1085,7 +1085,7 @@ Qed. Theorem load_store_same: load chunk m2 b ofs = Some (Val.load_result chunk v). Proof. - apply load_store_similar_2; auto. omega. + apply load_store_similar_2; auto. lia. Qed. Theorem load_store_other: @@ -1137,9 +1137,9 @@ Proof. destruct H. congruence. destruct (zle n 0) as [z | n0]. rewrite (Z_to_nat_neg _ z). auto. - destruct H. omegaContradiction. + destruct H. extlia. apply getN_setN_outside. rewrite encode_val_length. rewrite <- size_chunk_conv. - rewrite Z2Nat.id. auto. omega. + rewrite Z2Nat.id. auto. lia. auto. red; intros. eauto with mem. rewrite pred_dec_false. auto. @@ -1152,11 +1152,11 @@ Lemma setN_in: In (ZMap.get q (setN vl p c)) vl. Proof. induction vl; intros. - simpl in H. omegaContradiction. + simpl in H. extlia. simpl length in H. rewrite Nat2Z.inj_succ in H. simpl. destruct (zeq p q). subst q. rewrite setN_outside. rewrite ZMap.gss. - auto with coqlib. omega. - right. apply IHvl. omega. + auto with coqlib. lia. + right. apply IHvl. lia. Qed. Lemma getN_in: @@ -1165,10 +1165,10 @@ Lemma getN_in: In (ZMap.get q c) (getN n p c). Proof. induction n; intros. - simpl in H; omegaContradiction. + simpl in H; extlia. rewrite Nat2Z.inj_succ in H. simpl. destruct (zeq p q). subst q. auto. - right. apply IHn. omega. + right. apply IHn. lia. Qed. End STORE. @@ -1205,28 +1205,28 @@ Proof. split. rewrite V', SIZE'. apply decode_val_shape. destruct (zeq ofs' ofs). - subst ofs'. left; split. auto. unfold c'. simpl. - rewrite setN_outside by omega. apply ZMap.gss. + rewrite setN_outside by lia. apply ZMap.gss. - right. destruct (zlt ofs ofs'). (* If ofs < ofs': the load reads (at ofs') a continuation byte from the write. ofs ofs' ofs+|chunk| [-------------------] write [-------------------] read *) -+ left; split. omega. unfold c'. simpl. apply setN_in. ++ left; split. lia. unfold c'. simpl. apply setN_in. assert (Z.of_nat (length (mv1 :: mvl)) = size_chunk chunk). { rewrite <- ENC; rewrite encode_val_length. rewrite size_chunk_conv; auto. } - simpl length in H3. rewrite Nat2Z.inj_succ in H3. omega. + simpl length in H3. rewrite Nat2Z.inj_succ in H3. lia. (* If ofs > ofs': the load reads (at ofs) the first byte from the write. ofs' ofs ofs'+|chunk'| [-------------------] write [----------------] read *) -+ right; split. omega. replace mv1 with (ZMap.get ofs c'). ++ right; split. lia. replace mv1 with (ZMap.get ofs c'). apply getN_in. assert (size_chunk chunk' = Z.succ (Z.of_nat sz')). { rewrite size_chunk_conv. rewrite SIZE'. rewrite Nat2Z.inj_succ; auto. } - omega. - unfold c'. simpl. rewrite setN_outside by omega. apply ZMap.gss. + lia. + unfold c'. simpl. rewrite setN_outside by lia. apply ZMap.gss. Qed. Definition compat_pointer_chunks (chunk1 chunk2: memory_chunk) : Prop := @@ -1313,10 +1313,10 @@ Theorem load_store_pointer_mismatch: Proof. intros. exploit load_store_overlap; eauto. - generalize (size_chunk_pos chunk'); omega. - generalize (size_chunk_pos chunk); omega. + generalize (size_chunk_pos chunk'); lia. + generalize (size_chunk_pos chunk); lia. intros (mv1 & mvl & mv1' & mvl' & ENC & DEC & CASES). - destruct CASES as [(A & B) | [(A & B) | (A & B)]]; try omegaContradiction. + destruct CASES as [(A & B) | [(A & B) | (A & B)]]; try extlia. inv ENC; inv DEC; auto. - elim H1. apply compat_pointer_chunks_true; auto. - contradiction. @@ -1338,8 +1338,8 @@ Proof. destruct (valid_access_dec m chunk1 b ofs Writable); destruct (valid_access_dec m chunk2 b ofs Writable); auto. f_equal. apply mkmem_ext; auto. congruence. - elim n. apply valid_access_compat with chunk1; auto. omega. - elim n. apply valid_access_compat with chunk2; auto. omega. + elim n. apply valid_access_compat with chunk1; auto. lia. + elim n. apply valid_access_compat with chunk2; auto. lia. Qed. Theorem store_signed_unsigned_8: @@ -1385,7 +1385,7 @@ Proof. destruct (valid_access_dec m Mfloat64 b ofs Writable); try discriminate. destruct (valid_access_dec m Mfloat64al32 b ofs Writable). rewrite <- H. f_equal. apply mkmem_ext; auto. - elim n. apply valid_access_compat with Mfloat64; auto. simpl; omega. + elim n. apply valid_access_compat with Mfloat64; auto. simpl; lia. Qed. Theorem storev_float64al32: @@ -1548,7 +1548,7 @@ Proof. rewrite pred_dec_true. rewrite storebytes_mem_contents. decEq. rewrite PMap.gsspec. destruct (peq b' b). subst b'. - apply getN_setN_disjoint. rewrite Z2Nat.id by omega. intuition congruence. + apply getN_setN_disjoint. rewrite Z2Nat.id by lia. intuition congruence. auto. red; auto with mem. apply pred_dec_false. @@ -1593,8 +1593,8 @@ Lemma setN_concat: setN (bytes1 ++ bytes2) ofs c = setN bytes2 (ofs + Z.of_nat (length bytes1)) (setN bytes1 ofs c). Proof. induction bytes1; intros. - simpl. decEq. omega. - simpl length. rewrite Nat2Z.inj_succ. simpl. rewrite IHbytes1. decEq. omega. + simpl. decEq. lia. + simpl length. rewrite Nat2Z.inj_succ. simpl. rewrite IHbytes1. decEq. lia. Qed. Theorem storebytes_concat: @@ -1613,8 +1613,8 @@ Proof. elim n. rewrite app_length. rewrite Nat2Z.inj_add. red; intros. destruct (zlt ofs0 (ofs + Z.of_nat(length bytes1))). - apply r. omega. - eapply perm_storebytes_2; eauto. apply r0. omega. + apply r. lia. + eapply perm_storebytes_2; eauto. apply r0. lia. Qed. Theorem storebytes_split: @@ -1627,10 +1627,10 @@ Proof. intros. destruct (range_perm_storebytes m b ofs bytes1) as [m1 ST1]. red; intros. exploit storebytes_range_perm; eauto. rewrite app_length. - rewrite Nat2Z.inj_add. omega. + rewrite Nat2Z.inj_add. lia. destruct (range_perm_storebytes m1 b (ofs + Z.of_nat (length bytes1)) bytes2) as [m2' ST2]. red; intros. eapply perm_storebytes_1; eauto. exploit storebytes_range_perm. - eexact H. instantiate (1 := ofs0). rewrite app_length. rewrite Nat2Z.inj_add. omega. + eexact H. instantiate (1 := ofs0). rewrite app_length. rewrite Nat2Z.inj_add. lia. auto. assert (Some m2 = Some m2'). rewrite <- H. eapply storebytes_concat; eauto. @@ -1738,7 +1738,7 @@ Theorem perm_alloc_2: Proof. unfold perm; intros. injection ALLOC; intros. rewrite <- H1; simpl. subst b. rewrite PMap.gss. unfold proj_sumbool. rewrite zle_true. - rewrite zlt_true. simpl. auto with mem. omega. omega. + rewrite zlt_true. simpl. auto with mem. lia. lia. Qed. Theorem perm_alloc_inv: @@ -1782,7 +1782,7 @@ Theorem valid_access_alloc_same: valid_access m2 chunk b ofs Freeable. Proof. intros. constructor; auto with mem. - red; intros. apply perm_alloc_2. omega. + red; intros. apply perm_alloc_2. lia. Qed. Local Hint Resolve valid_access_alloc_other valid_access_alloc_same: mem. @@ -1797,11 +1797,11 @@ Proof. intros. inv H. generalize (size_chunk_pos chunk); intro. destruct (eq_block b' b). subst b'. - assert (perm m2 b ofs Cur p). apply H0. omega. - assert (perm m2 b (ofs + size_chunk chunk - 1) Cur p). apply H0. omega. + assert (perm m2 b ofs Cur p). apply H0. lia. + assert (perm m2 b (ofs + size_chunk chunk - 1) Cur p). apply H0. lia. exploit perm_alloc_inv. eexact H2. rewrite dec_eq_true. intro. exploit perm_alloc_inv. eexact H3. rewrite dec_eq_true. intro. - intuition omega. + intuition lia. split; auto. red; intros. exploit perm_alloc_inv. apply H0. eauto. rewrite dec_eq_false; auto. Qed. @@ -1848,7 +1848,7 @@ Theorem load_alloc_same': Proof. intros. assert (exists v, load chunk m2 b ofs = Some v). apply valid_access_load. constructor; auto. - red; intros. eapply perm_implies. apply perm_alloc_2. omega. auto with mem. + red; intros. eapply perm_implies. apply perm_alloc_2. lia. auto with mem. destruct H2 as [v LOAD]. rewrite LOAD. decEq. eapply load_alloc_same; eauto. Qed. @@ -1958,7 +1958,7 @@ Theorem perm_free_2: Proof. intros. rewrite free_result. unfold perm, unchecked_free; simpl. rewrite PMap.gss. unfold proj_sumbool. rewrite zle_true. rewrite zlt_true. - simpl. tauto. omega. omega. + simpl. tauto. lia. lia. Qed. Theorem perm_free_3: @@ -1991,7 +1991,7 @@ Theorem valid_access_free_1: Proof. intros. inv H. constructor; auto with mem. red; intros. eapply perm_free_1; eauto. - destruct (zlt lo hi). intuition. right. omega. + destruct (zlt lo hi). intuition. right. lia. Qed. Theorem valid_access_free_2: @@ -2003,9 +2003,9 @@ Proof. generalize (size_chunk_pos chunk); intros. destruct (zlt ofs lo). elim (perm_free_2 lo Cur p). - omega. apply H3. omega. + lia. apply H3. lia. elim (perm_free_2 ofs Cur p). - omega. apply H3. omega. + lia. apply H3. lia. Qed. Theorem valid_access_free_inv_1: @@ -2031,7 +2031,7 @@ Proof. destruct (zlt lo hi); auto. destruct (zle (ofs + size_chunk chunk) lo); auto. destruct (zle hi ofs); auto. - elim (valid_access_free_2 chunk ofs p); auto. omega. + elim (valid_access_free_2 chunk ofs p); auto. lia. Qed. Theorem load_free: @@ -2069,7 +2069,7 @@ Proof. red; intros. eapply perm_free_3; eauto. rewrite pred_dec_false; auto. red; intros. elim n0; red; intros. - eapply perm_free_1; eauto. destruct H; auto. right; omega. + eapply perm_free_1; eauto. destruct H; auto. right; lia. Qed. Theorem loadbytes_free_2: @@ -2139,7 +2139,7 @@ Proof. unfold drop_perm in DROP. destruct (range_perm_dec m b lo hi Cur Freeable); inv DROP. unfold perm. simpl. rewrite PMap.gss. unfold proj_sumbool. rewrite zle_true. rewrite zlt_true. simpl. constructor. - omega. omega. + lia. lia. Qed. Theorem perm_drop_2: @@ -2149,7 +2149,7 @@ Proof. unfold drop_perm in DROP. destruct (range_perm_dec m b lo hi Cur Freeable); inv DROP. revert H0. unfold perm; simpl. rewrite PMap.gss. unfold proj_sumbool. rewrite zle_true. rewrite zlt_true. simpl. auto. - omega. omega. + lia. lia. Qed. Theorem perm_drop_3: @@ -2159,7 +2159,7 @@ Proof. unfold drop_perm in DROP. destruct (range_perm_dec m b lo hi Cur Freeable); inv DROP. unfold perm; simpl. rewrite PMap.gsspec. destruct (peq b' b). subst b'. unfold proj_sumbool. destruct (zle lo ofs). destruct (zlt ofs hi). - byContradiction. intuition omega. + byContradiction. intuition lia. auto. auto. auto. Qed. @@ -2185,7 +2185,7 @@ Proof. destruct (eq_block b' b). subst b'. destruct (zlt ofs0 lo). eapply perm_drop_3; eauto. destruct (zle hi ofs0). eapply perm_drop_3; eauto. - apply perm_implies with p. eapply perm_drop_1; eauto. omega. + apply perm_implies with p. eapply perm_drop_1; eauto. lia. generalize (size_chunk_pos chunk); intros. intuition. eapply perm_drop_3; eauto. Qed. @@ -2227,7 +2227,7 @@ Proof. destruct (eq_block b' b). subst b'. destruct (zlt ofs0 lo). eapply perm_drop_3; eauto. destruct (zle hi ofs0). eapply perm_drop_3; eauto. - apply perm_implies with p. eapply perm_drop_1; eauto. omega. intuition. + apply perm_implies with p. eapply perm_drop_1; eauto. lia. intuition. eapply perm_drop_3; eauto. rewrite pred_dec_false; eauto. red; intros; elim n0; red; intros. @@ -2285,8 +2285,8 @@ Lemma range_perm_inj: range_perm m2 b2 (lo + delta) (hi + delta) k p. Proof. intros; red; intros. - replace ofs with ((ofs - delta) + delta) by omega. - eapply perm_inj; eauto. apply H0. omega. + replace ofs with ((ofs - delta) + delta) by lia. + eapply perm_inj; eauto. apply H0. lia. Qed. Lemma valid_access_inj: @@ -2298,7 +2298,7 @@ Lemma valid_access_inj: Proof. intros. destruct H1 as [A B]. constructor. replace (ofs + delta + size_chunk chunk) - with ((ofs + size_chunk chunk) + delta) by omega. + with ((ofs + size_chunk chunk) + delta) by lia. eapply range_perm_inj; eauto. apply Z.divide_add_r; auto. eapply mi_align; eauto with mem. Qed. @@ -2320,9 +2320,9 @@ Proof. rewrite Nat2Z.inj_succ in H1. constructor. eapply mi_memval; eauto. - apply H1. omega. - replace (ofs + delta + 1) with ((ofs + 1) + delta) by omega. - apply IHn. red; intros; apply H1; omega. + apply H1. lia. + replace (ofs + delta + 1) with ((ofs + 1) + delta) by lia. + apply IHn. red; intros; apply H1; lia. Qed. Lemma load_inj: @@ -2353,11 +2353,11 @@ Proof. destruct (range_perm_dec m1 b1 ofs (ofs + len) Cur Readable); inv H0. exists (getN (Z.to_nat len) (ofs + delta) (m2.(mem_contents)#b2)). split. apply pred_dec_true. - replace (ofs + delta + len) with ((ofs + len) + delta) by omega. + replace (ofs + delta + len) with ((ofs + len) + delta) by lia. eapply range_perm_inj; eauto with mem. apply getN_inj; auto. - destruct (zle 0 len). rewrite Z2Nat.id by omega. auto. - rewrite Z_to_nat_neg by omega. simpl. red; intros; omegaContradiction. + destruct (zle 0 len). rewrite Z2Nat.id by lia. auto. + rewrite Z_to_nat_neg by lia. simpl. red; intros; extlia. Qed. (** Preservation of stores. *) @@ -2372,11 +2372,11 @@ Lemma setN_inj: Proof. induction 1; intros; simpl. auto. - replace (p + delta + 1) with ((p + 1) + delta) by omega. + replace (p + delta + 1) with ((p + 1) + delta) by lia. apply IHlist_forall2; auto. intros. rewrite ZMap.gsspec at 1. destruct (ZIndexed.eq q0 p). subst q0. rewrite ZMap.gss. auto. - rewrite ZMap.gso. auto. unfold ZIndexed.t in *. omega. + rewrite ZMap.gso. auto. unfold ZIndexed.t in *. lia. Qed. Definition meminj_no_overlap (f: meminj) (m: mem) : Prop := @@ -2431,8 +2431,8 @@ Proof. assert (b2 <> b2 \/ ofs0 + delta0 <> (r - delta) + delta). eapply H1; eauto. eauto 6 with mem. exploit store_valid_access_3. eexact H0. intros [A B]. - eapply perm_implies. apply perm_cur_max. apply A. omega. auto with mem. - destruct H8. congruence. omega. + eapply perm_implies. apply perm_cur_max. apply A. lia. auto with mem. + destruct H8. congruence. lia. (* block <> b1, block <> b2 *) eapply mi_memval; eauto. eauto with mem. Qed. @@ -2479,8 +2479,8 @@ Proof. rewrite setN_outside. auto. rewrite encode_val_length. rewrite <- size_chunk_conv. destruct (zlt (ofs0 + delta) ofs); auto. - destruct (zle (ofs + size_chunk chunk) (ofs0 + delta)). omega. - byContradiction. eapply H0; eauto. omega. + destruct (zle (ofs + size_chunk chunk) (ofs0 + delta)). lia. + byContradiction. eapply H0; eauto. lia. eauto with mem. Qed. @@ -2501,7 +2501,7 @@ Proof. with ((ofs + Z.of_nat (length bytes1)) + delta). eapply range_perm_inj; eauto with mem. eapply storebytes_range_perm; eauto. - rewrite (list_forall2_length H3). omega. + rewrite (list_forall2_length H3). lia. destruct (range_perm_storebytes _ _ _ _ H4) as [n2 STORE]. exists n2; split. eauto. constructor. @@ -2532,9 +2532,9 @@ Proof. eapply H1; eauto 6 with mem. exploit storebytes_range_perm. eexact H0. instantiate (1 := r - delta). - rewrite (list_forall2_length H3). omega. + rewrite (list_forall2_length H3). lia. eauto 6 with mem. - destruct H9. congruence. omega. + destruct H9. congruence. lia. (* block <> b1, block <> b2 *) eauto. Qed. @@ -2581,8 +2581,8 @@ Proof. rewrite PMap.gsspec. destruct (peq b2 b). subst b2. rewrite setN_outside. auto. destruct (zlt (ofs0 + delta) ofs); auto. - destruct (zle (ofs + Z.of_nat (length bytes2)) (ofs0 + delta)). omega. - byContradiction. eapply H0; eauto. omega. + destruct (zle (ofs + Z.of_nat (length bytes2)) (ofs0 + delta)). lia. + byContradiction. eapply H0; eauto. lia. eauto with mem. Qed. @@ -2679,10 +2679,10 @@ Proof. intros. destruct (eq_block b0 b1). subst b0. assert (delta0 = delta) by congruence. subst delta0. assert (lo <= ofs < hi). - { eapply perm_alloc_3; eauto. apply H6. generalize (size_chunk_pos chunk); omega. } + { eapply perm_alloc_3; eauto. apply H6. generalize (size_chunk_pos chunk); lia. } assert (lo <= ofs + size_chunk chunk - 1 < hi). - { eapply perm_alloc_3; eauto. apply H6. generalize (size_chunk_pos chunk); omega. } - apply H2. omega. + { eapply perm_alloc_3; eauto. apply H6. generalize (size_chunk_pos chunk); lia. } + apply H2. lia. eapply mi_align0 with (ofs := ofs) (p := p); eauto. red; intros. eapply perm_alloc_4; eauto. (* mem_contents *) @@ -2727,7 +2727,7 @@ Proof. intros. eapply perm_free_1; eauto. destruct (eq_block b2 b); auto. subst b. right. assert (~ (lo <= ofs + delta < hi)). red; intros; eapply H1; eauto. - omega. + lia. constructor. (* perm *) auto. @@ -2772,8 +2772,8 @@ Proof. intros. assert ({ m2' | drop_perm m2 b2 (lo + delta) (hi + delta) p = Some m2' }). apply range_perm_drop_2. red; intros. - replace ofs with ((ofs - delta) + delta) by omega. - eapply perm_inj; eauto. eapply range_perm_drop_1; eauto. omega. + replace ofs with ((ofs - delta) + delta) by lia. + eapply perm_inj; eauto. eapply range_perm_drop_1; eauto. lia. destruct X as [m2' DROP]. exists m2'; split; auto. inv H. constructor. @@ -2787,9 +2787,9 @@ Proof. destruct (zlt (ofs + delta0) (lo + delta0)). eapply perm_drop_3; eauto. destruct (zle (hi + delta0) (ofs + delta0)). eapply perm_drop_3; eauto. assert (perm_order p p0). - eapply perm_drop_2. eexact H0. instantiate (1 := ofs). omega. eauto. + eapply perm_drop_2. eexact H0. instantiate (1 := ofs). lia. eauto. apply perm_implies with p; auto. - eapply perm_drop_1. eauto. omega. + eapply perm_drop_1. eauto. lia. (* b1 <> b0 *) eapply perm_drop_3; eauto. destruct (eq_block b3 b2); auto. @@ -2798,7 +2798,7 @@ Proof. exploit H1; eauto. instantiate (1 := ofs + delta0 - delta). apply perm_cur_max. apply perm_implies with Freeable. - eapply range_perm_drop_1; eauto. omega. auto with mem. + eapply range_perm_drop_1; eauto. lia. auto with mem. eapply perm_drop_4; eauto. eapply perm_max. apply perm_implies with p0. eauto. eauto with mem. intuition. @@ -2829,7 +2829,7 @@ Proof. destruct (eq_block b2 b); auto. subst b2. right. destruct (zlt (ofs + delta) lo); auto. destruct (zle hi (ofs + delta)); auto. - byContradiction. exploit H1; eauto. omega. + byContradiction. exploit H1; eauto. lia. (* align *) eapply mi_align0; eauto. (* contents *) @@ -2862,9 +2862,9 @@ Theorem extends_refl: forall m, extends m m. Proof. intros. constructor. auto. constructor. - intros. unfold inject_id in H; inv H. replace (ofs + 0) with ofs by omega. auto. + intros. unfold inject_id in H; inv H. replace (ofs + 0) with ofs by lia. auto. intros. unfold inject_id in H; inv H. apply Z.divide_0_r. - intros. unfold inject_id in H; inv H. replace (ofs + 0) with ofs by omega. + intros. unfold inject_id in H; inv H. replace (ofs + 0) with ofs by lia. apply memval_lessdef_refl. tauto. Qed. @@ -2877,7 +2877,7 @@ Theorem load_extends: Proof. intros. inv H. exploit load_inj; eauto. unfold inject_id; reflexivity. intros [v2 [A B]]. exists v2; split. - replace (ofs + 0) with ofs in A by omega. auto. + replace (ofs + 0) with ofs in A by lia. auto. rewrite val_inject_id in B. auto. Qed. @@ -2901,7 +2901,7 @@ Theorem loadbytes_extends: /\ list_forall2 memval_lessdef bytes1 bytes2. Proof. intros. inv H. - replace ofs with (ofs + 0) by omega. eapply loadbytes_inj; eauto. + replace ofs with (ofs + 0) by lia. eapply loadbytes_inj; eauto. Qed. Theorem store_within_extends: @@ -2920,7 +2920,7 @@ Proof. rewrite val_inject_id. eauto. intros [m2' [A B]]. exists m2'; split. - replace (ofs + 0) with ofs in A by omega. auto. + replace (ofs + 0) with ofs in A by lia. auto. constructor; auto. rewrite (nextblock_store _ _ _ _ _ _ H0). rewrite (nextblock_store _ _ _ _ _ _ A). @@ -2938,7 +2938,7 @@ Proof. intros. inversion H. constructor. rewrite (nextblock_store _ _ _ _ _ _ H0). auto. eapply store_outside_inj; eauto. - unfold inject_id; intros. inv H2. eapply H1; eauto. omega. + unfold inject_id; intros. inv H2. eapply H1; eauto. lia. intros. eauto using perm_store_2. Qed. @@ -2972,7 +2972,7 @@ Proof. unfold inject_id; reflexivity. intros [m2' [A B]]. exists m2'; split. - replace (ofs + 0) with ofs in A by omega. auto. + replace (ofs + 0) with ofs in A by lia. auto. constructor; auto. rewrite (nextblock_storebytes _ _ _ _ _ H0). rewrite (nextblock_storebytes _ _ _ _ _ A). @@ -2990,7 +2990,7 @@ Proof. intros. inversion H. constructor. rewrite (nextblock_storebytes _ _ _ _ _ H0). auto. eapply storebytes_outside_inj; eauto. - unfold inject_id; intros. inv H2. eapply H1; eauto. omega. + unfold inject_id; intros. inv H2. eapply H1; eauto. lia. intros. eauto using perm_storebytes_2. Qed. @@ -3022,12 +3022,12 @@ Proof. intros. eapply perm_implies with Freeable; auto with mem. eapply perm_alloc_2; eauto. - omega. + lia. intros. eapply perm_alloc_inv in H; eauto. generalize (perm_alloc_inv _ _ _ _ _ H0 b0 ofs Max Nonempty); intros PERM. destruct (eq_block b0 b). subst b0. - assert (EITHER: lo1 <= ofs < hi1 \/ ~(lo1 <= ofs < hi1)) by omega. + assert (EITHER: lo1 <= ofs < hi1 \/ ~(lo1 <= ofs < hi1)) by lia. destruct EITHER. left. apply perm_implies with Freeable; auto with mem. eapply perm_alloc_2; eauto. right; tauto. @@ -3059,7 +3059,7 @@ Proof. intros. inv H. constructor. rewrite (nextblock_free _ _ _ _ _ H0). auto. eapply free_right_inj; eauto. - unfold inject_id; intros. inv H. eapply H1; eauto. omega. + unfold inject_id; intros. inv H. eapply H1; eauto. lia. intros. eauto using perm_free_3. Qed. @@ -3074,7 +3074,7 @@ Proof. intros. inversion H. assert ({ m2': mem | free m2 b lo hi = Some m2' }). apply range_perm_free. red; intros. - replace ofs with (ofs + 0) by omega. + replace ofs with (ofs + 0) by lia. eapply perm_inj with (b1 := b); eauto. eapply free_range_perm; eauto. destruct X as [m2' FREE]. exists m2'; split; auto. @@ -3084,7 +3084,7 @@ Proof. eapply free_right_inj with (m1 := m1'); eauto. eapply free_left_inj; eauto. unfold inject_id; intros. inv H1. - eapply perm_free_2. eexact H0. instantiate (1 := ofs); omega. eauto. + eapply perm_free_2. eexact H0. instantiate (1 := ofs); lia. eauto. intros. exploit mext_perm_inv0; eauto using perm_free_3. intros [A|A]. eapply perm_free_inv in A; eauto. destruct A as [[A B]|A]; auto. subst b0. right; eapply perm_free_2; eauto. @@ -3103,7 +3103,7 @@ Theorem perm_extends: forall m1 m2 b ofs k p, extends m1 m2 -> perm m1 b ofs k p -> perm m2 b ofs k p. Proof. - intros. inv H. replace ofs with (ofs + 0) by omega. + intros. inv H. replace ofs with (ofs + 0) by lia. eapply perm_inj; eauto. Qed. @@ -3118,7 +3118,7 @@ Theorem valid_access_extends: forall m1 m2 chunk b ofs p, extends m1 m2 -> valid_access m1 chunk b ofs p -> valid_access m2 chunk b ofs p. Proof. - intros. inv H. replace ofs with (ofs + 0) by omega. + intros. inv H. replace ofs with (ofs + 0) by lia. eapply valid_access_inj; eauto. auto. Qed. @@ -3263,7 +3263,7 @@ Theorem weak_valid_pointer_inject: weak_valid_pointer m2 b2 (ofs + delta) = true. Proof. intros until 2. unfold weak_valid_pointer. rewrite !orb_true_iff. - replace (ofs + delta - 1) with ((ofs - 1) + delta) by omega. + replace (ofs + delta - 1) with ((ofs - 1) + delta) by lia. intros []; eauto using valid_pointer_inject. Qed. @@ -3281,8 +3281,8 @@ Proof. assert (perm m1 b1 (Ptrofs.unsigned ofs1) Max Nonempty) by eauto with mem. exploit mi_representable; eauto. intros [A B]. assert (0 <= delta <= Ptrofs.max_unsigned). - generalize (Ptrofs.unsigned_range ofs1). omega. - unfold Ptrofs.add. repeat rewrite Ptrofs.unsigned_repr; omega. + generalize (Ptrofs.unsigned_range ofs1). lia. + unfold Ptrofs.add. repeat rewrite Ptrofs.unsigned_repr; lia. Qed. Lemma address_inject': @@ -3293,7 +3293,7 @@ Lemma address_inject': Ptrofs.unsigned (Ptrofs.add ofs1 (Ptrofs.repr delta)) = Ptrofs.unsigned ofs1 + delta. Proof. intros. destruct H0. eapply address_inject; eauto. - apply H0. generalize (size_chunk_pos chunk). omega. + apply H0. generalize (size_chunk_pos chunk). lia. Qed. Theorem weak_valid_pointer_inject_no_overflow: @@ -3308,7 +3308,7 @@ Proof. exploit mi_representable; eauto. destruct H0; eauto with mem. intros [A B]. pose proof (Ptrofs.unsigned_range ofs). - rewrite Ptrofs.unsigned_repr; omega. + rewrite Ptrofs.unsigned_repr; lia. Qed. Theorem valid_pointer_inject_no_overflow: @@ -3348,7 +3348,7 @@ Proof. exploit mi_representable; eauto. destruct H0; eauto with mem. intros [A B]. pose proof (Ptrofs.unsigned_range ofs). - unfold Ptrofs.add. repeat rewrite Ptrofs.unsigned_repr; auto; omega. + unfold Ptrofs.add. repeat rewrite Ptrofs.unsigned_repr; auto; lia. Qed. Theorem inject_no_overlap: @@ -3383,8 +3383,8 @@ Proof. rewrite (address_inject' _ _ _ _ _ _ _ _ H H2 H4). inv H1. simpl in H5. inv H2. simpl in H1. eapply mi_no_overlap; eauto. - apply perm_cur_max. apply (H5 (Ptrofs.unsigned ofs1)). omega. - apply perm_cur_max. apply (H1 (Ptrofs.unsigned ofs2)). omega. + apply perm_cur_max. apply (H5 (Ptrofs.unsigned ofs1)). lia. + apply perm_cur_max. apply (H1 (Ptrofs.unsigned ofs2)). lia. Qed. Theorem disjoint_or_equal_inject: @@ -3403,16 +3403,16 @@ Proof. intros. destruct (eq_block b1 b2). assert (b1' = b2') by congruence. assert (delta1 = delta2) by congruence. subst. - destruct H5. congruence. right. destruct H5. left; congruence. right. omega. + destruct H5. congruence. right. destruct H5. left; congruence. right. lia. destruct (eq_block b1' b2'); auto. subst. right. right. set (i1 := (ofs1 + delta1, ofs1 + delta1 + sz)). set (i2 := (ofs2 + delta2, ofs2 + delta2 + sz)). change (snd i1 <= fst i2 \/ snd i2 <= fst i1). - apply Intv.range_disjoint'; simpl; try omega. + apply Intv.range_disjoint'; simpl; try lia. unfold Intv.disjoint, Intv.In; simpl; intros. red; intros. exploit mi_no_overlap; eauto. - instantiate (1 := x - delta1). apply H2. omega. - instantiate (1 := x - delta2). apply H3. omega. + instantiate (1 := x - delta1). apply H2. lia. + instantiate (1 := x - delta2). apply H3. lia. intuition. Qed. @@ -3427,9 +3427,9 @@ Theorem aligned_area_inject: (al | ofs + delta). Proof. intros. - assert (P: al > 0) by omega. - assert (Q: Z.abs al <= Z.abs sz). apply Zdivide_bounds; auto. omega. - rewrite Z.abs_eq in Q; try omega. rewrite Z.abs_eq in Q; try omega. + assert (P: al > 0) by lia. + assert (Q: Z.abs al <= Z.abs sz). apply Zdivide_bounds; auto. lia. + rewrite Z.abs_eq in Q; try lia. rewrite Z.abs_eq in Q; try lia. assert (R: exists chunk, al = align_chunk chunk /\ al = size_chunk chunk). destruct H0. subst; exists Mint8unsigned; auto. destruct H0. subst; exists Mint16unsigned; auto. @@ -3437,7 +3437,7 @@ Proof. subst; exists Mint64; auto. destruct R as [chunk [A B]]. assert (valid_access m chunk b ofs Nonempty). - split. red; intros; apply H3. omega. congruence. + split. red; intros; apply H3. lia. congruence. exploit valid_access_inject; eauto. intros [C D]. congruence. Qed. @@ -3794,7 +3794,7 @@ Proof. unfold f'; intros. destruct (eq_block b0 b1). inversion H8. subst b0 b3 delta0. elim (fresh_block_alloc _ _ _ _ _ H0). - eapply perm_valid_block with (ofs := ofs). apply H9. generalize (size_chunk_pos chunk); omega. + eapply perm_valid_block with (ofs := ofs). apply H9. generalize (size_chunk_pos chunk); lia. eauto. unfold f'; intros. destruct (eq_block b0 b1). inversion H8. subst b0 b3 delta0. @@ -3817,10 +3817,10 @@ Proof. congruence. inversion H10; subst b0 b1' delta1. destruct (eq_block b2 b2'); auto. subst b2'. right; red; intros. - eapply H6; eauto. omega. + eapply H6; eauto. lia. inversion H11; subst b3 b2' delta2. destruct (eq_block b1' b2); auto. subst b1'. right; red; intros. - eapply H6; eauto. omega. + eapply H6; eauto. lia. eauto. (* representable *) unfold f'; intros. @@ -3828,16 +3828,16 @@ Proof. subst. injection H9; intros; subst b' delta0. destruct H10. exploit perm_alloc_inv; eauto; rewrite dec_eq_true; intro. exploit H3. apply H4 with (k := Max) (p := Nonempty); eauto. - generalize (Ptrofs.unsigned_range_2 ofs). omega. + generalize (Ptrofs.unsigned_range_2 ofs). lia. exploit perm_alloc_inv; eauto; rewrite dec_eq_true; intro. exploit H3. apply H4 with (k := Max) (p := Nonempty); eauto. - generalize (Ptrofs.unsigned_range_2 ofs). omega. + generalize (Ptrofs.unsigned_range_2 ofs). lia. eapply mi_representable0; try eassumption. destruct H10; eauto using perm_alloc_4. (* perm inv *) intros. unfold f' in H9; destruct (eq_block b0 b1). inversion H9; clear H9; subst b0 b3 delta0. - assert (EITHER: lo <= ofs < hi \/ ~(lo <= ofs < hi)) by omega. + assert (EITHER: lo <= ofs < hi \/ ~(lo <= ofs < hi)) by lia. destruct EITHER. left. apply perm_implies with Freeable; auto with mem. eapply perm_alloc_2; eauto. right; intros A. eapply perm_alloc_inv in A; eauto. rewrite dec_eq_true in A. tauto. @@ -3868,10 +3868,10 @@ Proof. eapply alloc_right_inject; eauto. eauto. instantiate (1 := b2). eauto with mem. - instantiate (1 := 0). unfold Ptrofs.max_unsigned. generalize Ptrofs.modulus_pos; omega. + instantiate (1 := 0). unfold Ptrofs.max_unsigned. generalize Ptrofs.modulus_pos; lia. auto. intros. apply perm_implies with Freeable; auto with mem. - eapply perm_alloc_2; eauto. omega. + eapply perm_alloc_2; eauto. lia. red; intros. apply Z.divide_0_r. intros. apply (valid_not_valid_diff m2 b2 b2); eauto with mem. intros [f' [A [B [C D]]]]. @@ -3994,13 +3994,13 @@ Proof. simpl; rewrite H0; auto. intros. destruct (eq_block b1 b). subst b1. rewrite H1 in H2; inv H2. - exists lo, hi; split; auto with coqlib. omega. + exists lo, hi; split; auto with coqlib. lia. exploit mi_no_overlap. eexact H. eexact n. eauto. eauto. eapply perm_max. eapply perm_implies. eauto. auto with mem. instantiate (1 := ofs + delta0 - delta). apply perm_cur_max. apply perm_implies with Freeable; auto with mem. - eapply free_range_perm; eauto. omega. - intros [A|A]. congruence. omega. + eapply free_range_perm; eauto. lia. + intros [A|A]. congruence. lia. Qed. Lemma drop_outside_inject: forall f m1 m2 b lo hi p m2', @@ -4027,7 +4027,7 @@ Proof. (* perm *) destruct (f b1) as [[b' delta'] |] eqn:?; try discriminate. destruct (f' b') as [[b'' delta''] |] eqn:?; inv H. - replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') by omega. + replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') by lia. eauto. (* align *) destruct (f b1) as [[b' delta'] |] eqn:?; try discriminate. @@ -4035,12 +4035,12 @@ Proof. apply Z.divide_add_r. eapply mi_align0; eauto. eapply mi_align1 with (ofs := ofs + delta') (p := p); eauto. - red; intros. replace ofs0 with ((ofs0 - delta') + delta') by omega. - eapply mi_perm0; eauto. apply H0. omega. + red; intros. replace ofs0 with ((ofs0 - delta') + delta') by lia. + eapply mi_perm0; eauto. apply H0. lia. (* memval *) destruct (f b1) as [[b' delta'] |] eqn:?; try discriminate. destruct (f' b') as [[b'' delta''] |] eqn:?; inv H. - replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') by omega. + replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') by lia. eapply memval_inject_compose; eauto. Qed. @@ -4069,11 +4069,11 @@ Proof. exploit mi_no_overlap0; eauto. intros A. destruct (eq_block b1x b2x). subst b1x. destruct A. congruence. - assert (delta1y = delta2y) by congruence. right; omega. + assert (delta1y = delta2y) by congruence. right; lia. exploit mi_no_overlap1. eauto. eauto. eauto. eapply perm_inj. eauto. eexact H2. eauto. eapply perm_inj. eauto. eexact H3. eauto. - intuition omega. + intuition lia. (* representable *) intros. destruct (f b) as [[b1 delta1] |] eqn:?; try discriminate. @@ -4085,15 +4085,15 @@ Proof. exploit mi_representable1. eauto. instantiate (1 := ofs'). rewrite H. replace (Ptrofs.unsigned ofs + delta1 - 1) with - ((Ptrofs.unsigned ofs - 1) + delta1) by omega. + ((Ptrofs.unsigned ofs - 1) + delta1) by lia. destruct H0; eauto using perm_inj. - rewrite H. omega. + rewrite H. lia. (* perm inv *) intros. destruct (f b1) as [[b' delta'] |] eqn:?; try discriminate. destruct (f' b') as [[b'' delta''] |] eqn:?; try discriminate. inversion H; clear H; subst b'' delta. - replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') in H0 by omega. + replace (ofs + (delta' + delta'')) with ((ofs + delta') + delta'') in H0 by lia. exploit mi_perm_inv1; eauto. intros [A|A]. eapply mi_perm_inv0; eauto. right; red; intros. elim A. eapply perm_inj; eauto. @@ -4145,7 +4145,7 @@ Proof. (* inj *) replace f with (compose_meminj f inject_id). eapply mem_inj_compose; eauto. apply extensionality; intros. unfold compose_meminj, inject_id. - destruct (f x) as [[y delta] | ]; auto. decEq. decEq. omega. + destruct (f x) as [[y delta] | ]; auto. decEq. decEq. lia. (* unmapped *) eauto. (* mapped *) @@ -4210,7 +4210,7 @@ Proof. apply flat_inj_no_overlap. (* range *) unfold flat_inj; intros. - destruct (plt b (nextblock m)); inv H0. generalize (Ptrofs.unsigned_range_2 ofs); omega. + destruct (plt b (nextblock m)); inv H0. generalize (Ptrofs.unsigned_range_2 ofs); lia. (* perm inv *) unfold flat_inj; intros. destruct (plt b1 (nextblock m)); inv H0. @@ -4223,7 +4223,7 @@ Proof. intros; red; constructor. (* perm *) unfold flat_inj; intros. destruct (plt b1 thr); inv H. - replace (ofs + 0) with ofs by omega; auto. + replace (ofs + 0) with ofs by lia; auto. (* align *) unfold flat_inj; intros. destruct (plt b1 thr); inv H. apply Z.divide_0_r. (* mem_contents *) @@ -4243,7 +4243,7 @@ Proof. red. intros. apply Z.divide_0_r. intros. apply perm_implies with Freeable; auto with mem. - eapply perm_alloc_2; eauto. omega. + eapply perm_alloc_2; eauto. lia. unfold flat_inj. apply pred_dec_true. rewrite (alloc_result _ _ _ _ _ H). auto. Qed. @@ -4259,7 +4259,7 @@ Proof. intros; red. exploit store_mapped_inj. eauto. eauto. apply flat_inj_no_overlap. unfold flat_inj. apply pred_dec_true; auto. eauto. - replace (ofs + 0) with ofs by omega. + replace (ofs + 0) with ofs by lia. intros [m'' [A B]]. congruence. Qed. @@ -4306,7 +4306,7 @@ Lemma valid_block_unchanged_on: forall m m' b, unchanged_on m m' -> valid_block m b -> valid_block m' b. Proof. - unfold valid_block; intros. apply unchanged_on_nextblock in H. xomega. + unfold valid_block; intros. apply unchanged_on_nextblock in H. extlia. Qed. Lemma perm_unchanged_on: @@ -4349,7 +4349,7 @@ Proof. + unfold loadbytes. destruct H. destruct (range_perm_dec m b ofs (ofs + n) Cur Readable). rewrite pred_dec_true. f_equal. - apply getN_exten. intros. rewrite Z2Nat.id in H by omega. + apply getN_exten. intros. rewrite Z2Nat.id in H by lia. apply unchanged_on_contents0; auto. red; intros. apply unchanged_on_perm0; auto. rewrite pred_dec_false. auto. @@ -4367,7 +4367,7 @@ Proof. destruct (zle n 0). + erewrite loadbytes_empty in * by assumption. auto. + rewrite <- H1. apply loadbytes_unchanged_on_1; auto. - exploit loadbytes_range_perm; eauto. instantiate (1 := ofs). omega. + exploit loadbytes_range_perm; eauto. instantiate (1 := ofs). lia. intros. eauto with mem. Qed. @@ -4410,7 +4410,7 @@ Proof. rewrite encode_val_length. rewrite <- size_chunk_conv. destruct (zlt ofs0 ofs); auto. destruct (zlt ofs0 (ofs + size_chunk chunk)); auto. - elim (H0 ofs0). omega. auto. + elim (H0 ofs0). lia. auto. Qed. Lemma storebytes_unchanged_on: @@ -4426,7 +4426,7 @@ Proof. destruct (peq b0 b); auto. subst b0. apply setN_outside. destruct (zlt ofs0 ofs); auto. destruct (zlt ofs0 (ofs + Z.of_nat (length bytes))); auto. - elim (H0 ofs0). omega. auto. + elim (H0 ofs0). lia. auto. Qed. Lemma alloc_unchanged_on: @@ -4455,7 +4455,7 @@ Proof. - split; intros. eapply perm_free_1; eauto. destruct (eq_block b0 b); auto. destruct (zlt ofs lo); auto. destruct (zle hi ofs); auto. - subst b0. elim (H0 ofs). omega. auto. + subst b0. elim (H0 ofs). lia. auto. eapply perm_free_3; eauto. - unfold free in H. destruct (range_perm_dec m b lo hi Cur Freeable); inv H. simpl. auto. @@ -4473,7 +4473,7 @@ Proof. destruct (eq_block b0 b); auto. subst b0. assert (~ (lo <= ofs < hi)). { red; intros; eelim H0; eauto. } - right; omega. + right; lia. eapply perm_drop_4; eauto. - unfold drop_perm in H. destruct (range_perm_dec m b lo hi Cur Freeable); inv H; simpl. auto. diff --git a/common/Separation.v b/common/Separation.v index 27065d1f..0357b2bf 100644 --- a/common/Separation.v +++ b/common/Separation.v @@ -355,12 +355,12 @@ Proof. intros. rewrite <- sep_assoc. eapply sep_imp; eauto. split; simpl; intros. - intuition auto. -+ omega. -+ apply H5; omega. -+ omega. -+ apply H5; omega. -+ red; simpl; intros; omega. -- intuition omega. ++ lia. ++ apply H5; lia. ++ lia. ++ apply H5; lia. ++ red; simpl; intros; lia. +- intuition lia. Qed. Lemma range_drop_left: @@ -392,12 +392,12 @@ Proof. assert (mid <= align mid al) by (apply align_le; auto). split; simpl; intros. - intuition auto. -+ omega. -+ apply H7; omega. -+ omega. -+ apply H7; omega. -+ red; simpl; intros; omega. -- intuition omega. ++ lia. ++ apply H7; lia. ++ lia. ++ apply H7; lia. ++ red; simpl; intros; lia. +- intuition lia. Qed. Lemma range_preserved: @@ -493,7 +493,7 @@ Proof. split; [|split]. - assert (Mem.valid_access m chunk b ofs Freeable). { split; auto. red; auto. } - split. generalize (size_chunk_pos chunk). unfold Ptrofs.max_unsigned. omega. + split. generalize (size_chunk_pos chunk). unfold Ptrofs.max_unsigned. lia. split. auto. + destruct (Mem.valid_access_load m chunk b ofs) as [v LOAD]. eauto with mem. @@ -616,7 +616,7 @@ Next Obligation. assert (IMG: forall b1 b2 delta ofs k p, j b1 = Some (b2, delta) -> Mem.perm m0 b1 ofs k p -> img b2 (ofs + delta)). { intros. red. exists b1, delta; split; auto. - replace (ofs + delta - delta) with ofs by omega. + replace (ofs + delta - delta) with ofs by lia. eauto with mem. } destruct H. constructor. - destruct mi_inj. constructor; intros. @@ -668,7 +668,7 @@ Proof. intros; red; intros. eelim C; eauto. simpl. exists b1, delta; split; auto. destruct VALID as [V1 V2]. apply Mem.perm_cur_max. apply Mem.perm_implies with Writable; auto with mem. - apply V1. omega. + apply V1. lia. - red; simpl; intros. destruct H1 as (b0 & delta0 & U & V). eelim C; eauto. simpl. exists b0, delta0; eauto with mem. Qed. @@ -690,7 +690,7 @@ Lemma alloc_parallel_rule: /\ (forall b, b <> b1 -> j' b = j b). Proof. intros until delta; intros SEP ALLOC1 ALLOC2 ALIGN LO HI RANGE1 RANGE2 RANGE3. - assert (RANGE4: lo <= hi) by xomega. + assert (RANGE4: lo <= hi) by extlia. assert (FRESH1: ~Mem.valid_block m1 b1) by (eapply Mem.fresh_block_alloc; eauto). assert (FRESH2: ~Mem.valid_block m2 b2) by (eapply Mem.fresh_block_alloc; eauto). destruct SEP as (INJ & SP & DISJ). simpl in INJ. @@ -698,10 +698,10 @@ Proof. - eapply Mem.alloc_right_inject; eauto. - eexact ALLOC1. - instantiate (1 := b2). eauto with mem. -- instantiate (1 := delta). xomega. -- intros. assert (0 <= ofs < sz2) by (eapply Mem.perm_alloc_3; eauto). omega. +- instantiate (1 := delta). extlia. +- intros. assert (0 <= ofs < sz2) by (eapply Mem.perm_alloc_3; eauto). lia. - intros. apply Mem.perm_implies with Freeable; auto with mem. - eapply Mem.perm_alloc_2; eauto. xomega. + eapply Mem.perm_alloc_2; eauto. extlia. - red; intros. apply Z.divide_trans with 8; auto. exists (8 / align_chunk chunk). destruct chunk; reflexivity. - intros. elim FRESH2. eapply Mem.valid_block_inject_2; eauto. @@ -709,19 +709,19 @@ Proof. exists j'; split; auto. rewrite <- ! sep_assoc. split; [|split]. -+ simpl. intuition auto; try (unfold Ptrofs.max_unsigned in *; omega). ++ simpl. intuition auto; try (unfold Ptrofs.max_unsigned in *; lia). * apply Mem.perm_implies with Freeable; auto with mem. - eapply Mem.perm_alloc_2; eauto. omega. + eapply Mem.perm_alloc_2; eauto. lia. * apply Mem.perm_implies with Freeable; auto with mem. - eapply Mem.perm_alloc_2; eauto. omega. -* red; simpl; intros. destruct H1, H2. omega. + eapply Mem.perm_alloc_2; eauto. lia. +* red; simpl; intros. destruct H1, H2. lia. * red; simpl; intros. assert (b = b2) by tauto. subst b. assert (0 <= ofs < lo \/ hi <= ofs < sz2) by tauto. clear H1. destruct H2 as (b0 & delta0 & D & E). eapply Mem.perm_alloc_inv in E; eauto. destruct (eq_block b0 b1). - subst b0. rewrite J2 in D. inversion D; clear D; subst delta0. xomega. + subst b0. rewrite J2 in D. inversion D; clear D; subst delta0. extlia. rewrite J3 in D by auto. elim FRESH2. eapply Mem.valid_block_inject_2; eauto. + apply (m_invar P) with m2; auto. eapply Mem.alloc_unchanged_on; eauto. + red; simpl; intros. @@ -753,11 +753,11 @@ Proof. simpl in E. assert (PERM: Mem.range_perm m2 b2 0 sz2 Cur Freeable). { red; intros. - destruct (zlt ofs lo). apply J; omega. - destruct (zle hi ofs). apply K; omega. - replace ofs with ((ofs - delta) + delta) by omega. + destruct (zlt ofs lo). apply J; lia. + destruct (zle hi ofs). apply K; lia. + replace ofs with ((ofs - delta) + delta) by lia. eapply Mem.perm_inject; eauto. - eapply Mem.free_range_perm; eauto. xomega. + eapply Mem.free_range_perm; eauto. extlia. } destruct (Mem.range_perm_free _ _ _ _ PERM) as [m2' FREE]. exists m2'; split; auto. split; [|split]. @@ -768,16 +768,16 @@ Proof. destruct (zle hi (ofs + delta0)). intuition auto. destruct (eq_block b0 b1). * subst b0. rewrite H1 in H; inversion H; clear H; subst delta0. - eelim (Mem.perm_free_2 m1); eauto. xomega. + eelim (Mem.perm_free_2 m1); eauto. extlia. * exploit Mem.mi_no_overlap; eauto. apply Mem.perm_max with k. apply Mem.perm_implies with p; auto with mem. eapply Mem.perm_free_3; eauto. apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable; auto with mem. eapply (Mem.free_range_perm m1); eauto. - instantiate (1 := ofs + delta0 - delta). xomega. - intros [X|X]. congruence. omega. + instantiate (1 := ofs + delta0 - delta). extlia. + intros [X|X]. congruence. lia. + simpl. exists b0, delta0; split; auto. - replace (ofs + delta0 - delta0) with ofs by omega. + replace (ofs + delta0 - delta0) with ofs by lia. apply Mem.perm_max with k. apply Mem.perm_implies with p; auto with mem. eapply Mem.perm_free_3; eauto. - apply (m_invar P) with m2; auto. @@ -787,7 +787,7 @@ Proof. destruct (zle hi i). intuition auto. right; exists b1, delta; split; auto. apply Mem.perm_cur_max. apply Mem.perm_implies with Freeable; auto with mem. - eapply Mem.free_range_perm; eauto. xomega. + eapply Mem.free_range_perm; eauto. extlia. - red; simpl; intros. eelim C; eauto. simpl. right. destruct H as (b0 & delta0 & U & V). exists b0, delta0; split; auto. diff --git a/common/Smallstep.v b/common/Smallstep.v index 27ad0a2d..5ac67c96 100644 --- a/common/Smallstep.v +++ b/common/Smallstep.v @@ -893,8 +893,8 @@ Proof. exploit (sd_traces DET). eexact H3. intros L2. assert (t1 = t0 /\ t2 = t3). destruct t1. inv MT. auto. - destruct t1; simpl in L1; try omegaContradiction. - destruct t0. inv MT. destruct t0; simpl in L2; try omegaContradiction. + destruct t1; simpl in L1; try extlia. + destruct t0. inv MT. destruct t0; simpl in L2; try extlia. simpl in H5. split. congruence. congruence. destruct H1; subst. assert (s2 = s4) by (eapply sd_determ_2; eauto). subst s4. @@ -974,7 +974,7 @@ Proof. destruct C as [P | [P Q]]; auto using lex_ord_left. + exploit sd_determ_3. eauto. eexact A. eauto. intros [P Q]; subst t s1'0. exists (i, n), s2; split; auto. - right; split. apply star_refl. apply lex_ord_right. omega. + right; split. apply star_refl. apply lex_ord_right. lia. - exact public_preserved. Qed. @@ -1256,7 +1256,7 @@ Proof. subst t. assert (EITHER: t1 = E0 \/ t2 = E0). unfold Eapp in H2; rewrite app_length in H2. - destruct t1; auto. destruct t2; auto. simpl in H2; omegaContradiction. + destruct t1; auto. destruct t2; auto. simpl in H2; extlia. destruct EITHER; subst. exploit IHstar; eauto. intros [s2x [s2y [A [B C]]]]. exists s2x; exists s2y; intuition. eapply star_left; eauto. @@ -1305,7 +1305,7 @@ Proof. - (* 1 L2 makes one or several transitions *) assert (EITHER: t = E0 \/ (length t = 1)%nat). { exploit L3_single_events; eauto. - destruct t; auto. destruct t; auto. simpl. intros. omegaContradiction. } + destruct t; auto. destruct t; auto. simpl. intros. extlia. } destruct EITHER. + (* 1.1 these are silent transitions *) subst t. exploit (bsim_E0_plus S12); eauto. @@ -1473,7 +1473,7 @@ Remark not_silent_length: forall t1 t2, (length (t1 ** t2) <= 1)%nat -> t1 = E0 \/ t2 = E0. Proof. unfold Eapp, E0; intros. rewrite app_length in H. - destruct t1; destruct t2; auto. simpl in H. omegaContradiction. + destruct t1; destruct t2; auto. simpl in H. extlia. Qed. Lemma f2b_determinacy_inv: @@ -1622,7 +1622,7 @@ Proof. intros [[EQ1 [EQ2 EQ3]] | [NOT1 [NOT2 MT]]]. + (* 2.1 L2 makes a silent transition: remain in "before" state *) subst. simpl in *. exists (F2BI_before n0); exists s1; split. - right; split. apply star_refl. constructor. omega. + right; split. apply star_refl. constructor. lia. econstructor; eauto. eapply star_right; eauto. + (* 2.2 L2 make a non-silent transition *) exploit not_silent_length. eapply (sr_traces L1_receptive); eauto. intros [EQ | EQ]. @@ -1650,7 +1650,7 @@ Proof. exploit f2b_determinacy_inv. eexact H2. eexact STEP2. intros [[EQ1 [EQ2 EQ3]] | [NOT1 [NOT2 MT]]]. subst. exists (F2BI_after n); exists s1; split. - right; split. apply star_refl. constructor; omega. + right; split. apply star_refl. constructor; lia. eapply f2b_match_after'; eauto. congruence. Qed. @@ -1763,7 +1763,7 @@ Proof. destruct IHstar as [s2x [A B]]. exists s2x; split; auto. eapply plus_left. eauto. apply plus_star; eauto. auto. destruct t1. simpl in *. subst t. exists s2; split; auto. apply plus_one; auto. - simpl in LEN. omegaContradiction. + simpl in LEN. extlia. Qed. Lemma ffs_simulation: @@ -1955,7 +1955,7 @@ Proof. assert (t2 = ev :: nil). inv H1; simpl in H0; tauto. subst t2. exists (t, s0). constructor; auto. simpl; auto. (* single-event *) - red. intros. inv H0; simpl; omega. + red. intros. inv H0; simpl; lia. Qed. (** * Connections with big-step semantics *) diff --git a/common/Subtyping.v b/common/Subtyping.v index 26b282e0..f1047d45 100644 --- a/common/Subtyping.v +++ b/common/Subtyping.v @@ -222,7 +222,7 @@ Definition weight_bounds (ob: option bounds) : nat := Lemma weight_bounds_1: forall lo hi s, weight_bounds (Some (B lo hi s)) < weight_bounds None. Proof. - intros; simpl. generalize (T.weight_range hi); omega. + intros; simpl. generalize (T.weight_range hi); lia. Qed. Lemma weight_bounds_2: @@ -233,8 +233,8 @@ Proof. intros; simpl. generalize (T.weight_sub _ _ s1) (T.weight_sub _ _ s2) (T.weight_sub _ _ H) (T.weight_sub _ _ H0); intros. destruct H1. - assert (T.weight lo2 < T.weight lo1) by (apply T.weight_sub_strict; auto). omega. - assert (T.weight hi1 < T.weight hi2) by (apply T.weight_sub_strict; auto). omega. + assert (T.weight lo2 < T.weight lo1) by (apply T.weight_sub_strict; auto). lia. + assert (T.weight hi1 < T.weight hi2) by (apply T.weight_sub_strict; auto). lia. Qed. Hint Resolve T.sub_refl: ty. @@ -250,11 +250,11 @@ Lemma weight_type_move: Proof. unfold type_move; intros. destruct (peq r1 r2). - inv H. split; auto. split; intros. omega. discriminate. + inv H. split; auto. split; intros. lia. discriminate. destruct (te_typ e)!r1 as [[lo1 hi1 s1]|] eqn:E1; destruct (te_typ e)!r2 as [[lo2 hi2 s2]|] eqn:E2. - destruct (T.sub_dec hi1 lo2). - inv H. split; auto. split; intros. omega. discriminate. + inv H. split; auto. split; intros. lia. discriminate. destruct (T.sub_dec lo1 hi2); try discriminate. set (lo2' := T.lub lo1 lo2) in *. set (hi1' := T.glb hi1 hi2) in *. @@ -264,45 +264,45 @@ Proof. set (b2 := B lo2' hi2 (T.lub_min lo1 lo2 hi2 s s2)) in *. Local Opaque weight_bounds. destruct (T.eq lo2 lo2'); destruct (T.eq hi1 hi1'); inversion H; clear H; subst changed e'; simpl. -+ split; auto. split; intros. omega. discriminate. ++ split; auto. split; intros. lia. discriminate. + assert (weight_bounds (Some b1) < weight_bounds (Some (B lo1 hi1 s1))) by (apply weight_bounds_2; auto with ty). split; auto. split; intros. - rewrite PTree.gsspec. destruct (peq r r1). subst r. rewrite E1. omega. omega. - rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E2. omega. + rewrite PTree.gsspec. destruct (peq r r1). subst r. rewrite E1. lia. lia. + rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E2. lia. + assert (weight_bounds (Some b2) < weight_bounds (Some (B lo2 hi2 s2))) by (apply weight_bounds_2; auto with ty). split; auto. split; intros. - rewrite PTree.gsspec. destruct (peq r r2). subst r. rewrite E2. omega. omega. - rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E1. omega. + rewrite PTree.gsspec. destruct (peq r r2). subst r. rewrite E2. lia. lia. + rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E1. lia. + assert (weight_bounds (Some b1) < weight_bounds (Some (B lo1 hi1 s1))) by (apply weight_bounds_2; auto with ty). assert (weight_bounds (Some b2) < weight_bounds (Some (B lo2 hi2 s2))) by (apply weight_bounds_2; auto with ty). split; auto. split; intros. rewrite ! PTree.gsspec. - destruct (peq r r2). subst r. rewrite E2. omega. - destruct (peq r r1). subst r. rewrite E1. omega. - omega. - rewrite PTree.gss. rewrite PTree.gso by auto. rewrite PTree.gss. omega. + destruct (peq r r2). subst r. rewrite E2. lia. + destruct (peq r r1). subst r. rewrite E1. lia. + lia. + rewrite PTree.gss. rewrite PTree.gso by auto. rewrite PTree.gss. lia. - set (b2 := B lo1 (T.high_bound lo1) (T.high_bound_sub lo1)) in *. assert (weight_bounds (Some b2) < weight_bounds None) by (apply weight_bounds_1). inv H; simpl. split. destruct (T.sub_dec hi1 lo1); auto. split; intros. - rewrite PTree.gsspec. destruct (peq r r2). subst r; rewrite E2; omega. omega. - rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E1. omega. + rewrite PTree.gsspec. destruct (peq r r2). subst r; rewrite E2; lia. lia. + rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E1. lia. - set (b1 := B (T.low_bound hi2) hi2 (T.low_bound_sub hi2)) in *. assert (weight_bounds (Some b1) < weight_bounds None) by (apply weight_bounds_1). inv H; simpl. split. destruct (T.sub_dec hi2 lo2); auto. split; intros. - rewrite PTree.gsspec. destruct (peq r r1). subst r; rewrite E1; omega. omega. - rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E2. omega. + rewrite PTree.gsspec. destruct (peq r r1). subst r; rewrite E1; lia. lia. + rewrite PTree.gss. rewrite PTree.gso by auto. rewrite E2. lia. -- inv H. split; auto. simpl; split; intros. omega. congruence. +- inv H. split; auto. simpl; split; intros. lia. congruence. Qed. Definition weight_constraints (b: PTree.t bounds) (cstr: list constraint) : nat := @@ -312,7 +312,7 @@ Remark weight_constraints_tighter: forall b1 b2, (forall r, weight_bounds b1!r <= weight_bounds b2!r) -> forall q, weight_constraints b1 q <= weight_constraints b2 q. Proof. - induction q; simpl. omega. generalize (H (fst a)) (H (snd a)); omega. + induction q; simpl. lia. generalize (H (fst a)) (H (snd a)); lia. Qed. Lemma weight_solve_rec: @@ -323,8 +323,8 @@ Lemma weight_solve_rec: <= weight_constraints e.(te_typ) e.(te_sub) + weight_constraints e.(te_typ) q. Proof. induction q; simpl; intros. -- inv H. split. intros; omega. replace (changed' && negb changed') with false. - omega. destruct changed'; auto. +- inv H. split. intros; lia. replace (changed' && negb changed') with false. + lia. destruct changed'; auto. - destruct a as [r1 r2]; monadInv H; simpl. rename x into changed1. rename x0 into e1. exploit weight_type_move; eauto. intros [A [B C]]. @@ -336,7 +336,7 @@ Proof. assert (Q: weight_constraints (te_typ e1) (te_sub e1) <= weight_constraints (te_typ e1) (te_sub e) + weight_bounds (te_typ e1)!r1 + weight_bounds (te_typ e1)!r2). - { destruct A as [Q|Q]; rewrite Q. omega. simpl. omega. } + { destruct A as [Q|Q]; rewrite Q. lia. simpl. lia. } assert (R: weight_constraints (te_typ e1) q <= weight_constraints (te_typ e) q) by (apply weight_constraints_tighter; auto). set (ch1 := if changed' && negb (changed || changed1) then 1 else 0) in *. @@ -344,11 +344,11 @@ Proof. destruct changed1. assert (ch2 <= ch1 + 1). { unfold ch2, ch1. rewrite orb_true_r. simpl. rewrite andb_false_r. - destruct (changed' && negb changed); omega. } - exploit C; eauto. omega. + destruct (changed' && negb changed); lia. } + exploit C; eauto. lia. assert (ch2 <= ch1). - { unfold ch2, ch1. rewrite orb_false_r. omega. } - generalize (B r1) (B r2); omega. + { unfold ch2, ch1. rewrite orb_false_r. lia. } + generalize (B r1) (B r2); lia. Qed. Definition weight_typenv (e: typenv) : nat := @@ -364,7 +364,7 @@ Function solve_constraints (e: typenv) {measure weight_typenv e}: res typenv := end. Proof. intros. exploit weight_solve_rec; eauto. simpl. intros [A B]. - unfold weight_typenv. omega. + unfold weight_typenv. lia. Qed. Definition typassign := positive -> T.t. diff --git a/common/Switch.v b/common/Switch.v index 5a6d4c63..748aa459 100644 --- a/common/Switch.v +++ b/common/Switch.v @@ -235,8 +235,8 @@ Proof. destruct (split_lt n cases) as [lc rc] eqn:SEQ. rewrite (IHcases lc rc) by auto. destruct (zlt key n); intros EQ; inv EQ; simpl. -+ destruct (zeq v key). rewrite zlt_true by omega. auto. auto. -+ destruct (zeq v key). rewrite zlt_false by omega. auto. auto. ++ destruct (zeq v key). rewrite zlt_true by lia. auto. auto. ++ destruct (zeq v key). rewrite zlt_false by lia. auto. auto. Qed. Lemma split_between_prop: @@ -269,12 +269,12 @@ Lemma validate_jumptable_correct_rec: list_nth_z tbl v = Some(ZMap.get (base + v) cases). Proof. induction tbl; simpl; intros. -- unfold list_length_z in H0. simpl in H0. omegaContradiction. +- unfold list_length_z in H0. simpl in H0. extlia. - InvBooleans. rewrite list_length_z_cons in H0. apply beq_nat_true in H1. destruct (zeq v 0). - + replace (base + v) with base by omega. congruence. - + replace (base + v) with (Z.succ base + Z.pred v) by omega. - apply IHtbl. auto. omega. + + replace (base + v) with base by lia. congruence. + + replace (base + v) with (Z.succ base + Z.pred v) by lia. + apply IHtbl. auto. lia. Qed. Lemma validate_jumptable_correct: @@ -288,12 +288,12 @@ Lemma validate_jumptable_correct: Proof. intros. rewrite (validate_jumptable_correct_rec cases tbl ofs); auto. -- f_equal. f_equal. rewrite Z.mod_small. omega. - destruct (zle ofs v). omega. +- f_equal. f_equal. rewrite Z.mod_small. lia. + destruct (zle ofs v). lia. assert (M: ((v - ofs) + 1 * modulus) mod modulus = (v - ofs) + modulus). - { rewrite Z.mod_small. omega. omega. } - rewrite Z_mod_plus in M by auto. rewrite M in H0. omega. -- generalize (Z_mod_lt (v - ofs) modulus modulus_pos). omega. + { rewrite Z.mod_small. lia. lia. } + rewrite Z_mod_plus in M by auto. rewrite M in H0. lia. +- generalize (Z_mod_lt (v - ofs) modulus modulus_pos). lia. Qed. Lemma validate_correct_rec: @@ -309,7 +309,7 @@ Proof. destruct cases as [ | [key1 act1] cases1]; intros. + apply beq_nat_true in H. subst act. reflexivity. + InvBooleans. apply beq_nat_true in H2. subst. simpl. - destruct (zeq v hi). auto. omegaContradiction. + destruct (zeq v hi). auto. extlia. - (* eq node *) destruct (split_eq key cases) as [optact others] eqn:EQ. intros. destruct optact as [act1|]; InvBooleans; try discriminate. @@ -319,19 +319,19 @@ Proof. + congruence. + eapply IHt; eauto. unfold refine_low_bound, refine_high_bound. split. - destruct (zeq key lo); omega. - destruct (zeq key hi); omega. + destruct (zeq key lo); lia. + destruct (zeq key hi); lia. - (* lt node *) destruct (split_lt key cases) as [lcases rcases] eqn:EQ; intros; InvBooleans. rewrite (split_lt_prop v default _ _ _ _ EQ). destruct (zlt v key). - eapply IHt1. eauto. omega. - eapply IHt2. eauto. omega. + eapply IHt1. eauto. lia. + eapply IHt2. eauto. lia. - (* jumptable node *) destruct (split_between default ofs sz cases) as [ins outs] eqn:EQ; intros; InvBooleans. rewrite (split_between_prop v _ _ _ _ _ _ EQ). - assert (0 <= (v - ofs) mod modulus < modulus) by (apply Z_mod_lt; omega). + assert (0 <= (v - ofs) mod modulus < modulus) by (apply Z_mod_lt; lia). destruct (zlt ((v - ofs) mod modulus) sz). - rewrite Z.mod_small by omega. eapply validate_jumptable_correct; eauto. + rewrite Z.mod_small by lia. eapply validate_jumptable_correct; eauto. eapply IHt; eauto. Qed. @@ -346,7 +346,7 @@ Theorem validate_switch_correct: Proof. unfold validate_switch, table_tree_agree; split. eapply validate_wf; eauto. - intros; eapply validate_correct_rec; eauto. omega. + intros; eapply validate_correct_rec; eauto. lia. Qed. End COMPTREE. diff --git a/common/Unityping.v b/common/Unityping.v index 28bcfb5c..878e5943 100644 --- a/common/Unityping.v +++ b/common/Unityping.v @@ -126,12 +126,12 @@ Lemma length_move: length e'.(te_equ) + (if changed then 1 else 0) <= S(length e.(te_equ)). Proof. unfold move; intros. - destruct (peq r1 r2). inv H. omega. + destruct (peq r1 r2). inv H. lia. destruct e.(te_typ)!r1 as [ty1|]; destruct e.(te_typ)!r2 as [ty2|]; inv H; simpl. - destruct (T.eq ty1 ty2); inv H1. omega. - omega. - omega. - omega. + destruct (T.eq ty1 ty2); inv H1. lia. + lia. + lia. + lia. Qed. Lemma length_solve_rec: @@ -140,14 +140,14 @@ Lemma length_solve_rec: length e'.(te_equ) + (if ch' && negb ch then 1 else 0) <= length e.(te_equ) + length q. Proof. induction q; simpl; intros. -- inv H. replace (ch' && negb ch') with false. omega. destruct ch'; auto. +- inv H. replace (ch' && negb ch') with false. lia. destruct ch'; auto. - destruct a as [r1 r2]; monadInv H. rename x0 into e0. rename x into ch0. exploit IHq; eauto. intros A. exploit length_move; eauto. intros B. set (X := (if ch' && negb (ch || ch0) then 1 else 0)) in *. set (Y := (if ch0 then 1 else 0)) in *. set (Z := (if ch' && negb ch then 1 else 0)) in *. - cut (Z <= X + Y). intros. omega. + cut (Z <= X + Y). intros. lia. unfold X, Y, Z. destruct ch'; destruct ch; destruct ch0; simpl; auto. Qed. @@ -164,7 +164,7 @@ Function solve_constraints (e: typenv) {measure weight_typenv e}: res typenv := end. Proof. intros. exploit length_solve_rec; eauto. simpl. intros. - unfold weight_typenv. omega. + unfold weight_typenv. lia. Qed. Definition typassign := positive -> T.t. diff --git a/common/Values.v b/common/Values.v index 68a2054b..c5b07e2f 100644 --- a/common/Values.v +++ b/common/Values.v @@ -1024,10 +1024,10 @@ Lemma load_result_rettype: forall chunk v, has_rettype (load_result chunk v) (rettype_of_chunk chunk). Proof. intros. unfold has_rettype; destruct chunk; destruct v; simpl; auto. -- rewrite Int.sign_ext_idem by omega; auto. -- rewrite Int.zero_ext_idem by omega; auto. -- rewrite Int.sign_ext_idem by omega; auto. -- rewrite Int.zero_ext_idem by omega; auto. +- rewrite Int.sign_ext_idem by lia; auto. +- rewrite Int.zero_ext_idem by lia; auto. +- rewrite Int.sign_ext_idem by lia; auto. +- rewrite Int.zero_ext_idem by lia; auto. - destruct Archi.ptr64 eqn:SF; simpl; auto. - destruct Archi.ptr64 eqn:SF; simpl; auto. - destruct Archi.ptr64 eqn:SF; simpl; auto. @@ -1053,14 +1053,14 @@ Theorem cast8unsigned_and: forall x, zero_ext 8 x = and x (Vint(Int.repr 255)). Proof. destruct x; simpl; auto. decEq. - change 255 with (two_p 8 - 1). apply Int.zero_ext_and. omega. + change 255 with (two_p 8 - 1). apply Int.zero_ext_and. lia. Qed. Theorem cast16unsigned_and: forall x, zero_ext 16 x = and x (Vint(Int.repr 65535)). Proof. destruct x; simpl; auto. decEq. - change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. omega. + change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. lia. Qed. Theorem bool_of_val_of_bool: @@ -1297,7 +1297,7 @@ Proof. unfold divs. rewrite Int.eq_false; try discriminate. simpl. rewrite (Int.eq_false Int.one Int.mone); try discriminate. rewrite andb_false_intro2; auto. f_equal. f_equal. - rewrite Int.divs_one; auto. replace Int.zwordsize with 32; auto. omega. + rewrite Int.divs_one; auto. replace Int.zwordsize with 32; auto. lia. Qed. Theorem divu_pow2: @@ -1424,7 +1424,7 @@ Proof. destruct (Int.ltu i0 (Int.repr 31)) eqn:?; inv H1. exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 31)) with 31. intros. assert (Int.ltu i0 Int.iwordsize = true). - unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. omega. + unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. lia. simpl. rewrite H0. simpl. decEq. rewrite Int.shrx_carry; auto. Qed. @@ -1439,7 +1439,7 @@ Proof. destruct (Int.ltu i0 (Int.repr 31)) eqn:?; inv H1. exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 31)) with 31. intros. assert (Int.ltu i0 Int.iwordsize = true). - unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. omega. + unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. lia. exists i; exists i0; intuition. rewrite Int.shrx_shr; auto. destruct (Int.lt i Int.zero); simpl; rewrite H0; auto. Qed. @@ -1462,12 +1462,12 @@ Proof. replace (Int.ltu (Int.sub (Int.repr 32) n) Int.iwordsize) with true. simpl. replace (Int.ltu n Int.iwordsize) with true. f_equal; apply Int.shrx_shr_2; assumption. - symmetry; apply zlt_true. change (Int.unsigned n < 32); omega. + symmetry; apply zlt_true. change (Int.unsigned n < 32); lia. symmetry; apply zlt_true. unfold Int.sub. change (Int.unsigned (Int.repr 32)) with 32. assert (Int.unsigned n <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned n), H0. auto. } rewrite Int.unsigned_repr. - change (Int.unsigned Int.iwordsize) with 32; omega. - assert (32 < Int.max_unsigned) by reflexivity. omega. + change (Int.unsigned Int.iwordsize) with 32; lia. + assert (32 < Int.max_unsigned) by reflexivity. lia. Qed. Theorem or_rolm: @@ -1657,7 +1657,7 @@ Proof. rewrite (Int64.eq_false Int64.one Int64.mone); try discriminate. rewrite andb_false_intro2; auto. simpl. f_equal. f_equal. apply Int64.divs_one. - replace Int64.zwordsize with 64; auto. omega. + replace Int64.zwordsize with 64; auto. lia. Qed. Theorem divlu_pow2: @@ -1700,7 +1700,7 @@ Proof. destruct (Int.ltu i0 (Int.repr 63)) eqn:?; inv H1. exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 63)) with 63. intros. assert (Int.ltu i0 Int64.iwordsize' = true). - unfold Int.ltu. apply zlt_true. change (Int.unsigned Int64.iwordsize') with 64. omega. + unfold Int.ltu. apply zlt_true. change (Int.unsigned Int64.iwordsize') with 64. lia. simpl. rewrite H0. simpl. decEq. rewrite Int64.shrx'_carry; auto. Qed. @@ -1721,12 +1721,12 @@ Proof. replace (Int.ltu (Int.sub (Int.repr 64) n) Int64.iwordsize') with true. simpl. replace (Int.ltu n Int64.iwordsize') with true. f_equal; apply Int64.shrx'_shr_2; assumption. - symmetry; apply zlt_true. change (Int.unsigned n < 64); omega. + symmetry; apply zlt_true. change (Int.unsigned n < 64); lia. symmetry; apply zlt_true. unfold Int.sub. change (Int.unsigned (Int.repr 64)) with 64. assert (Int.unsigned n <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned n), H0. auto. } rewrite Int.unsigned_repr. - change (Int.unsigned Int64.iwordsize') with 64; omega. - assert (64 < Int.max_unsigned) by reflexivity. omega. + change (Int.unsigned Int64.iwordsize') with 64; lia. + assert (64 < Int.max_unsigned) by reflexivity. lia. Qed. Theorem negate_cmp_bool: diff --git a/flocq/Core/Raux.v b/flocq/Core/Raux.v index 58d1a630..455190dc 100644 --- a/flocq/Core/Raux.v +++ b/flocq/Core/Raux.v @@ -18,7 +18,7 @@ COPYING file for more details. *) (** * Missing definitions/lemmas *) -Require Import Psatz. +Require Export Psatz. Require Export Reals ZArith. Require Export Zaux. diff --git a/lib/Coqlib.v b/lib/Coqlib.v index 02c5d07f..948f128e 100644 --- a/lib/Coqlib.v +++ b/lib/Coqlib.v @@ -22,6 +22,7 @@ Require Export ZArith. Require Export Znumtheory. Require Export List. Require Export Bool. +Require Export Lia. Global Set Asymmetric Patterns. @@ -45,11 +46,7 @@ Ltac decEq := cut (A <> B); [intro; congruence | try discriminate] end. -Ltac byContradiction := - cut False; [contradiction|idtac]. - -Ltac omegaContradiction := - cut False; [contradiction|omega]. +Ltac byContradiction := exfalso. Lemma modusponens: forall (P Q: Prop), P -> (P -> Q) -> Q. Proof. auto. Qed. @@ -180,8 +177,7 @@ Proof (Pos.lt_irrefl). Hint Resolve Ple_refl Plt_Ple Ple_succ Plt_strict: coqlib. -Ltac xomega := unfold Plt, Ple in *; zify; omega. -Ltac xomegaContradiction := exfalso; xomega. +Ltac extlia := unfold Plt, Ple in *; lia. (** Peano recursion over positive numbers. *) @@ -284,7 +280,7 @@ Lemma zlt_true: Proof. intros. case (zlt x y); intros. auto. - omegaContradiction. + extlia. Qed. Lemma zlt_false: @@ -292,7 +288,7 @@ Lemma zlt_false: x >= y -> (if zlt x y then a else b) = b. Proof. intros. case (zlt x y); intros. - omegaContradiction. + extlia. auto. Qed. @@ -304,7 +300,7 @@ Lemma zle_true: Proof. intros. case (zle x y); intros. auto. - omegaContradiction. + extlia. Qed. Lemma zle_false: @@ -312,7 +308,7 @@ Lemma zle_false: x > y -> (if zle x y then a else b) = b. Proof. intros. case (zle x y); intros. - omegaContradiction. + extlia. auto. Qed. @@ -323,54 +319,54 @@ Proof. reflexivity. Qed. Lemma two_power_nat_pos : forall n : nat, two_power_nat n > 0. Proof. - induction n. rewrite two_power_nat_O. omega. - rewrite two_power_nat_S. omega. + induction n. rewrite two_power_nat_O. lia. + rewrite two_power_nat_S. lia. Qed. Lemma two_power_nat_two_p: forall x, two_power_nat x = two_p (Z.of_nat x). Proof. induction x. auto. - rewrite two_power_nat_S. rewrite Nat2Z.inj_succ. rewrite two_p_S. omega. omega. + rewrite two_power_nat_S. rewrite Nat2Z.inj_succ. rewrite two_p_S. lia. lia. Qed. Lemma two_p_monotone: forall x y, 0 <= x <= y -> two_p x <= two_p y. Proof. intros. - replace (two_p x) with (two_p x * 1) by omega. - replace y with (x + (y - x)) by omega. - rewrite two_p_is_exp; try omega. + replace (two_p x) with (two_p x * 1) by lia. + replace y with (x + (y - x)) by lia. + rewrite two_p_is_exp; try lia. apply Zmult_le_compat_l. - assert (two_p (y - x) > 0). apply two_p_gt_ZERO. omega. omega. - assert (two_p x > 0). apply two_p_gt_ZERO. omega. omega. + assert (two_p (y - x) > 0). apply two_p_gt_ZERO. lia. lia. + assert (two_p x > 0). apply two_p_gt_ZERO. lia. lia. Qed. Lemma two_p_monotone_strict: forall x y, 0 <= x < y -> two_p x < two_p y. Proof. - intros. assert (two_p x <= two_p (y - 1)). apply two_p_monotone; omega. - assert (two_p (y - 1) > 0). apply two_p_gt_ZERO. omega. - replace y with (Z.succ (y - 1)) by omega. rewrite two_p_S. omega. omega. + intros. assert (two_p x <= two_p (y - 1)). apply two_p_monotone; lia. + assert (two_p (y - 1) > 0). apply two_p_gt_ZERO. lia. + replace y with (Z.succ (y - 1)) by lia. rewrite two_p_S. lia. lia. Qed. Lemma two_p_strict: forall x, x >= 0 -> x < two_p x. Proof. intros x0 GT. pattern x0. apply natlike_ind. - simpl. omega. - intros. rewrite two_p_S; auto. generalize (two_p_gt_ZERO x H). omega. - omega. + simpl. lia. + intros. rewrite two_p_S; auto. generalize (two_p_gt_ZERO x H). lia. + lia. Qed. Lemma two_p_strict_2: forall x, x >= 0 -> 2 * x - 1 < two_p x. Proof. - intros. assert (x = 0 \/ x - 1 >= 0) by omega. destruct H0. + intros. assert (x = 0 \/ x - 1 >= 0) by lia. destruct H0. subst. vm_compute. auto. replace (two_p x) with (2 * two_p (x - 1)). - generalize (two_p_strict _ H0). omega. - rewrite <- two_p_S. decEq. omega. omega. + generalize (two_p_strict _ H0). lia. + rewrite <- two_p_S. decEq. lia. lia. Qed. (** Properties of [Zmin] and [Zmax] *) @@ -401,12 +397,12 @@ Qed. Lemma Zmax_bound_l: forall x y z, x <= y -> x <= Z.max y z. Proof. - intros. generalize (Z.le_max_l y z). omega. + intros. generalize (Z.le_max_l y z). lia. Qed. Lemma Zmax_bound_r: forall x y z, x <= z -> x <= Z.max y z. Proof. - intros. generalize (Z.le_max_r y z). omega. + intros. generalize (Z.le_max_r y z). lia. Qed. (** Properties of Euclidean division and modulus. *) @@ -416,7 +412,7 @@ Lemma Zmod_unique: x = a * y + b -> 0 <= b < y -> x mod y = b. Proof. intros. subst x. rewrite Z.add_comm. - rewrite Z_mod_plus. apply Z.mod_small. auto. omega. + rewrite Z_mod_plus. apply Z.mod_small. auto. lia. Qed. Lemma Zdiv_unique: @@ -424,14 +420,14 @@ Lemma Zdiv_unique: x = a * y + b -> 0 <= b < y -> x / y = a. Proof. intros. subst x. rewrite Z.add_comm. - rewrite Z_div_plus. rewrite (Zdiv_small b y H0). omega. omega. + rewrite Z_div_plus. rewrite (Zdiv_small b y H0). lia. lia. Qed. Lemma Zdiv_Zdiv: forall a b c, b > 0 -> c > 0 -> (a / b) / c = a / (b * c). Proof. - intros. apply Z.div_div; omega. + intros. apply Z.div_div; lia. Qed. Lemma Zdiv_interval_1: @@ -445,14 +441,14 @@ Proof. set (q := a/b) in *. set (r := a mod b) in *. split. assert (lo < (q + 1)). - apply Zmult_lt_reg_r with b. omega. - apply Z.le_lt_trans with a. omega. + apply Zmult_lt_reg_r with b. lia. + apply Z.le_lt_trans with a. lia. replace ((q + 1) * b) with (b * q + b) by ring. - omega. - omega. - apply Zmult_lt_reg_r with b. omega. + lia. + lia. + apply Zmult_lt_reg_r with b. lia. replace (q * b) with (b * q) by ring. - omega. + lia. Qed. Lemma Zdiv_interval_2: @@ -462,13 +458,13 @@ Lemma Zdiv_interval_2: Proof. intros. assert (lo <= a / b < hi+1). - apply Zdiv_interval_1. omega. omega. auto. - assert (lo * b <= lo * 1) by (apply Z.mul_le_mono_nonpos_l; omega). + apply Zdiv_interval_1. lia. lia. auto. + assert (lo * b <= lo * 1) by (apply Z.mul_le_mono_nonpos_l; lia). replace (lo * 1) with lo in H3 by ring. - assert ((hi + 1) * 1 <= (hi + 1) * b) by (apply Z.mul_le_mono_nonneg_l; omega). + assert ((hi + 1) * 1 <= (hi + 1) * b) by (apply Z.mul_le_mono_nonneg_l; lia). replace ((hi + 1) * 1) with (hi + 1) in H4 by ring. - omega. - omega. + lia. + lia. Qed. Lemma Zmod_recombine: @@ -476,7 +472,7 @@ Lemma Zmod_recombine: a > 0 -> b > 0 -> x mod (a * b) = ((x/b) mod a) * b + (x mod b). Proof. - intros. rewrite (Z.mul_comm a b). rewrite Z.rem_mul_r by omega. ring. + intros. rewrite (Z.mul_comm a b). rewrite Z.rem_mul_r by lia. ring. Qed. (** Properties of divisibility. *) @@ -486,9 +482,9 @@ Lemma Zdivide_interval: 0 < c -> 0 <= a < b -> (c | a) -> (c | b) -> 0 <= a <= b - c. Proof. intros. destruct H1 as [x EQ1]. destruct H2 as [y EQ2]. subst. destruct H0. - split. omega. exploit Zmult_lt_reg_r; eauto. intros. + split. lia. exploit Zmult_lt_reg_r; eauto. intros. replace (y * c - c) with ((y - 1) * c) by ring. - apply Zmult_le_compat_r; omega. + apply Zmult_le_compat_r; lia. Qed. (** Conversion from [Z] to [nat]. *) @@ -503,8 +499,8 @@ Lemma Z_to_nat_max: forall z, Z.of_nat (Z.to_nat z) = Z.max z 0. Proof. intros. destruct (zle 0 z). -- rewrite Z2Nat.id by auto. xomega. -- rewrite Z_to_nat_neg by omega. xomega. +- rewrite Z2Nat.id by auto. extlia. +- rewrite Z_to_nat_neg by lia. extlia. Qed. (** Alignment: [align n amount] returns the smallest multiple of [amount] @@ -519,8 +515,8 @@ Proof. generalize (Z_div_mod_eq (x + y - 1) y H). intro. replace ((x + y - 1) / y * y) with ((x + y - 1) - (x + y - 1) mod y). - generalize (Z_mod_lt (x + y - 1) y H). omega. - rewrite Z.mul_comm. omega. + generalize (Z_mod_lt (x + y - 1) y H). lia. + rewrite Z.mul_comm. lia. Qed. Lemma align_divides: forall x y, y > 0 -> (y | align x y). @@ -599,8 +595,8 @@ Remark list_length_z_aux_shift: list_length_z_aux l n = list_length_z_aux l m + (n - m). Proof. induction l; intros; simpl. - omega. - replace (n - m) with (Z.succ n - Z.succ m) by omega. auto. + lia. + replace (n - m) with (Z.succ n - Z.succ m) by lia. auto. Qed. Definition list_length_z (A: Type) (l: list A) : Z := @@ -611,15 +607,15 @@ Lemma list_length_z_cons: list_length_z (hd :: tl) = list_length_z tl + 1. Proof. intros. unfold list_length_z. simpl. - rewrite (list_length_z_aux_shift tl 1 0). omega. + rewrite (list_length_z_aux_shift tl 1 0). lia. Qed. Lemma list_length_z_pos: forall (A: Type) (l: list A), list_length_z l >= 0. Proof. - induction l; simpl. unfold list_length_z; simpl. omega. - rewrite list_length_z_cons. omega. + induction l; simpl. unfold list_length_z; simpl. lia. + rewrite list_length_z_cons. lia. Qed. Lemma list_length_z_map: @@ -663,8 +659,8 @@ Proof. induction l; simpl; intros. discriminate. rewrite list_length_z_cons. destruct (zeq n 0). - generalize (list_length_z_pos l); omega. - exploit IHl; eauto. omega. + generalize (list_length_z_pos l); lia. + exploit IHl; eauto. lia. Qed. (** Properties of [List.incl] (list inclusion). *) diff --git a/lib/Decidableplus.v b/lib/Decidableplus.v index 66dffb3a..73f080b6 100644 --- a/lib/Decidableplus.v +++ b/lib/Decidableplus.v @@ -126,14 +126,14 @@ Program Instance Decidable_ge_Z : forall (x y: Z), Decidable (x >= y) := { Decidable_witness := Z.geb x y }. Next Obligation. - rewrite Z.geb_le. intuition omega. + rewrite Z.geb_le. intuition lia. Qed. Program Instance Decidable_gt_Z : forall (x y: Z), Decidable (x > y) := { Decidable_witness := Z.gtb x y }. Next Obligation. - rewrite Z.gtb_lt. intuition omega. + rewrite Z.gtb_lt. intuition lia. Qed. Program Instance Decidable_divides : forall (x y: Z), Decidable (x | y) := { @@ -146,7 +146,7 @@ Next Obligation. destruct (Z.eq_dec x 0). subst x. rewrite Z.mul_0_r in EQ. subst y. reflexivity. assert (k = y / x). - { apply Zdiv_unique_full with 0. red; omega. rewrite EQ; ring. } + { apply Zdiv_unique_full with 0. red; lia. rewrite EQ; ring. } congruence. Qed. diff --git a/lib/Floats.v b/lib/Floats.v index b9050017..9d0d1668 100644 --- a/lib/Floats.v +++ b/lib/Floats.v @@ -108,7 +108,7 @@ Proof. { apply Digits.Zdigits_le_Zpower. rewrite <- H. rewrite Z.abs_eq; tauto. } destruct (zeq p' 0). - rewrite e. simpl; auto. -- rewrite Z2Pos.id by omega. omega. +- rewrite Z2Pos.id by lia. lia. Qed. (** Transform a Nan payload to a quiet Nan payload. *) @@ -117,7 +117,7 @@ Definition quiet_nan_64_payload (p: positive) := Z.to_pos (P_mod_two_p (Pos.lor p ((iter_nat xO 51 1%positive))) 52%nat). Lemma quiet_nan_64_proof: forall p, nan_pl 53 (quiet_nan_64_payload p) = true. -Proof. intros; apply normalized_nan; auto; omega. Qed. +Proof. intros; apply normalized_nan; auto; lia. Qed. Definition quiet_nan_64 (sp: bool * positive) : {x :float | is_nan _ _ x = true} := let (s, p) := sp in @@ -129,7 +129,7 @@ Definition quiet_nan_32_payload (p: positive) := Z.to_pos (P_mod_two_p (Pos.lor p ((iter_nat xO 22 1%positive))) 23%nat). Lemma quiet_nan_32_proof: forall p, nan_pl 24 (quiet_nan_32_payload p) = true. -Proof. intros; apply normalized_nan; auto; omega. Qed. +Proof. intros; apply normalized_nan; auto; lia. Qed. Definition quiet_nan_32 (sp: bool * positive) : {x :float32 | is_nan _ _ x = true} := let (s, p) := sp in @@ -163,7 +163,7 @@ Proof. rewrite Z.ltb_lt in *. unfold Pos.shiftl_nat, nat_rect, Digits.digits2_pos. fold (Digits.digits2_pos p). - zify; omega. + zify; lia. Qed. Definition expand_nan s p H : {x | is_nan _ _ x = true} := @@ -336,7 +336,7 @@ Ltac smart_omega := compute_this Int64.modulus; compute_this Int64.half_modulus; compute_this Int64.max_unsigned; compute_this (Z.pow_pos 2 1024); compute_this (Z.pow_pos 2 53); compute_this (Z.pow_pos 2 52); compute_this (Z.pow_pos 2 32); - zify; omega. + zify; lia. (** Commutativity properties of addition and multiplication. *) @@ -432,7 +432,7 @@ Proof. intros; unfold of_bits, to_bits, bits_of_b64, b64_of_bits. rewrite Int64.unsigned_repr, binary_float_of_bits_of_binary_float; [reflexivity|]. generalize (bits_of_binary_float_range 52 11 __ __ f). - change (2^(52+11+1)) with (Int64.max_unsigned + 1). omega. + change (2^(52+11+1)) with (Int64.max_unsigned + 1). lia. Qed. Theorem to_of_bits: @@ -476,7 +476,7 @@ Proof. rewrite BofZ_plus by auto. f_equal. unfold Int.ltu in H. destruct zlt in H; try discriminate. - unfold y, Int.sub. rewrite Int.signed_repr. omega. + unfold y, Int.sub. rewrite Int.signed_repr. lia. compute_this (Int.unsigned ox8000_0000); smart_omega. Qed. @@ -498,8 +498,8 @@ Proof. change (Int.and ox7FFF_FFFF ox8000_0000) with Int.zero. rewrite ! Int.and_zero; auto. } assert (RNG: 0 <= Int.unsigned lo < two_p 31). - { unfold lo. change ox7FFF_FFFF with (Int.repr (two_p 31 - 1)). rewrite <- Int.zero_ext_and by omega. - apply Int.zero_ext_range. compute_this Int.zwordsize. omega. } + { unfold lo. change ox7FFF_FFFF with (Int.repr (two_p 31 - 1)). rewrite <- Int.zero_ext_and by lia. + apply Int.zero_ext_range. compute_this Int.zwordsize. lia. } assert (B: forall i, 0 <= i < Int.zwordsize -> Int.testbit ox8000_0000 i = if zeq i 31 then true else false). { intros; unfold Int.testbit. change (Int.unsigned ox8000_0000) with (2^31). destruct (zeq i 31). subst i; auto. apply Z.pow2_bits_false; auto. } @@ -512,12 +512,12 @@ Proof. assert (SU: - Int.signed hi = Int.unsigned hi). { destruct EITHER as [EQ|EQ]; rewrite EQ; reflexivity. } unfold Z.sub; rewrite SU, <- E. - unfold Int.add; rewrite Int.unsigned_repr, Int.signed_eq_unsigned. omega. - - assert (Int.max_signed = two_p 31 - 1) by reflexivity. omega. + unfold Int.add; rewrite Int.unsigned_repr, Int.signed_eq_unsigned. lia. + - assert (Int.max_signed = two_p 31 - 1) by reflexivity. lia. - assert (Int.unsigned hi = 0 \/ Int.unsigned hi = two_p 31) by (destruct EITHER as [EQ|EQ]; rewrite EQ; [left|right]; reflexivity). assert (Int.max_unsigned = two_p 31 + two_p 31 - 1) by reflexivity. - omega. + lia. Qed. Theorem to_intu_to_int_1: @@ -540,14 +540,14 @@ Proof. { rewrite ZofB_correct in C. destruct (is_finite _ _ x) eqn:FINx; congruence. } destruct (zeq p 0). subst p; smart_omega. - destruct (ZofB_range_pos 53 1024 __ __ x p C) as [P Q]. omega. + destruct (ZofB_range_pos 53 1024 __ __ x p C) as [P Q]. lia. assert (CMP: Bcompare _ _ x y = Some Lt). { unfold cmp, cmp_of_comparison, compare in H. destruct (Bcompare _ _ x y) as [[]|]; auto; discriminate. } rewrite Bcompare_correct in CMP by auto. inv CMP. apply Rcompare_Lt_inv in H1. rewrite EQy in H1. assert (p < Int.unsigned ox8000_0000). { apply lt_IZR. apply Rle_lt_trans with (1 := P) (2 := H1). } - change Int.max_signed with (Int.unsigned ox8000_0000 - 1). omega. + change Int.max_signed with (Int.unsigned ox8000_0000 - 1). lia. Qed. Theorem to_intu_to_int_2: @@ -579,7 +579,7 @@ Proof. compute_this (Int.unsigned ox8000_0000). smart_omega. apply Rge_le; auto. } - unfold to_int; rewrite EQ. simpl. unfold Int.sub. rewrite Int.unsigned_repr by omega. auto. + unfold to_int; rewrite EQ. simpl. unfold Int.sub. rewrite Int.unsigned_repr by lia. auto. Qed. (** Conversions from ints to floats can be defined as bitwise manipulations @@ -598,8 +598,8 @@ Proof. - f_equal. rewrite Int64.ofwords_add'. reflexivity. - apply split_join_bits. generalize (Int.unsigned_range x). - compute_this Int.modulus; compute_this (2^52); omega. - compute_this (2^11); omega. + compute_this Int.modulus; compute_this (2^52); lia. + compute_this (2^11); lia. Qed. Lemma from_words_value: @@ -637,7 +637,7 @@ Theorem of_intu_from_words: Proof. intros. pose proof (Int.unsigned_range x). rewrite ! from_words_eq. unfold sub. rewrite BofZ_minus. - unfold of_intu. apply (f_equal (BofZ 53 1024 __ __)). rewrite Int.unsigned_zero. omega. + unfold of_intu. apply (f_equal (BofZ 53 1024 __ __)). rewrite Int.unsigned_zero. lia. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto; rewrite Int.unsigned_zero; smart_omega. Qed. @@ -664,7 +664,7 @@ Proof. rewrite ! from_words_eq. rewrite ox8000_0000_signed_unsigned. change (Int.unsigned ox8000_0000) with Int.half_modulus. unfold sub. rewrite BofZ_minus. - unfold of_int. apply f_equal. omega. + unfold of_int. apply f_equal. lia. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto; smart_omega. Qed. @@ -680,8 +680,8 @@ Proof. - f_equal. rewrite Int64.ofwords_add'. reflexivity. - apply split_join_bits. generalize (Int.unsigned_range x). - compute_this Int.modulus; compute_this (2^52); omega. - compute_this (2^11); omega. + compute_this Int.modulus; compute_this (2^52); lia. + compute_this (2^11); lia. Qed. Lemma from_words_value': @@ -711,11 +711,11 @@ Proof. destruct (BofZ_representable 53 1024 __ __ (2^84 + Int.unsigned x * 2^32)) as (D & E & F). replace (2^84 + Int.unsigned x * 2^32) with ((2^52 + Int.unsigned x) * 2^32) by ring. - apply integer_representable_n2p; auto. smart_omega. omega. omega. + apply integer_representable_n2p; auto. smart_omega. lia. lia. apply B2R_Bsign_inj; auto. - rewrite A, D. rewrite <- IZR_Zpower by omega. rewrite <- plus_IZR. auto. + rewrite A, D. rewrite <- IZR_Zpower by lia. rewrite <- plus_IZR. auto. rewrite C, F. symmetry. apply Zlt_bool_false. - compute_this (2^84); compute_this (2^32); omega. + compute_this (2^84); compute_this (2^32); lia. Qed. Theorem of_longu_from_words: @@ -742,12 +742,12 @@ Proof. rewrite <- (Int64.ofwords_recompose l) at 1. rewrite Int64.ofwords_add'. fold xh; fold xl. compute_this (two_p 32); compute_this p20; ring. apply integer_representable_n2p; auto. - compute_this p20; smart_omega. omega. omega. + compute_this p20; smart_omega. lia. lia. apply integer_representable_n; auto; smart_omega. replace (2^84 + xh * 2^32) with ((2^52 + xh) * 2^32) by ring. - apply integer_representable_n2p; auto. smart_omega. omega. omega. + apply integer_representable_n2p; auto. smart_omega. lia. lia. change (2^84 + p20 * 2^32) with ((2^52 + 1048576) * 2^32). - apply integer_representable_n2p; auto. omega. omega. + apply integer_representable_n2p; auto. lia. lia. Qed. Theorem of_long_from_words: @@ -776,15 +776,15 @@ Proof. rewrite <- (Int64.ofwords_recompose l) at 1. rewrite Int64.ofwords_add''. fold xh; fold xl. compute_this (two_p 32); ring. apply integer_representable_n2p; auto. - compute_this (2^20); smart_omega. omega. omega. + compute_this (2^20); smart_omega. lia. lia. apply integer_representable_n; auto; smart_omega. replace (2^84 + (xh + Int.half_modulus) * 2^32) with ((2^52 + xh + Int.half_modulus) * 2^32) by (compute_this Int.half_modulus; ring). - apply integer_representable_n2p; auto. smart_omega. omega. omega. + apply integer_representable_n2p; auto. smart_omega. lia. lia. change (2^84 + p * 2^32) with ((2^52 + p) * 2^32). apply integer_representable_n2p; auto. - compute_this p; smart_omega. omega. + compute_this p; smart_omega. lia. Qed. (** Conversions from 64-bit integers can be expressed in terms of @@ -806,7 +806,7 @@ Proof. assert (DECOMP: x = yh * 2^32 + yl). { unfold x. rewrite <- (Int64.ofwords_recompose l). apply Int64.ofwords_add'. } rewrite BofZ_mult. rewrite BofZ_plus. rewrite DECOMP; auto. - apply integer_representable_n2p; auto. smart_omega. omega. omega. + apply integer_representable_n2p; auto. smart_omega. lia. lia. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto; smart_omega. @@ -829,7 +829,7 @@ Proof. assert (DECOMP: x = yh * 2^32 + yl). { unfold x. rewrite <- (Int64.ofwords_recompose l), Int64.ofwords_add''. auto. } rewrite BofZ_mult. rewrite BofZ_plus. rewrite DECOMP; auto. - apply integer_representable_n2p; auto. smart_omega. omega. omega. + apply integer_representable_n2p; auto. smart_omega. lia. lia. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto; smart_omega. apply integer_representable_n; auto. compute; intuition congruence. @@ -871,53 +871,53 @@ Proof. { intros; unfold n; autorewrite with ints; auto. rewrite Int64.unsigned_one. rewrite Int64.bits_one. compute_this Int64.zwordsize. destruct (zeq i 0); simpl proj_sumbool. - rewrite zlt_true by omega. rewrite andb_true_r. subst i; auto. + rewrite zlt_true by lia. rewrite andb_true_r. subst i; auto. rewrite andb_false_r, orb_false_r. - destruct (zeq i 63). subst i. apply zlt_false; omega. - apply zlt_true; omega. } + destruct (zeq i 63). subst i. apply zlt_false; lia. + apply zlt_true; lia. } assert (NB2: forall i, 0 <= i -> Z.testbit (Int64.signed n * 2^1) i = if zeq i 0 then false else if zeq i 1 then Int64.testbit x 1 || Int64.testbit x 0 else Int64.testbit x i). - { intros. rewrite Z.mul_pow2_bits by omega. destruct (zeq i 0). - apply Z.testbit_neg_r; omega. - rewrite Int64.bits_signed by omega. compute_this Int64.zwordsize. + { intros. rewrite Z.mul_pow2_bits by lia. destruct (zeq i 0). + apply Z.testbit_neg_r; lia. + rewrite Int64.bits_signed by lia. compute_this Int64.zwordsize. destruct (zlt (i-1) 64). - rewrite NB by omega. destruct (zeq i 1). + rewrite NB by lia. destruct (zeq i 1). subst. rewrite dec_eq_true by auto. auto. - rewrite dec_eq_false by omega. destruct (zeq (i - 1) 63). - symmetry. apply Int64.bits_above. compute_this Int64.zwordsize; omega. - f_equal; omega. - rewrite NB by omega. rewrite dec_eq_false by omega. rewrite dec_eq_true by auto. - rewrite dec_eq_false by omega. symmetry. apply Int64.bits_above. compute_this Int64.zwordsize; omega. + rewrite dec_eq_false by lia. destruct (zeq (i - 1) 63). + symmetry. apply Int64.bits_above. compute_this Int64.zwordsize; lia. + f_equal; lia. + rewrite NB by lia. rewrite dec_eq_false by lia. rewrite dec_eq_true by auto. + rewrite dec_eq_false by lia. symmetry. apply Int64.bits_above. compute_this Int64.zwordsize; lia. } assert (EQ: Int64.signed n * 2 = int_round_odd (Int64.unsigned x) 1). { - symmetry. apply int_round_odd_bits. omega. - intros. rewrite NB2 by omega. replace i with 0 by omega. auto. - rewrite NB2 by omega. rewrite dec_eq_false by omega. rewrite dec_eq_true. + symmetry. apply int_round_odd_bits. lia. + intros. rewrite NB2 by lia. replace i with 0 by lia. auto. + rewrite NB2 by lia. rewrite dec_eq_false by lia. rewrite dec_eq_true. rewrite orb_comm. unfold Int64.testbit. change (2^1) with 2. destruct (Z.testbit (Int64.unsigned x) 0) eqn:B0; - [rewrite Z.testbit_true in B0 by omega|rewrite Z.testbit_false in B0 by omega]; + [rewrite Z.testbit_true in B0 by lia|rewrite Z.testbit_false in B0 by lia]; change (2^0) with 1 in B0; rewrite Zdiv_1_r in B0; rewrite B0; auto. - intros. rewrite NB2 by omega. rewrite ! dec_eq_false by omega. auto. + intros. rewrite NB2 by lia. rewrite ! dec_eq_false by lia. auto. } unfold mul, of_long, of_longu. rewrite BofZ_mult_2p. - change (2^1) with 2. rewrite EQ. apply BofZ_round_odd with (p := 1). -+ omega. ++ lia. + apply Z.le_trans with Int64.modulus; trivial. smart_omega. -+ omega. -+ apply Z.le_trans with (2^63). compute; intuition congruence. xomega. ++ lia. ++ apply Z.le_trans with (2^63). compute; intuition congruence. extlia. - apply Z.le_trans with Int64.modulus; trivial. pose proof (Int64.signed_range n). compute_this Int64.min_signed; compute_this Int64.max_signed; - compute_this Int64.modulus; xomega. + compute_this Int64.modulus; extlia. - assert (2^63 <= int_round_odd (Int64.unsigned x) 1). - { change (2^63) with (int_round_odd (2^63) 1). apply int_round_odd_le; omega. } - rewrite <- EQ in H1. compute_this (2^63). compute_this (2^53). xomega. -- omega. + { change (2^63) with (int_round_odd (2^63) 1). apply int_round_odd_le; lia. } + rewrite <- EQ in H1. compute_this (2^63). compute_this (2^53). extlia. +- lia. Qed. (** Conversions to/from 32-bit integers can be implemented by going through 64-bit integers. *) @@ -931,8 +931,8 @@ Proof. intros. exploit ZofB_range_inversion; eauto. intros (A & B & C). unfold ZofB_range; rewrite C. replace (min2 <=? n) with true. replace (n <=? max2) with true. auto. - symmetry; apply Z.leb_le; omega. - symmetry; apply Z.leb_le; omega. + symmetry; apply Z.leb_le; lia. + symmetry; apply Z.leb_le; lia. Qed. Theorem to_int_to_long: @@ -954,7 +954,7 @@ Proof. exploit ZofB_range_inversion; eauto. intros (A & B & C). replace (ZofB_range 53 1024 f 0 Int64.max_unsigned) with (Some z). simpl. rewrite Int.unsigned_repr; auto. - symmetry; eapply ZofB_range_widen; eauto. omega. compute; congruence. + symmetry; eapply ZofB_range_widen; eauto. lia. compute; congruence. Qed. Theorem to_intu_to_long: @@ -1183,7 +1183,7 @@ Theorem cmp_double: forall f1 f2 c, cmp c f1 f2 = Float.cmp c (to_double f1) (to_double f2). Proof. unfold cmp, Float.cmp; intros. f_equal. symmetry. apply Bcompare_Bconv_widen. - red; omega. omega. omega. + red; lia. lia. lia. Qed. (** Properties of conversions to/from in-memory representation. @@ -1195,7 +1195,7 @@ Proof. intros; unfold of_bits, to_bits, bits_of_b32, b32_of_bits. rewrite Int.unsigned_repr, binary_float_of_bits_of_binary_float; [reflexivity|]. generalize (bits_of_binary_float_range 23 8 __ __ f). - change (2^(23+8+1)) with (Int.max_unsigned + 1). omega. + change (2^(23+8+1)) with (Int.max_unsigned + 1). lia. Qed. Theorem to_of_bits: @@ -1235,7 +1235,7 @@ Proof. unfold to_int in H. destruct (ZofB_range _ _ f Int.min_signed Int.max_signed) as [n'|] eqn:E; inv H. unfold Float.to_int, to_double, Float.of_single. - erewrite ZofB_range_Bconv; eauto. auto. omega. omega. omega. omega. + erewrite ZofB_range_Bconv; eauto. auto. lia. lia. lia. lia. Qed. Theorem to_intu_double: @@ -1245,7 +1245,7 @@ Proof. unfold to_intu in H. destruct (ZofB_range _ _ f 0 Int.max_unsigned) as [n'|] eqn:E; inv H. unfold Float.to_intu, to_double, Float.of_single. - erewrite ZofB_range_Bconv; eauto. auto. omega. omega. omega. omega. + erewrite ZofB_range_Bconv; eauto. auto. lia. lia. lia. lia. Qed. Theorem to_long_double: @@ -1255,7 +1255,7 @@ Proof. unfold to_long in H. destruct (ZofB_range _ _ f Int64.min_signed Int64.max_signed) as [n'|] eqn:E; inv H. unfold Float.to_long, to_double, Float.of_single. - erewrite ZofB_range_Bconv; eauto. auto. omega. omega. omega. omega. + erewrite ZofB_range_Bconv; eauto. auto. lia. lia. lia. lia. Qed. Theorem to_longu_double: @@ -1265,7 +1265,7 @@ Proof. unfold to_longu in H. destruct (ZofB_range _ _ f 0 Int64.max_unsigned) as [n'|] eqn:E; inv H. unfold Float.to_longu, to_double, Float.of_single. - erewrite ZofB_range_Bconv; eauto. auto. omega. omega. omega. omega. + erewrite ZofB_range_Bconv; eauto. auto. lia. lia. lia. lia. Qed. (** Conversions from 64-bit integers to single-precision floats can be expressed @@ -1280,37 +1280,37 @@ Proof. intros. assert (POS: 0 < 2^p) by (apply (Zpower_gt_0 radix2); auto). assert (A: Z.land n (2^p-1) = n mod 2^p). - { rewrite <- Z.land_ones by auto. f_equal. rewrite Z.ones_equiv. omega. } + { rewrite <- Z.land_ones by auto. f_equal. rewrite Z.ones_equiv. lia. } rewrite A. assert (B: 0 <= n mod 2^p < 2^p). - { apply Z_mod_lt. omega. } + { apply Z_mod_lt. lia. } set (m := n mod 2^p + (2^p-1)) in *. assert (C: m / 2^p = if zeq (n mod 2^p) 0 then 0 else 1). { unfold m. destruct (zeq (n mod 2^p) 0). - rewrite e. apply Z.div_small. omega. - eapply Coqlib.Zdiv_unique with (n mod 2^p - 1). ring. omega. } + rewrite e. apply Z.div_small. lia. + eapply Coqlib.Zdiv_unique with (n mod 2^p - 1). ring. lia. } assert (D: Z.testbit m p = if zeq (n mod 2^p) 0 then false else true). { destruct (zeq (n mod 2^p) 0). apply Z.testbit_false; auto. rewrite C; auto. apply Z.testbit_true; auto. rewrite C; auto. } assert (E: forall i, p < i -> Z.testbit m i = false). - { intros. apply Z.testbit_false. omega. + { intros. apply Z.testbit_false. lia. replace (m / 2^i) with 0. auto. symmetry. apply Z.div_small. - unfold m. split. omega. apply Z.lt_le_trans with (2 * 2^p). omega. - change 2 with (2^1) at 1. rewrite <- (Zpower_plus radix2) by omega. - apply Zpower_le. omega. } + unfold m. split. lia. apply Z.lt_le_trans with (2 * 2^p). lia. + change 2 with (2^1) at 1. rewrite <- (Zpower_plus radix2) by lia. + apply Zpower_le. lia. } assert (F: forall i, 0 <= i -> Z.testbit (-2^p) i = if zlt i p then false else true). { intros. rewrite Z.bits_opp by auto. rewrite <- Z.ones_equiv. destruct (zlt i p). - rewrite Z.ones_spec_low by omega. auto. - rewrite Z.ones_spec_high by omega. auto. } + rewrite Z.ones_spec_low by lia. auto. + rewrite Z.ones_spec_high by lia. auto. } apply int_round_odd_bits; auto. - - intros. rewrite Z.land_spec, F, zlt_true by omega. apply andb_false_r. - - rewrite Z.land_spec, Z.lor_spec, D, F, zlt_false, andb_true_r by omega. + - intros. rewrite Z.land_spec, F, zlt_true by lia. apply andb_false_r. + - rewrite Z.land_spec, Z.lor_spec, D, F, zlt_false, andb_true_r by lia. destruct (Z.eqb (n mod 2^p) 0) eqn:Z. rewrite Z.eqb_eq in Z. rewrite Z, zeq_true. apply orb_false_r. rewrite Z.eqb_neq in Z. rewrite zeq_false by auto. apply orb_true_r. - - intros. rewrite Z.land_spec, Z.lor_spec, E, F, zlt_false, andb_true_r by omega. + - intros. rewrite Z.land_spec, Z.lor_spec, E, F, zlt_false, andb_true_r by lia. apply orb_false_r. Qed. @@ -1319,22 +1319,22 @@ Lemma of_long_round_odd: 2^36 <= Z.abs n < 2^64 -> BofZ 24 128 __ __ n = Bconv _ _ 24 128 __ __ conv_nan mode_NE (BofZ 53 1024 __ __ (Z.land (Z.lor n ((Z.land n 2047) + 2047)) (-2048))). Proof. - intros. rewrite <- (int_round_odd_plus 11) by omega. + intros. rewrite <- (int_round_odd_plus 11) by lia. assert (-2^64 <= int_round_odd n 11). - { change (-2^64) with (int_round_odd (-2^64) 11). apply int_round_odd_le; xomega. } + { change (-2^64) with (int_round_odd (-2^64) 11). apply int_round_odd_le; extlia. } assert (int_round_odd n 11 <= 2^64). - { change (2^64) with (int_round_odd (2^64) 11). apply int_round_odd_le; xomega. } + { change (2^64) with (int_round_odd (2^64) 11). apply int_round_odd_le; extlia. } rewrite Bconv_BofZ. apply BofZ_round_odd with (p := 11). - omega. - apply Z.le_trans with (2^64). omega. compute; intuition congruence. - omega. + lia. + apply Z.le_trans with (2^64). lia. compute; intuition congruence. + lia. exact (proj1 H). - unfold int_round_odd. apply integer_representable_n2p_wide. auto. omega. + unfold int_round_odd. apply integer_representable_n2p_wide. auto. lia. unfold int_round_odd in H0, H1. split; (apply Zmult_le_reg_r with (2^11); [compute; auto | assumption]). - omega. - omega. + lia. + lia. Qed. Theorem of_longu_double_1: @@ -1343,7 +1343,7 @@ Theorem of_longu_double_1: of_longu n = of_double (Float.of_longu n). Proof. intros. symmetry; apply Bconv_BofZ. apply integer_representable_n; auto. - pose proof (Int64.unsigned_range n); omega. + pose proof (Int64.unsigned_range n); lia. Qed. Theorem of_longu_double_2: @@ -1361,14 +1361,14 @@ Proof. unfold of_double, Float.to_single. instantiate (1 := Float.to_single_nan). f_equal. unfold Float.of_longu. f_equal. set (n' := Z.land (Z.lor (Int64.unsigned n) (Z.land (Int64.unsigned n) 2047 + 2047)) (-2048)). - assert (int_round_odd (Int64.unsigned n) 11 = n') by (apply int_round_odd_plus; omega). + assert (int_round_odd (Int64.unsigned n) 11 = n') by (apply int_round_odd_plus; lia). assert (0 <= n'). - { rewrite <- H1. change 0 with (int_round_odd 0 11). apply int_round_odd_le; omega. } + { rewrite <- H1. change 0 with (int_round_odd 0 11). apply int_round_odd_le; lia. } assert (n' < Int64.modulus). { apply Z.le_lt_trans with (int_round_odd (Int64.modulus - 1) 11). - rewrite <- H1. apply int_round_odd_le; omega. + rewrite <- H1. apply int_round_odd_le; lia. compute; auto. } - rewrite <- (Int64.unsigned_repr n') by (unfold Int64.max_unsigned; omega). + rewrite <- (Int64.unsigned_repr n') by (unfold Int64.max_unsigned; lia). f_equal. Int64.bit_solve. rewrite Int64.testbit_repr by auto. unfold n'. rewrite Z.land_spec, Z.lor_spec. f_equal. f_equal. unfold Int64.testbit. rewrite Int64.add_unsigned. @@ -1377,11 +1377,11 @@ Proof. Int64.unsigned (Int64.repr 2047))) i). rewrite Int64.testbit_repr by auto. f_equal. f_equal. unfold Int64.and. symmetry. apply Int64.unsigned_repr. change 2047 with (Z.ones 11). - rewrite Z.land_ones by omega. + rewrite Z.land_ones by lia. exploit (Z_mod_lt (Int64.unsigned n) (2^11)). compute; auto. - assert (2^11 < Int64.max_unsigned) by (compute; auto). omega. + assert (2^11 < Int64.max_unsigned) by (compute; auto). lia. apply Int64.same_bits_eqm; auto. exists (-1); auto. - split. xomega. change (2^64) with Int64.modulus. xomega. + split. extlia. change (2^64) with Int64.modulus. extlia. Qed. Theorem of_long_double_1: @@ -1389,7 +1389,7 @@ Theorem of_long_double_1: Z.abs (Int64.signed n) <= 2^53 -> of_long n = of_double (Float.of_long n). Proof. - intros. symmetry; apply Bconv_BofZ. apply integer_representable_n; auto. xomega. + intros. symmetry; apply Bconv_BofZ. apply integer_representable_n; auto. extlia. Qed. Theorem of_long_double_2: @@ -1407,34 +1407,34 @@ Proof. unfold of_double, Float.to_single. instantiate (1 := Float.to_single_nan). f_equal. unfold Float.of_long. f_equal. set (n' := Z.land (Z.lor (Int64.signed n) (Z.land (Int64.signed n) 2047 + 2047)) (-2048)). - assert (int_round_odd (Int64.signed n) 11 = n') by (apply int_round_odd_plus; omega). + assert (int_round_odd (Int64.signed n) 11 = n') by (apply int_round_odd_plus; lia). assert (Int64.min_signed <= n'). - { rewrite <- H1. change Int64.min_signed with (int_round_odd Int64.min_signed 11). apply int_round_odd_le; omega. } + { rewrite <- H1. change Int64.min_signed with (int_round_odd Int64.min_signed 11). apply int_round_odd_le; lia. } assert (n' <= Int64.max_signed). { apply Z.le_trans with (int_round_odd Int64.max_signed 11). - rewrite <- H1. apply int_round_odd_le; omega. + rewrite <- H1. apply int_round_odd_le; lia. compute; intuition congruence. } - rewrite <- (Int64.signed_repr n') by omega. + rewrite <- (Int64.signed_repr n') by lia. f_equal. Int64.bit_solve. rewrite Int64.testbit_repr by auto. unfold n'. rewrite Z.land_spec, Z.lor_spec. f_equal. f_equal. - rewrite Int64.bits_signed by omega. rewrite zlt_true by omega. auto. + rewrite Int64.bits_signed by lia. rewrite zlt_true by lia. auto. unfold Int64.testbit. rewrite Int64.add_unsigned. fold (Int64.testbit (Int64.repr (Int64.unsigned (Int64.and n (Int64.repr 2047)) + Int64.unsigned (Int64.repr 2047))) i). rewrite Int64.testbit_repr by auto. f_equal. f_equal. unfold Int64.and. change (Int64.unsigned (Int64.repr 2047)) with 2047. - change 2047 with (Z.ones 11). rewrite ! Z.land_ones by omega. + change 2047 with (Z.ones 11). rewrite ! Z.land_ones by lia. rewrite Int64.unsigned_repr. apply eqmod_mod_eq. - apply Z.lt_gt. apply (Zpower_gt_0 radix2); omega. + apply Z.lt_gt. apply (Zpower_gt_0 radix2); lia. apply eqmod_divides with (2^64). apply Int64.eqm_signed_unsigned. exists (2^(64-11)); auto. exploit (Z_mod_lt (Int64.unsigned n) (2^11)). compute; auto. - assert (2^11 < Int64.max_unsigned) by (compute; auto). omega. + assert (2^11 < Int64.max_unsigned) by (compute; auto). lia. apply Int64.same_bits_eqm; auto. exists (-1); auto. split. auto. assert (-2^64 < Int64.min_signed) by (compute; auto). assert (Int64.max_signed < 2^64) by (compute; auto). - xomega. + extlia. Qed. End Float32. diff --git a/lib/IEEE754_extra.v b/lib/IEEE754_extra.v index 5e35a191..580d4f90 100644 --- a/lib/IEEE754_extra.v +++ b/lib/IEEE754_extra.v @@ -119,7 +119,7 @@ Definition integer_representable (n: Z): Prop := Let int_upper_bound_eq: 2^emax - 2^(emax - prec) = (2^prec - 1) * 2^(emax - prec). Proof. red in prec_gt_0_. - ring_simplify. rewrite <- (Zpower_plus radix2) by omega. f_equal. f_equal. omega. + ring_simplify. rewrite <- (Zpower_plus radix2) by lia. f_equal. f_equal. lia. Qed. Lemma integer_representable_n2p: @@ -130,14 +130,14 @@ Proof. intros; split. - red in prec_gt_0_. replace (Z.abs (n * 2^p)) with (Z.abs n * 2^p). rewrite int_upper_bound_eq. - apply Zmult_le_compat. zify; omega. apply (Zpower_le radix2); omega. - zify; omega. apply (Zpower_ge_0 radix2). + apply Zmult_le_compat. zify; lia. apply (Zpower_le radix2); lia. + zify; lia. apply (Zpower_ge_0 radix2). rewrite Z.abs_mul. f_equal. rewrite Z.abs_eq. auto. apply (Zpower_ge_0 radix2). - apply generic_format_FLT. exists (Float radix2 n p). unfold F2R; simpl. rewrite <- IZR_Zpower by auto. apply mult_IZR. - simpl; zify; omega. - unfold emin, Fexp; red in prec_gt_0_; omega. + simpl; zify; lia. + unfold emin, Fexp; red in prec_gt_0_; lia. Qed. Lemma integer_representable_2p: @@ -149,19 +149,19 @@ Proof. - red in prec_gt_0_. rewrite Z.abs_eq by (apply (Zpower_ge_0 radix2)). apply Z.le_trans with (2^(emax-1)). - apply (Zpower_le radix2); omega. + apply (Zpower_le radix2); lia. assert (2^emax = 2^(emax-1)*2). - { change 2 with (2^1) at 3. rewrite <- (Zpower_plus radix2) by omega. - f_equal. omega. } + { change 2 with (2^1) at 3. rewrite <- (Zpower_plus radix2) by lia. + f_equal. lia. } assert (2^(emax - prec) <= 2^(emax - 1)). - { apply (Zpower_le radix2). omega. } - omega. + { apply (Zpower_le radix2). lia. } + lia. - red in prec_gt_0_. apply generic_format_FLT. exists (Float radix2 1 p). unfold F2R; simpl. - rewrite Rmult_1_l. rewrite <- IZR_Zpower. auto. omega. - simpl Z.abs. change 1 with (2^0). apply (Zpower_lt radix2). omega. auto. - unfold emin, Fexp; omega. + rewrite Rmult_1_l. rewrite <- IZR_Zpower. auto. lia. + simpl Z.abs. change 1 with (2^0). apply (Zpower_lt radix2). lia. auto. + unfold emin, Fexp; lia. Qed. Lemma integer_representable_opp: @@ -178,12 +178,12 @@ Lemma integer_representable_n2p_wide: Proof. intros. red in prec_gt_0_. destruct (Z.eq_dec n (2^prec)); [idtac | destruct (Z.eq_dec n (-2^prec))]. -- rewrite e. rewrite <- (Zpower_plus radix2) by omega. - apply integer_representable_2p. omega. +- rewrite e. rewrite <- (Zpower_plus radix2) by lia. + apply integer_representable_2p. lia. - rewrite e. rewrite <- Zopp_mult_distr_l. apply integer_representable_opp. - rewrite <- (Zpower_plus radix2) by omega. - apply integer_representable_2p. omega. -- apply integer_representable_n2p; omega. + rewrite <- (Zpower_plus radix2) by lia. + apply integer_representable_2p. lia. +- apply integer_representable_n2p; lia. Qed. Lemma integer_representable_n: @@ -191,7 +191,7 @@ Lemma integer_representable_n: Proof. red in prec_gt_0_. intros. replace n with (n * 2^0) by (change (2^0) with 1; ring). - apply integer_representable_n2p_wide. auto. omega. omega. + apply integer_representable_n2p_wide. auto. lia. lia. Qed. Lemma round_int_no_overflow: @@ -205,14 +205,14 @@ Proof. apply round_le_generic. apply fexp_correct; auto. apply valid_rnd_N. apply generic_format_FLT. exists (Float radix2 (2^prec-1) (emax-prec)). rewrite int_upper_bound_eq. unfold F2R; simpl. - rewrite <- IZR_Zpower by omega. rewrite <- mult_IZR. auto. - assert (0 < 2^prec) by (apply (Zpower_gt_0 radix2); omega). - unfold Fnum; simpl; zify; omega. - unfold emin, Fexp; omega. + rewrite <- IZR_Zpower by lia. rewrite <- mult_IZR. auto. + assert (0 < 2^prec) by (apply (Zpower_gt_0 radix2); lia). + unfold Fnum; simpl; zify; lia. + unfold emin, Fexp; lia. rewrite <- abs_IZR. apply IZR_le. auto. - rewrite <- IZR_Zpower by omega. apply IZR_lt. simpl. - assert (0 < 2^(emax-prec)) by (apply (Zpower_gt_0 radix2); omega). - omega. + rewrite <- IZR_Zpower by lia. apply IZR_lt. simpl. + assert (0 < 2^(emax-prec)) by (apply (Zpower_gt_0 radix2); lia). + lia. apply fexp_correct. auto. Qed. @@ -299,8 +299,8 @@ Proof. { apply round_le_generic. apply fexp_correct. auto. apply valid_rnd_N. apply (integer_representable_opp 1). apply (integer_representable_2p 0). - red in prec_gt_0_; omega. - apply IZR_le; omega. + red in prec_gt_0_; lia. + apply IZR_le; lia. } lra. Qed. @@ -335,7 +335,7 @@ Proof. rewrite R, W, C, F. rewrite Rcompare_IZR. unfold Z.ltb at 3. generalize (Zcompare_spec (p + q) 0); intros SPEC; inversion SPEC; auto. - assert (EITHER: 0 <= p \/ 0 <= q) by omega. + assert (EITHER: 0 <= p \/ 0 <= q) by lia. destruct EITHER; [apply andb_false_intro1 | apply andb_false_intro2]; apply Zlt_bool_false; auto. - intros P (U & V). @@ -343,8 +343,8 @@ Proof. rewrite P, U, C. f_equal. rewrite C, F in V. generalize (Zlt_bool_spec p 0) (Zlt_bool_spec q 0). rewrite <- V. intros SPEC1 SPEC2; inversion SPEC1; inversion SPEC2; try congruence; symmetry. - apply Zlt_bool_true; omega. - apply Zlt_bool_false; omega. + apply Zlt_bool_true; lia. + apply Zlt_bool_false; lia. Qed. Theorem BofZ_minus: @@ -365,7 +365,7 @@ Proof. rewrite R, W, C, F. rewrite Rcompare_IZR. unfold Z.ltb at 3. generalize (Zcompare_spec (p - q) 0); intros SPEC; inversion SPEC; auto. - assert (EITHER: 0 <= p \/ q < 0) by omega. + assert (EITHER: 0 <= p \/ q < 0) by lia. destruct EITHER; [apply andb_false_intro1 | apply andb_false_intro2]. rewrite Zlt_bool_false; auto. rewrite Zlt_bool_true; auto. @@ -375,8 +375,8 @@ Proof. generalize (Zlt_bool_spec p 0) (Zlt_bool_spec q 0). rewrite V. intros SPEC1 SPEC2; inversion SPEC1; inversion SPEC2; symmetry. rewrite <- H3 in H1; discriminate. - apply Zlt_bool_true; omega. - apply Zlt_bool_false; omega. + apply Zlt_bool_true; lia. + apply Zlt_bool_false; lia. rewrite <- H3 in H1; discriminate. Qed. @@ -389,10 +389,10 @@ Proof. intros. assert (SIGN: xorb (p <? 0) (q <? 0) = (p * q <? 0)). { - rewrite (Zlt_bool_false q) by omega. + rewrite (Zlt_bool_false q) by lia. generalize (Zlt_bool_spec p 0); intros SPEC; inversion SPEC; simpl; symmetry. - apply Zlt_bool_true. rewrite Z.mul_comm. apply Z.mul_pos_neg; omega. - apply Zlt_bool_false. apply Zsame_sign_imp; omega. + apply Zlt_bool_true. rewrite Z.mul_comm. apply Z.mul_pos_neg; lia. + apply Zlt_bool_false. apply Zsame_sign_imp; lia. } destruct (BofZ_representable p) as (A & B & C); auto. destruct (BofZ_representable q) as (D & E & F); auto. @@ -420,10 +420,10 @@ Proof. destruct (Z.eq_dec x 0). - subst x. apply BofZ_mult. apply integer_representable_n. - generalize (Zpower_ge_0 radix2 prec). simpl; omega. + generalize (Zpower_ge_0 radix2 prec). simpl; lia. apply integer_representable_2p. auto. apply (Zpower_gt_0 radix2). - omega. + lia. - assert (IZR x <> 0%R) by (apply (IZR_neq _ _ n)). destruct (BofZ_finite x H) as (A & B & C). destruct (BofZ_representable (2^p)) as (D & E & F). @@ -432,16 +432,16 @@ Proof. cexp radix2 fexp (IZR x) + p). { unfold cexp, fexp. rewrite mult_IZR. - change (2^p) with (radix2^p). rewrite IZR_Zpower by omega. + change (2^p) with (radix2^p). rewrite IZR_Zpower by lia. rewrite mag_mult_bpow by auto. assert (prec + 1 <= mag radix2 (IZR x)). { rewrite <- (mag_abs radix2 (IZR x)). rewrite <- (mag_bpow radix2 prec). apply mag_le. - apply bpow_gt_0. rewrite <- IZR_Zpower by (red in prec_gt_0_;omega). + apply bpow_gt_0. rewrite <- IZR_Zpower by (red in prec_gt_0_;lia). rewrite <- abs_IZR. apply IZR_le; auto. } unfold FLT_exp. - unfold emin; red in prec_gt_0_; zify; omega. + unfold emin; red in prec_gt_0_; zify; lia. } assert (forall m, round radix2 fexp m (IZR x) * IZR (2^p) = round radix2 fexp m (IZR (x * 2^p)))%R. @@ -451,11 +451,11 @@ Proof. set (a := IZR x); set (b := bpow radix2 (- cexp radix2 fexp a)). replace (a * IZR (2^p) * (b * bpow radix2 (-p)))%R with (a * b)%R. unfold F2R; simpl. rewrite Rmult_assoc. f_equal. - rewrite bpow_plus. f_equal. apply (IZR_Zpower radix2). omega. + rewrite bpow_plus. f_equal. apply (IZR_Zpower radix2). lia. transitivity ((a * b) * (IZR (2^p) * bpow radix2 (-p)))%R. rewrite (IZR_Zpower radix2). rewrite <- bpow_plus. - replace (p + -p) with 0 by omega. change (bpow radix2 0) with 1%R. ring. - omega. + replace (p + -p) with 0 by lia. change (bpow radix2 0) with 1%R. ring. + lia. ring. } assert (forall m x, @@ -468,11 +468,11 @@ Proof. } assert (xorb (x <? 0) (2^p <? 0) = (x * 2^p <? 0)). { - assert (0 < 2^p) by (apply (Zpower_gt_0 radix2); omega). - rewrite (Zlt_bool_false (2^p)) by omega. rewrite xorb_false_r. + assert (0 < 2^p) by (apply (Zpower_gt_0 radix2); lia). + rewrite (Zlt_bool_false (2^p)) by lia. rewrite xorb_false_r. symmetry. generalize (Zlt_bool_spec x 0); intros SPEC; inversion SPEC. apply Zlt_bool_true. apply Z.mul_neg_pos; auto. - apply Zlt_bool_false. apply Z.mul_nonneg_nonneg; omega. + apply Zlt_bool_false. apply Z.mul_nonneg_nonneg; lia. } generalize (Bmult_correct _ _ _ Hmax nan mode_NE (BofZ x) (BofZ (2^p))) (BofZ_correct (x * 2^p)). @@ -496,10 +496,10 @@ Lemma round_odd_flt: round radix2 fexp (Znearest choice) x. Proof. intros. apply round_N_odd. auto. apply fexp_correct; auto. - apply exists_NE_FLT. right; omega. - apply FLT_exp_valid. red; omega. - apply exists_NE_FLT. right; omega. - unfold fexp, FLT_exp; intros. zify; omega. + apply exists_NE_FLT. right; lia. + apply FLT_exp_valid. red; lia. + apply exists_NE_FLT. right; lia. + unfold fexp, FLT_exp; intros. zify; lia. Qed. Corollary round_odd_fix: @@ -522,8 +522,8 @@ Proof. cexp radix2 (FIX_exp p) x). { unfold cexp, FLT_exp, FIX_exp. - replace (mag radix2 x - prec') with p by (unfold prec'; omega). - apply Z.max_l. unfold emin', emin. red in prec_gt_0_; omega. + replace (mag radix2 x - prec') with p by (unfold prec'; lia). + apply Z.max_l. unfold emin', emin. red in prec_gt_0_; lia. } assert (RND: round radix2 (FIX_exp p) Zrnd_odd x = round radix2 (FLT_exp emin' prec') Zrnd_odd x). @@ -532,9 +532,9 @@ Proof. } rewrite RND. apply round_odd_flt. auto. - unfold prec'. red in prec_gt_0_; omega. - unfold prec'. omega. - unfold emin'. omega. + unfold prec'. red in prec_gt_0_; lia. + unfold prec'. lia. + unfold emin'. lia. Qed. Definition int_round_odd (x: Z) (p: Z) := @@ -546,22 +546,22 @@ Lemma Zrnd_odd_int: int_round_odd n p. Proof. clear. intros. - assert (0 < 2^p) by (apply (Zpower_gt_0 radix2); omega). - assert (n = (n / 2^p) * 2^p + n mod 2^p) by (rewrite Z.mul_comm; apply Z.div_mod; omega). - assert (0 <= n mod 2^p < 2^p) by (apply Z_mod_lt; omega). + assert (0 < 2^p) by (apply (Zpower_gt_0 radix2); lia). + assert (n = (n / 2^p) * 2^p + n mod 2^p) by (rewrite Z.mul_comm; apply Z.div_mod; lia). + assert (0 <= n mod 2^p < 2^p) by (apply Z_mod_lt; lia). unfold int_round_odd. set (q := n / 2^p) in *; set (r := n mod 2^p) in *. f_equal. pose proof (bpow_gt_0 radix2 (-p)). assert (bpow radix2 p * bpow radix2 (-p) = 1)%R. - { rewrite <- bpow_plus. replace (p + -p) with 0 by omega. auto. } + { rewrite <- bpow_plus. replace (p + -p) with 0 by lia. auto. } assert (IZR n * bpow radix2 (-p) = IZR q + IZR r * bpow radix2 (-p))%R. { rewrite H1. rewrite plus_IZR, mult_IZR. change (IZR (2^p)) with (IZR (radix2^p)). - rewrite IZR_Zpower by omega. ring_simplify. + rewrite IZR_Zpower by lia. ring_simplify. rewrite Rmult_assoc. rewrite H4. ring. } assert (0 <= IZR r < bpow radix2 p)%R. - { split. apply IZR_le; omega. - rewrite <- IZR_Zpower by omega. apply IZR_lt; tauto. } + { split. apply IZR_le; lia. + rewrite <- IZR_Zpower by lia. apply IZR_lt; tauto. } assert (0 <= IZR r * bpow radix2 (-p) < 1)%R. { generalize (bpow_gt_0 radix2 (-p)). intros. split. apply Rmult_le_pos; lra. @@ -615,15 +615,15 @@ Proof. assert (DIV: (2^p | 2^emax - 2^(emax - prec))). { rewrite int_upper_bound_eq. apply Z.divide_mul_r. exists (2^(emax - prec - p)). red in prec_gt_0_. - rewrite <- (Zpower_plus radix2) by omega. f_equal; omega. } + rewrite <- (Zpower_plus radix2) by lia. f_equal; lia. } assert (YRANGE: Z.abs (int_round_odd x p) <= 2^emax - 2^(emax-prec)). { apply Z.abs_le. split. replace (-(2^emax - 2^(emax-prec))) with (int_round_odd (-(2^emax - 2^(emax-prec))) p). - apply int_round_odd_le; zify; omega. - apply int_round_odd_exact. omega. apply Z.divide_opp_r. auto. + apply int_round_odd_le; zify; lia. + apply int_round_odd_exact. lia. apply Z.divide_opp_r. auto. replace (2^emax - 2^(emax-prec)) with (int_round_odd (2^emax - 2^(emax-prec)) p). - apply int_round_odd_le; zify; omega. - apply int_round_odd_exact. omega. auto. } + apply int_round_odd_le; zify; lia. + apply int_round_odd_exact. lia. auto. } destruct (BofZ_finite x XRANGE) as (X1 & X2 & X3). destruct (BofZ_finite (int_round_odd x p) YRANGE) as (Y1 & Y2 & Y3). apply BofZ_finite_equal; auto. @@ -631,12 +631,12 @@ Proof. assert (IZR (int_round_odd x p) = round radix2 (FIX_exp p) Zrnd_odd (IZR x)). { unfold round, scaled_mantissa, cexp, FIX_exp. - rewrite <- Zrnd_odd_int by omega. - unfold F2R; simpl. rewrite mult_IZR. f_equal. apply (IZR_Zpower radix2). omega. + rewrite <- Zrnd_odd_int by lia. + unfold F2R; simpl. rewrite mult_IZR. f_equal. apply (IZR_Zpower radix2). lia. } - rewrite H. symmetry. apply round_odd_fix. auto. omega. + rewrite H. symmetry. apply round_odd_fix. auto. lia. rewrite <- IZR_Zpower. rewrite <- abs_IZR. apply IZR_le; auto. - red in prec_gt_0_; omega. + red in prec_gt_0_; lia. Qed. Lemma int_round_odd_shifts: @@ -667,17 +667,17 @@ Proof. apply Z.bits_inj'. intros. generalize (Zcompare_spec n p); intros SPEC; inversion SPEC. - rewrite BELOW by auto. apply Z.shiftl_spec_low; auto. -- subst n. rewrite AT. rewrite Z.shiftl_spec_high by omega. - replace (p - p) with 0 by omega. +- subst n. rewrite AT. rewrite Z.shiftl_spec_high by lia. + replace (p - p) with 0 by lia. destruct (x mod 2^p =? 0). - + rewrite Z.shiftr_spec by omega. f_equal; omega. + + rewrite Z.shiftr_spec by lia. f_equal; lia. + rewrite Z.lor_spec. apply orb_true_r. -- rewrite ABOVE by auto. rewrite Z.shiftl_spec_high by omega. +- rewrite ABOVE by auto. rewrite Z.shiftl_spec_high by lia. destruct (x mod 2^p =? 0). - rewrite Z.shiftr_spec by omega. f_equal; omega. - rewrite Z.lor_spec, Z.shiftr_spec by omega. - change 1 with (Z.ones 1). rewrite Z.ones_spec_high by omega. rewrite orb_false_r. - f_equal; omega. + rewrite Z.shiftr_spec by lia. f_equal; lia. + rewrite Z.lor_spec, Z.shiftr_spec by lia. + change 1 with (Z.ones 1). rewrite Z.ones_spec_high by lia. rewrite orb_false_r. + f_equal; lia. Qed. (** ** Conversion from a FP number to an integer *) @@ -709,7 +709,7 @@ Proof. } rewrite EQ. f_equal. generalize (Zpower_pos_gt_0 2 p (eq_refl _)); intros. - rewrite Ztrunc_floor. symmetry. apply Zfloor_div. omega. + rewrite Ztrunc_floor. symmetry. apply Zfloor_div. lia. apply Rmult_le_pos. apply IZR_le. compute; congruence. apply Rlt_le. apply Rinv_0_lt_compat. apply IZR_lt. auto. Qed. @@ -727,7 +727,7 @@ Proof. assert (-x < 0)%R. { apply Rlt_le_trans with (IZR (Zfloor (-x)) + 1)%R. apply Zfloor_ub. rewrite <- plus_IZR. - apply IZR_le. omega. } + apply IZR_le. lia. } lra. Qed. @@ -741,7 +741,7 @@ Proof. - rewrite Ztrunc_ceil in H by (apply Rlt_le; auto). split. + apply (Ropp_lt_cancel (-(1))). rewrite Ropp_involutive. replace 1%R with (IZR (Zfloor (-x)) + 1)%R. apply Zfloor_ub. - unfold Zceil in H. replace (Zfloor (-x)) with 0 by omega. simpl. apply Rplus_0_l. + unfold Zceil in H. replace (Zfloor (-x)) with 0 by lia. simpl. apply Rplus_0_l. + apply Rlt_le_trans with 0%R; auto. apply Rle_0_1. Qed. @@ -758,10 +758,10 @@ Proof. intros. rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; inversion H. set (x := B2R prec emax f) in *. set (y := (-x)%R). assert (A: (IZR (Ztrunc y) <= y < IZR (Ztrunc y + 1)%Z)%R). - { apply Ztrunc_range_pos. unfold y. rewrite Ztrunc_opp. omega. } + { apply Ztrunc_range_pos. unfold y. rewrite Ztrunc_opp. lia. } destruct A as [B C]. unfold y in B, C. rewrite Ztrunc_opp in B, C. - replace (- Ztrunc x + 1) with (- (Ztrunc x - 1)) in C by omega. + replace (- Ztrunc x + 1) with (- (Ztrunc x - 1)) in C by lia. rewrite opp_IZR in B, C. lra. Qed. @@ -777,7 +777,7 @@ Theorem ZofB_range_nonneg: Proof. intros. destruct (Z.eq_dec n 0). - subst n. apply ZofB_range_zero. auto. -- destruct (ZofB_range_pos f n) as (A & B). auto. omega. +- destruct (ZofB_range_pos f n) as (A & B). auto. lia. split; auto. apply Rlt_le_trans with 0%R. simpl; lra. apply Rle_trans with (IZR n); auto. apply IZR_le; auto. Qed. @@ -796,7 +796,7 @@ Qed. Remark Zfloor_minus: forall x n, Zfloor (x - IZR n) = Zfloor x - n. Proof. - intros. apply Zfloor_imp. replace (Zfloor x - n + 1) with ((Zfloor x + 1) - n) by omega. + intros. apply Zfloor_imp. replace (Zfloor x - n + 1) with ((Zfloor x + 1) - n) by lia. rewrite ! minus_IZR. unfold Rminus. split. apply Rplus_le_compat_r. apply Zfloor_lb. apply Rplus_lt_compat_r. rewrite plus_IZR. apply Zfloor_ub. @@ -809,11 +809,11 @@ Theorem ZofB_minus: Proof. intros. assert (Q: -2^prec <= q <= 2^prec). - { split; auto. generalize (Zpower_ge_0 radix2 prec); simpl; omega. } - assert (RANGE: (-1 < B2R _ _ f < IZR (p + 1)%Z)%R) by (apply ZofB_range_nonneg; auto; omega). + { split; auto. generalize (Zpower_ge_0 radix2 prec); simpl; lia. } + assert (RANGE: (-1 < B2R _ _ f < IZR (p + 1)%Z)%R) by (apply ZofB_range_nonneg; auto; lia). rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; try discriminate. assert (PQ2: (IZR (p + 1) <= IZR q * 2)%R). - { rewrite <- mult_IZR. apply IZR_le. omega. } + { rewrite <- mult_IZR. apply IZR_le. lia. } assert (EXACT: round radix2 fexp (round_mode m) (B2R _ _ f - IZR q)%R = (B2R _ _ f - IZR q)%R). { apply round_generic. apply valid_rnd_round_mode. apply sterbenz_aux. now apply FLT_exp_valid. apply FLT_exp_monotone. apply generic_format_B2R. @@ -828,7 +828,7 @@ Proof. - rewrite A. fold emin; fold fexp. rewrite EXACT. apply Rle_lt_trans with (bpow radix2 prec). apply Rle_trans with (IZR q). apply Rabs_le. lra. - rewrite <- IZR_Zpower. apply IZR_le; auto. red in prec_gt_0_; omega. + rewrite <- IZR_Zpower. apply IZR_le; auto. red in prec_gt_0_; lia. apply bpow_lt. auto. Qed. @@ -874,8 +874,8 @@ Proof. intros. destruct (ZofB_range_inversion _ _ _ _ H) as (A & B & C). set (f' := Bminus prec emax prec_gt_0_ Hmax minus_nan m f (BofZ q)). assert (D: ZofB f' = Some (p - q)). - { apply ZofB_minus. auto. omega. auto. auto. } - unfold ZofB_range. rewrite D. rewrite Zle_bool_true by omega. rewrite Zle_bool_true by omega. auto. + { apply ZofB_minus. auto. lia. auto. auto. } + unfold ZofB_range. rewrite D. rewrite Zle_bool_true by lia. rewrite Zle_bool_true by lia. auto. Qed. (** ** Algebraic identities *) @@ -961,7 +961,7 @@ Theorem Bmult2_Bplus: Proof. intros until f; intros NAN. destruct (BofZ_representable 2) as (A & B & C). - apply (integer_representable_2p 1). red in prec_gt_0_; omega. + apply (integer_representable_2p 1). red in prec_gt_0_; lia. pose proof (Bmult_correct _ _ _ Hmax mult_nan mode f (BofZ 2%Z)). fold emin in H. rewrite A, B, C in H. rewrite xorb_false_r in H. destruct (is_finite _ _ f) eqn:FIN. @@ -979,7 +979,7 @@ Proof. replace 0%R with (@F2R radix2 {| Fnum := 0%Z; Fexp := e |}). rewrite Rcompare_F2R. destruct s; auto. unfold F2R. simpl. ring. - apply IZR_lt. omega. + apply IZR_lt. lia. destruct (Bmult prec emax prec_gt_0_ Hmax mult_nan mode f (BofZ 2)); reflexivity || discriminate. + destruct H0 as (P & Q). apply B2FF_inj. rewrite P, H. auto. - destruct f as [sf|sf|sf pf Hf|sf mf ef Hf]; try discriminate. @@ -1000,11 +1000,11 @@ Proof. assert (REC: forall n, Z.pos (nat_rect _ xH (fun _ => xO) n) = 2 ^ (Z.of_nat n)). { induction n. reflexivity. simpl nat_rect. transitivity (2 * Z.pos (nat_rect _ xH (fun _ => xO) n)). reflexivity. - rewrite Nat2Z.inj_succ. rewrite IHn. unfold Z.succ. rewrite Zpower_plus by omega. + rewrite Nat2Z.inj_succ. rewrite IHn. unfold Z.succ. rewrite Zpower_plus by lia. change (2 ^ 1) with 2. ring. } red in prec_gt_0_. - unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by omega. rewrite REC. - rewrite Zabs2Nat.id_abs. rewrite Z.abs_eq by omega. auto. + unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by lia. rewrite REC. + rewrite Zabs2Nat.id_abs. rewrite Z.abs_eq by lia. auto. Qed. Remark Bexact_inverse_mantissa_digits2_pos: @@ -1013,11 +1013,11 @@ Proof. assert (DIGITS: forall n, digits2_pos (nat_rect _ xH (fun _ => xO) n) = Pos.of_nat (n+1)). { induction n; simpl. auto. rewrite IHn. destruct n; auto. } red in prec_gt_0_. - unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by omega. rewrite DIGITS. - rewrite Zabs2Nat.abs_nat_nonneg, Z2Nat.inj_sub by omega. + unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by lia. rewrite DIGITS. + rewrite Zabs2Nat.abs_nat_nonneg, Z2Nat.inj_sub by lia. destruct prec; try discriminate. rewrite Nat.sub_add. simpl. rewrite Pos2Nat.id. auto. - simpl. zify; omega. + simpl. zify; lia. Qed. Remark bounded_Bexact_inverse: @@ -1028,8 +1028,8 @@ Proof. rewrite <- Zeq_is_eq_bool. rewrite <- Zle_is_le_bool. rewrite Bexact_inverse_mantissa_digits2_pos. split. -- intros; split. unfold FLT_exp. unfold emin in H. zify; omega. omega. -- intros [A B]. unfold FLT_exp in A. unfold emin. zify; omega. +- intros; split. unfold FLT_exp. unfold emin in H. zify; lia. lia. +- intros [A B]. unfold FLT_exp in A. unfold emin. zify; lia. Qed. Program Definition Bexact_inverse (f: binary_float) : option binary_float := @@ -1045,7 +1045,7 @@ Program Definition Bexact_inverse (f: binary_float) : option binary_float := end. Next Obligation. rewrite <- bounded_Bexact_inverse in B. rewrite <- bounded_Bexact_inverse. - unfold emin in *. omega. + unfold emin in *. lia. Qed. Lemma Bexact_inverse_correct: @@ -1067,9 +1067,9 @@ Proof with (try discriminate). rewrite <- ! cond_Ropp_mult_l. red in prec_gt_0_. replace (IZR (2 ^ (prec - 1))) with (bpow radix2 (prec - 1)) - by (symmetry; apply (IZR_Zpower radix2); omega). + by (symmetry; apply (IZR_Zpower radix2); lia). rewrite <- ! bpow_plus. - replace (prec - 1 + e') with (- (prec - 1 + e)) by (unfold e'; omega). + replace (prec - 1 + e') with (- (prec - 1 + e)) by (unfold e'; lia). rewrite bpow_opp. unfold cond_Ropp; destruct s; auto. rewrite Ropp_inv_permute. auto. apply Rgt_not_eq. apply bpow_gt_0. split. simpl. apply F2R_neq_0. destruct s; simpl in H; discriminate. @@ -1163,9 +1163,9 @@ Proof. assert (C: 0 <= Z.log2_up base) by apply Z.log2_up_nonneg. destruct (Z.log2_spec base) as [D E]; auto. destruct (Z.log2_up_spec base) as [F G]. apply radix_gt_1. - assert (K: 0 <= 2 ^ Z.log2 base) by (apply Z.pow_nonneg; omega). - rewrite ! (Z.mul_comm n). rewrite ! Z.pow_mul_r by omega. - split; apply Z.pow_le_mono_l; omega. + assert (K: 0 <= 2 ^ Z.log2 base) by (apply Z.pow_nonneg; lia). + rewrite ! (Z.mul_comm n). rewrite ! Z.pow_mul_r by lia. + split; apply Z.pow_le_mono_l; lia. Qed. Lemma bpow_log_pos: @@ -1174,8 +1174,8 @@ Lemma bpow_log_pos: (bpow radix2 (n * Z.log2 base)%Z <= bpow base n)%R. Proof. intros. rewrite <- ! IZR_Zpower. apply IZR_le; apply Zpower_log; auto. - omega. - rewrite Z.mul_comm; apply Zmult_gt_0_le_0_compat. omega. apply Z.log2_nonneg. + lia. + rewrite Z.mul_comm; apply Zmult_gt_0_le_0_compat. lia. apply Z.log2_nonneg. Qed. Lemma bpow_log_neg: @@ -1183,10 +1183,10 @@ Lemma bpow_log_neg: n < 0 -> (bpow base n <= bpow radix2 (n * Z.log2 base)%Z)%R. Proof. - intros. set (m := -n). replace n with (-m) by (unfold m; omega). + intros. set (m := -n). replace n with (-m) by (unfold m; lia). rewrite ! Z.mul_opp_l, ! bpow_opp. apply Rinv_le. apply bpow_gt_0. - apply bpow_log_pos. unfold m; omega. + apply bpow_log_pos. unfold m; lia. Qed. (** Overflow and underflow conditions. *) @@ -1203,12 +1203,12 @@ Proof. rewrite <- (Rmult_1_l (bpow radix2 emax)). apply Rmult_le_compat. apply Rle_0_1. apply bpow_ge_0. - apply IZR_le. zify; omega. + apply IZR_le. zify; lia. eapply Rle_trans. eapply bpow_le. eassumption. apply bpow_log_pos; auto. apply generic_format_FLT. exists (Float radix2 1 emax). unfold F2R; simpl. ring. simpl. apply (Zpower_gt_1 radix2); auto. - simpl. unfold emin; red in prec_gt_0_; omega. + simpl. unfold emin; red in prec_gt_0_; lia. Qed. Lemma round_NE_underflows: @@ -1221,10 +1221,10 @@ Proof. assert (A: round radix2 fexp (round_mode mode_NE) eps = 0%R). { unfold round. simpl. assert (E: cexp radix2 fexp eps = emin). - { unfold cexp, eps. rewrite mag_bpow. unfold fexp, FLT_exp. zify; red in prec_gt_0_; omega. } + { unfold cexp, eps. rewrite mag_bpow. unfold fexp, FLT_exp. zify; red in prec_gt_0_; lia. } unfold scaled_mantissa; rewrite E. assert (P: (eps * bpow radix2 (-emin) = / 2)%R). - { unfold eps. rewrite <- bpow_plus. replace (emin - 1 + -emin) with (-1) by omega. auto. } + { unfold eps. rewrite <- bpow_plus. replace (emin - 1 + -emin) with (-1) by lia. auto. } rewrite P. unfold Znearest. assert (F: Zfloor (/ 2)%R = 0). { apply Zfloor_imp. simpl. lra. } @@ -1244,18 +1244,18 @@ Lemma round_integer_underflow: round radix2 fexp (round_mode mode_NE) (IZR (Zpos m) * bpow base e) = 0%R. Proof. intros. apply round_NE_underflows. split. -- apply Rmult_le_pos. apply IZR_le. zify; omega. apply bpow_ge_0. +- apply Rmult_le_pos. apply IZR_le. zify; lia. apply bpow_ge_0. - apply Rle_trans with (bpow radix2 (Z.log2_up (Z.pos m) + e * Z.log2 base)). + rewrite bpow_plus. apply Rmult_le_compat. - apply IZR_le; zify; omega. + apply IZR_le; zify; lia. apply bpow_ge_0. rewrite <- IZR_Zpower. apply IZR_le. destruct (Z.eq_dec (Z.pos m) 1). - rewrite e0. simpl. omega. - apply Z.log2_up_spec. zify; omega. + rewrite e0. simpl. lia. + apply Z.log2_up_spec. zify; lia. apply Z.log2_up_nonneg. apply bpow_log_neg. auto. -+ apply bpow_le. omega. ++ apply bpow_le. lia. Qed. (** Correctness of Bparse *) @@ -1281,20 +1281,20 @@ Proof. - (* e = Zpos e *) destruct (Z.ltb_spec (Z.pos e * Z.log2 (Z.pos b)) emax). + (* no overflow *) - rewrite pos_pow_spec. rewrite <- IZR_Zpower by (zify; omega). rewrite <- mult_IZR. + rewrite pos_pow_spec. rewrite <- IZR_Zpower by (zify; lia). rewrite <- mult_IZR. replace false with (Z.pos m * Z.pos b ^ Z.pos e <? 0). exact (BofZ_correct (Z.pos m * Z.pos b ^ Z.pos e)). - rewrite Z.ltb_ge. rewrite Z.mul_comm. apply Zmult_gt_0_le_0_compat. zify; omega. apply (Zpower_ge_0 base). + rewrite Z.ltb_ge. rewrite Z.mul_comm. apply Zmult_gt_0_le_0_compat. zify; lia. apply (Zpower_ge_0 base). + (* overflow *) rewrite Rlt_bool_false. auto. eapply Rle_trans; [idtac|apply Rle_abs]. - apply (round_integer_overflow base). zify; omega. auto. + apply (round_integer_overflow base). zify; lia. auto. - (* e = Zneg e *) destruct (Z.ltb_spec (Z.neg e * Z.log2 (Z.pos b) + Z.log2_up (Z.pos m)) emin). + (* undeflow *) rewrite round_integer_underflow; auto. rewrite Rlt_bool_true. auto. replace (Rabs 0)%R with 0%R. apply bpow_gt_0. apply (abs_IZR 0). - zify; omega. + zify; lia. + (* no underflow *) generalize (Bdiv_correct_aux prec emax prec_gt_0_ Hmax mode_NE false m 0 false (pos_pow b e) 0). set (f := let '(mz, ez, lz) := Fdiv_core_binary prec emax (Z.pos m) 0 (Z.pos (pos_pow b e)) 0 @@ -1384,13 +1384,13 @@ Proof. apply Rlt_le_trans with (bpow radix2 emax1). rewrite F2R_cond_Zopp. rewrite abs_cond_Ropp. rewrite <- F2R_Zabs. simpl Z.abs. eapply bounded_lt_emax; eauto. - apply bpow_le. omega. + apply bpow_le. lia. } assert (EQ: round radix2 fexp2 (round_mode m) (B2R prec1 emax1 f) = B2R prec1 emax1 f). { apply round_generic. apply valid_rnd_round_mode. eapply generic_inclusion_le. 5: apply generic_format_B2R. apply fexp_correct; auto. apply fexp_correct; auto. - instantiate (1 := emax2). intros. unfold fexp2, FLT_exp. unfold emin2. zify; omega. + instantiate (1 := emax2). intros. unfold fexp2, FLT_exp. unfold emin2. zify; lia. apply Rlt_le; auto. } rewrite EQ. rewrite Rlt_bool_true by auto. auto. @@ -1444,7 +1444,7 @@ Proof. intros. destruct (ZofB_range_inversion _ _ _ _ _ _ H3) as (A & B & C). unfold ZofB_range. erewrite ZofB_Bconv by eauto. - rewrite ! Zle_bool_true by omega. auto. + rewrite ! Zle_bool_true by lia. auto. Qed. (** Change of format (to higher precision) and comparison. *) diff --git a/lib/Integers.v b/lib/Integers.v index 8990c78d..03f19c98 100644 --- a/lib/Integers.v +++ b/lib/Integers.v @@ -72,7 +72,7 @@ Definition min_signed : Z := - half_modulus. Remark wordsize_pos: zwordsize > 0. Proof. - unfold zwordsize, wordsize. generalize WS.wordsize_not_zero. omega. + unfold zwordsize, wordsize. generalize WS.wordsize_not_zero. lia. Qed. Remark modulus_power: modulus = two_p zwordsize. @@ -83,12 +83,12 @@ Qed. Remark modulus_gt_one: modulus > 1. Proof. rewrite modulus_power. apply Z.lt_gt. apply (two_p_monotone_strict 0). - generalize wordsize_pos; omega. + generalize wordsize_pos; lia. Qed. Remark modulus_pos: modulus > 0. Proof. - generalize modulus_gt_one; omega. + generalize modulus_gt_one; lia. Qed. Hint Resolve modulus_pos: ints. @@ -321,16 +321,16 @@ Proof. unfold half_modulus. rewrite modulus_power. set (ws1 := zwordsize - 1). replace (zwordsize) with (Z.succ ws1). - rewrite two_p_S. rewrite Z.mul_comm. apply Z_div_mult. omega. - unfold ws1. generalize wordsize_pos; omega. - unfold ws1. omega. + rewrite two_p_S. rewrite Z.mul_comm. apply Z_div_mult. lia. + unfold ws1. generalize wordsize_pos; lia. + unfold ws1. lia. Qed. Remark half_modulus_modulus: modulus = 2 * half_modulus. Proof. rewrite half_modulus_power. rewrite modulus_power. - rewrite <- two_p_S. apply f_equal. omega. - generalize wordsize_pos; omega. + rewrite <- two_p_S. apply f_equal. lia. + generalize wordsize_pos; lia. Qed. (** Relative positions, from greatest to smallest: @@ -346,38 +346,38 @@ Qed. Remark half_modulus_pos: half_modulus > 0. Proof. - rewrite half_modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; omega. + rewrite half_modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; lia. Qed. Remark min_signed_neg: min_signed < 0. Proof. - unfold min_signed. generalize half_modulus_pos. omega. + unfold min_signed. generalize half_modulus_pos. lia. Qed. Remark max_signed_pos: max_signed >= 0. Proof. - unfold max_signed. generalize half_modulus_pos. omega. + unfold max_signed. generalize half_modulus_pos. lia. Qed. Remark wordsize_max_unsigned: zwordsize <= max_unsigned. Proof. assert (zwordsize < modulus). rewrite modulus_power. apply two_p_strict. - generalize wordsize_pos. omega. - unfold max_unsigned. omega. + generalize wordsize_pos. lia. + unfold max_unsigned. lia. Qed. Remark two_wordsize_max_unsigned: 2 * zwordsize - 1 <= max_unsigned. Proof. assert (2 * zwordsize - 1 < modulus). - rewrite modulus_power. apply two_p_strict_2. generalize wordsize_pos; omega. - unfold max_unsigned; omega. + rewrite modulus_power. apply two_p_strict_2. generalize wordsize_pos; lia. + unfold max_unsigned; lia. Qed. Remark max_signed_unsigned: max_signed < max_unsigned. Proof. unfold max_signed, max_unsigned. rewrite half_modulus_modulus. - generalize half_modulus_pos. omega. + generalize half_modulus_pos. lia. Qed. Lemma unsigned_repr_eq: @@ -495,7 +495,7 @@ Qed. Theorem unsigned_range: forall i, 0 <= unsigned i < modulus. Proof. - destruct i. simpl. omega. + destruct i. simpl. lia. Qed. Hint Resolve unsigned_range: ints. @@ -503,7 +503,7 @@ Theorem unsigned_range_2: forall i, 0 <= unsigned i <= max_unsigned. Proof. intro; unfold max_unsigned. - generalize (unsigned_range i). omega. + generalize (unsigned_range i). lia. Qed. Hint Resolve unsigned_range_2: ints. @@ -513,16 +513,16 @@ Proof. intros. unfold signed. generalize (unsigned_range i). set (n := unsigned i). intros. case (zlt n half_modulus); intro. - unfold max_signed. generalize min_signed_neg. omega. + unfold max_signed. generalize min_signed_neg. lia. unfold min_signed, max_signed. - rewrite half_modulus_modulus in *. omega. + rewrite half_modulus_modulus in *. lia. Qed. Theorem repr_unsigned: forall i, repr (unsigned i) = i. Proof. destruct i; simpl. unfold repr. apply mkint_eq. - rewrite Z_mod_modulus_eq. apply Z.mod_small; omega. + rewrite Z_mod_modulus_eq. apply Z.mod_small; lia. Qed. Hint Resolve repr_unsigned: ints. @@ -545,7 +545,7 @@ Theorem unsigned_repr: forall z, 0 <= z <= max_unsigned -> unsigned (repr z) = z. Proof. intros. rewrite unsigned_repr_eq. - apply Z.mod_small. unfold max_unsigned in H. omega. + apply Z.mod_small. unfold max_unsigned in H. lia. Qed. Hint Resolve unsigned_repr: ints. @@ -554,25 +554,25 @@ Theorem signed_repr: Proof. intros. unfold signed. destruct (zle 0 z). replace (unsigned (repr z)) with z. - rewrite zlt_true. auto. unfold max_signed in H. omega. - symmetry. apply unsigned_repr. generalize max_signed_unsigned. omega. + rewrite zlt_true. auto. unfold max_signed in H. lia. + symmetry. apply unsigned_repr. generalize max_signed_unsigned. lia. pose (z' := z + modulus). replace (repr z) with (repr z'). replace (unsigned (repr z')) with z'. - rewrite zlt_false. unfold z'. omega. + rewrite zlt_false. unfold z'. lia. unfold z'. unfold min_signed in H. - rewrite half_modulus_modulus. omega. + rewrite half_modulus_modulus. lia. symmetry. apply unsigned_repr. unfold z', max_unsigned. unfold min_signed, max_signed in H. - rewrite half_modulus_modulus. omega. - apply eqm_samerepr. unfold z'; red. exists 1. omega. + rewrite half_modulus_modulus. lia. + apply eqm_samerepr. unfold z'; red. exists 1. lia. Qed. Theorem signed_eq_unsigned: forall x, unsigned x <= max_signed -> signed x = unsigned x. Proof. intros. unfold signed. destruct (zlt (unsigned x) half_modulus). - auto. unfold max_signed in H. omegaContradiction. + auto. unfold max_signed in H. extlia. Qed. Theorem signed_positive: @@ -580,7 +580,7 @@ Theorem signed_positive: Proof. intros. unfold signed, max_signed. generalize (unsigned_range x) half_modulus_modulus half_modulus_pos; intros. - destruct (zlt (unsigned x) half_modulus); omega. + destruct (zlt (unsigned x) half_modulus); lia. Qed. (** ** Properties of zero, one, minus one *) @@ -592,11 +592,11 @@ Qed. Theorem unsigned_one: unsigned one = 1. Proof. - unfold one; rewrite unsigned_repr_eq. apply Z.mod_small. split. omega. + unfold one; rewrite unsigned_repr_eq. apply Z.mod_small. split. lia. unfold modulus. replace wordsize with (S(Init.Nat.pred wordsize)). rewrite two_power_nat_S. generalize (two_power_nat_pos (Init.Nat.pred wordsize)). - omega. - generalize wordsize_pos. unfold zwordsize. omega. + lia. + generalize wordsize_pos. unfold zwordsize. lia. Qed. Theorem unsigned_mone: unsigned mone = modulus - 1. @@ -604,25 +604,25 @@ Proof. unfold mone; rewrite unsigned_repr_eq. replace (-1) with ((modulus - 1) + (-1) * modulus). rewrite Z_mod_plus_full. apply Z.mod_small. - generalize modulus_pos. omega. omega. + generalize modulus_pos. lia. lia. Qed. Theorem signed_zero: signed zero = 0. Proof. - unfold signed. rewrite unsigned_zero. apply zlt_true. generalize half_modulus_pos; omega. + unfold signed. rewrite unsigned_zero. apply zlt_true. generalize half_modulus_pos; lia. Qed. Theorem signed_one: zwordsize > 1 -> signed one = 1. Proof. intros. unfold signed. rewrite unsigned_one. apply zlt_true. - change 1 with (two_p 0). rewrite half_modulus_power. apply two_p_monotone_strict. omega. + change 1 with (two_p 0). rewrite half_modulus_power. apply two_p_monotone_strict. lia. Qed. Theorem signed_mone: signed mone = -1. Proof. unfold signed. rewrite unsigned_mone. - rewrite zlt_false. omega. - rewrite half_modulus_modulus. generalize half_modulus_pos. omega. + rewrite zlt_false. lia. + rewrite half_modulus_modulus. generalize half_modulus_pos. lia. Qed. Theorem one_not_zero: one <> zero. @@ -636,7 +636,7 @@ Theorem unsigned_repr_wordsize: unsigned iwordsize = zwordsize. Proof. unfold iwordsize; rewrite unsigned_repr_eq. apply Z.mod_small. - generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; omega. + generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; lia. Qed. (** ** Properties of equality *) @@ -695,7 +695,7 @@ Proof. Qed. Theorem add_commut: forall x y, add x y = add y x. -Proof. intros; unfold add. decEq. omega. Qed. +Proof. intros; unfold add. decEq. lia. Qed. Theorem add_zero: forall x, add x zero = x. Proof. @@ -729,7 +729,7 @@ Theorem add_neg_zero: forall x, add x (neg x) = zero. Proof. intros; unfold add, neg, zero. apply eqm_samerepr. replace 0 with (unsigned x + (- (unsigned x))). - auto with ints. omega. + auto with ints. lia. Qed. Theorem unsigned_add_carry: @@ -741,8 +741,8 @@ Proof. rewrite unsigned_repr_eq. generalize (unsigned_range x) (unsigned_range y). intros. destruct (zlt (unsigned x + unsigned y) modulus). - rewrite unsigned_zero. apply Zmod_unique with 0. omega. omega. - rewrite unsigned_one. apply Zmod_unique with 1. omega. omega. + rewrite unsigned_zero. apply Zmod_unique with 0. lia. lia. + rewrite unsigned_one. apply Zmod_unique with 1. lia. lia. Qed. Corollary unsigned_add_either: @@ -753,8 +753,8 @@ Proof. intros. rewrite unsigned_add_carry. unfold add_carry. rewrite unsigned_zero. rewrite Z.add_0_r. destruct (zlt (unsigned x + unsigned y) modulus). - rewrite unsigned_zero. left; omega. - rewrite unsigned_one. right; omega. + rewrite unsigned_zero. left; lia. + rewrite unsigned_one. right; lia. Qed. (** ** Properties of negation *) @@ -773,7 +773,7 @@ Theorem neg_involutive: forall x, neg (neg x) = x. Proof. intros; unfold neg. apply eqm_repr_eq. eapply eqm_trans. apply eqm_neg. - apply eqm_unsigned_repr_l. apply eqm_refl. apply eqm_refl2. omega. + apply eqm_unsigned_repr_l. apply eqm_refl. apply eqm_refl2. lia. Qed. Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y). @@ -783,7 +783,7 @@ Proof. auto with ints. replace (- (unsigned x + unsigned y)) with ((- unsigned x) + (- unsigned y)). - auto with ints. omega. + auto with ints. lia. Qed. (** ** Properties of subtraction *) @@ -791,7 +791,7 @@ Qed. Theorem sub_zero_l: forall x, sub x zero = x. Proof. intros; unfold sub. rewrite unsigned_zero. - replace (unsigned x - 0) with (unsigned x) by omega. apply repr_unsigned. + replace (unsigned x - 0) with (unsigned x) by lia. apply repr_unsigned. Qed. Theorem sub_zero_r: forall x, sub zero x = neg x. @@ -807,7 +807,7 @@ Qed. Theorem sub_idem: forall x, sub x x = zero. Proof. - intros; unfold sub. unfold zero. decEq. omega. + intros; unfold sub. unfold zero. decEq. lia. Qed. Theorem sub_add_l: forall x y z, sub (add x y) z = add (sub x z) y. @@ -850,8 +850,8 @@ Proof. rewrite unsigned_repr_eq. generalize (unsigned_range x) (unsigned_range y). intros. destruct (zlt (unsigned x - unsigned y) 0). - rewrite unsigned_one. apply Zmod_unique with (-1). omega. omega. - rewrite unsigned_zero. apply Zmod_unique with 0. omega. omega. + rewrite unsigned_one. apply Zmod_unique with (-1). lia. lia. + rewrite unsigned_zero. apply Zmod_unique with 0. lia. lia. Qed. (** ** Properties of multiplication *) @@ -878,9 +878,9 @@ Theorem mul_mone: forall x, mul x mone = neg x. Proof. intros; unfold mul, neg. rewrite unsigned_mone. apply eqm_samerepr. - replace (-unsigned x) with (0 - unsigned x) by omega. + replace (-unsigned x) with (0 - unsigned x) by lia. replace (unsigned x * (modulus - 1)) with (unsigned x * modulus - unsigned x) by ring. - apply eqm_sub. exists (unsigned x). omega. apply eqm_refl. + apply eqm_sub. exists (unsigned x). lia. apply eqm_refl. Qed. Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z). @@ -955,7 +955,7 @@ Proof. generalize (unsigned_range y); intro. assert (unsigned y <> 0). red; intro. elim H. rewrite <- (repr_unsigned y). unfold zero. congruence. - unfold y'. omega. + unfold y'. lia. auto with ints. Qed. @@ -1025,7 +1025,7 @@ Proof. assert (Z.quot x' one = x'). symmetry. apply Zquot_unique_full with 0. red. change (Z.abs one) with 1. - destruct (zle 0 x'). left. omega. right. omega. + destruct (zle 0 x'). left. lia. right. lia. unfold one; ring. congruence. Qed. @@ -1053,12 +1053,12 @@ Proof. assert (unsigned d <> 0). { red; intros. elim H. rewrite <- (repr_unsigned d). rewrite H0; auto. } assert (0 < D). - { unfold D. generalize (unsigned_range d); intros. omega. } + { unfold D. generalize (unsigned_range d); intros. lia. } assert (0 <= Q <= max_unsigned). { unfold Q. apply Zdiv_interval_2. rewrite <- E1; apply unsigned_range_2. - omega. unfold max_unsigned; generalize modulus_pos; omega. omega. } - omega. + lia. unfold max_unsigned; generalize modulus_pos; lia. lia. } + lia. Qed. Lemma unsigned_signed: @@ -1067,8 +1067,8 @@ Proof. intros. unfold lt. rewrite signed_zero. unfold signed. generalize (unsigned_range n). rewrite half_modulus_modulus. intros. destruct (zlt (unsigned n) half_modulus). -- rewrite zlt_false by omega. auto. -- rewrite zlt_true by omega. ring. +- rewrite zlt_false by lia. auto. +- rewrite zlt_true by lia. ring. Qed. Theorem divmods2_divs_mods: @@ -1096,24 +1096,24 @@ Proof. - (* D = 1 *) rewrite e. rewrite Z.quot_1_r; auto. - (* D = -1 *) - rewrite e. change (-1) with (Z.opp 1). rewrite Z.quot_opp_r by omega. + rewrite e. change (-1) with (Z.opp 1). rewrite Z.quot_opp_r by lia. rewrite Z.quot_1_r. assert (N <> min_signed). { red; intros; destruct H0. + elim H0. rewrite <- (repr_signed n). rewrite <- H2. rewrite H4. auto. + elim H0. rewrite <- (repr_signed d). unfold D in e; rewrite e; auto. } - unfold min_signed, max_signed in *. omega. + unfold min_signed, max_signed in *. lia. - (* |D| > 1 *) assert (Z.abs (Z.quot N D) < half_modulus). - { rewrite <- Z.quot_abs by omega. apply Zquot_lt_upper_bound. - xomega. xomega. + { rewrite <- Z.quot_abs by lia. apply Zquot_lt_upper_bound. + extlia. extlia. apply Z.le_lt_trans with (half_modulus * 1). - rewrite Z.mul_1_r. unfold min_signed, max_signed in H3; xomega. - apply Zmult_lt_compat_l. generalize half_modulus_pos; omega. xomega. } + rewrite Z.mul_1_r. unfold min_signed, max_signed in H3; extlia. + apply Zmult_lt_compat_l. generalize half_modulus_pos; lia. extlia. } rewrite Z.abs_lt in H4. - unfold min_signed, max_signed; omega. + unfold min_signed, max_signed; lia. } - unfold proj_sumbool; rewrite ! zle_true by omega; simpl. + unfold proj_sumbool; rewrite ! zle_true by lia; simpl. unfold Q, R; rewrite H2; auto. Qed. @@ -1164,7 +1164,7 @@ Qed. Lemma bits_mone: forall i, 0 <= i < zwordsize -> testbit mone i = true. Proof. - intros. unfold mone. rewrite testbit_repr; auto. apply Ztestbit_m1. omega. + intros. unfold mone. rewrite testbit_repr; auto. apply Ztestbit_m1. lia. Qed. Hint Rewrite bits_zero bits_mone : ints. @@ -1181,7 +1181,7 @@ Proof. unfold zwordsize, ws1, wordsize. destruct WS.wordsize as [] eqn:E. elim WS.wordsize_not_zero; auto. - rewrite Nat2Z.inj_succ. simpl. omega. + rewrite Nat2Z.inj_succ. simpl. lia. assert (half_modulus = two_power_nat ws1). rewrite two_power_nat_two_p. rewrite <- H. apply half_modulus_power. rewrite H; rewrite H0. @@ -1195,11 +1195,11 @@ Lemma bits_signed: Proof. intros. destruct (zlt i zwordsize). - - apply same_bits_eqm. apply eqm_signed_unsigned. omega. + - apply same_bits_eqm. apply eqm_signed_unsigned. lia. - unfold signed. rewrite sign_bit_of_unsigned. destruct (zlt (unsigned x) half_modulus). + apply Ztestbit_above with wordsize. apply unsigned_range. auto. + apply Ztestbit_above_neg with wordsize. - fold modulus. generalize (unsigned_range x). omega. auto. + fold modulus. generalize (unsigned_range x). lia. auto. Qed. Lemma bits_le: @@ -1207,9 +1207,9 @@ Lemma bits_le: (forall i, 0 <= i < zwordsize -> testbit x i = true -> testbit y i = true) -> unsigned x <= unsigned y. Proof. - intros. apply Ztestbit_le. generalize (unsigned_range y); omega. + intros. apply Ztestbit_le. generalize (unsigned_range y); lia. intros. fold (testbit y i). destruct (zlt i zwordsize). - apply H. omega. auto. + apply H. lia. auto. fold (testbit x i) in H1. rewrite bits_above in H1; auto. congruence. Qed. @@ -1477,10 +1477,10 @@ Lemma unsigned_not: forall x, unsigned (not x) = max_unsigned - unsigned x. Proof. intros. transitivity (unsigned (repr(-unsigned x - 1))). - f_equal. bit_solve. rewrite testbit_repr; auto. symmetry. apply Z_one_complement. omega. + f_equal. bit_solve. rewrite testbit_repr; auto. symmetry. apply Z_one_complement. lia. rewrite unsigned_repr_eq. apply Zmod_unique with (-1). - unfold max_unsigned. omega. - generalize (unsigned_range x). unfold max_unsigned. omega. + unfold max_unsigned. lia. + generalize (unsigned_range x). unfold max_unsigned. lia. Qed. Theorem not_neg: @@ -1490,9 +1490,9 @@ Proof. rewrite <- (repr_unsigned x) at 1. unfold add. rewrite !testbit_repr; auto. transitivity (Z.testbit (-unsigned x - 1) i). - symmetry. apply Z_one_complement. omega. + symmetry. apply Z_one_complement. lia. apply same_bits_eqm; auto. - replace (-unsigned x - 1) with (-unsigned x + (-1)) by omega. + replace (-unsigned x - 1) with (-unsigned x + (-1)) by lia. apply eqm_add. unfold neg. apply eqm_unsigned_repr. rewrite unsigned_mone. exists (-1). ring. @@ -1534,9 +1534,9 @@ Proof. replace (unsigned (xor b one)) with (1 - unsigned b). destruct (zlt (unsigned x - unsigned y - unsigned b)). rewrite zlt_true. rewrite xor_zero_l; auto. - unfold max_unsigned; omega. + unfold max_unsigned; lia. rewrite zlt_false. rewrite xor_idem; auto. - unfold max_unsigned; omega. + unfold max_unsigned; lia. destruct H; subst b. rewrite xor_zero_l. rewrite unsigned_one, unsigned_zero; auto. rewrite xor_idem. rewrite unsigned_one, unsigned_zero; auto. @@ -1555,16 +1555,16 @@ Proof. rewrite (Zdecomp x) in *. rewrite (Zdecomp y) in *. transitivity (Z.testbit (Zshiftin (Z.odd x || Z.odd y) (Z.div2 x + Z.div2 y)) i). - f_equal. rewrite !Zshiftin_spec. - exploit (EXCL 0). omega. rewrite !Ztestbit_shiftin_base. intros. + exploit (EXCL 0). lia. rewrite !Ztestbit_shiftin_base. intros. Opaque Z.mul. destruct (Z.odd x); destruct (Z.odd y); simpl in *; discriminate || ring. - rewrite !Ztestbit_shiftin; auto. destruct (zeq i 0). + auto. - + apply IND. omega. intros. - exploit (EXCL (Z.succ j)). omega. + + apply IND. lia. intros. + exploit (EXCL (Z.succ j)). lia. rewrite !Ztestbit_shiftin_succ. auto. - omega. omega. + lia. lia. Qed. Theorem add_is_or: @@ -1573,10 +1573,10 @@ Theorem add_is_or: add x y = or x y. Proof. bit_solve. unfold add. rewrite testbit_repr; auto. - apply Z_add_is_or. omega. + apply Z_add_is_or. lia. intros. assert (testbit (and x y) j = testbit zero j) by congruence. - autorewrite with ints in H2. assumption. omega. + autorewrite with ints in H2. assumption. lia. Qed. Theorem xor_is_or: @@ -1622,7 +1622,7 @@ Proof. intros. unfold shl. rewrite testbit_repr; auto. destruct (zlt i (unsigned y)). apply Z.shiftl_spec_low. auto. - apply Z.shiftl_spec_high. omega. omega. + apply Z.shiftl_spec_high. lia. lia. Qed. Lemma bits_shru: @@ -1636,7 +1636,7 @@ Proof. destruct (zlt (i + unsigned y) zwordsize). auto. apply bits_above; auto. - omega. + lia. Qed. Lemma bits_shr: @@ -1647,15 +1647,15 @@ Lemma bits_shr: Proof. intros. unfold shr. rewrite testbit_repr; auto. rewrite Z.shiftr_spec. apply bits_signed. - generalize (unsigned_range y); omega. - omega. + generalize (unsigned_range y); lia. + lia. Qed. Hint Rewrite bits_shl bits_shru bits_shr: ints. Theorem shl_zero: forall x, shl x zero = x. Proof. - bit_solve. rewrite unsigned_zero. rewrite zlt_false. f_equal; omega. omega. + bit_solve. rewrite unsigned_zero. rewrite zlt_false. f_equal; lia. lia. Qed. Lemma bitwise_binop_shl: @@ -1667,7 +1667,7 @@ Proof. intros. apply same_bits_eq; intros. rewrite H; auto. rewrite !bits_shl; auto. destruct (zlt i (unsigned n)); auto. - rewrite H; auto. generalize (unsigned_range n); omega. + rewrite H; auto. generalize (unsigned_range n); lia. Qed. Theorem and_shl: @@ -1695,7 +1695,7 @@ Lemma ltu_inv: forall x y, ltu x y = true -> 0 <= unsigned x < unsigned y. Proof. unfold ltu; intros. destruct (zlt (unsigned x) (unsigned y)). - split; auto. generalize (unsigned_range x); omega. + split; auto. generalize (unsigned_range x); lia. discriminate. Qed. @@ -1716,15 +1716,15 @@ Proof. generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros. assert (unsigned (add y z) = unsigned y + unsigned z). unfold add. apply unsigned_repr. - generalize two_wordsize_max_unsigned; omega. + generalize two_wordsize_max_unsigned; lia. apply same_bits_eq; intros. rewrite bits_shl; auto. destruct (zlt i (unsigned z)). - - rewrite bits_shl; auto. rewrite zlt_true. auto. omega. + - rewrite bits_shl; auto. rewrite zlt_true. auto. lia. - rewrite bits_shl. destruct (zlt (i - unsigned z) (unsigned y)). - + rewrite bits_shl; auto. rewrite zlt_true. auto. omega. - + rewrite bits_shl; auto. rewrite zlt_false. f_equal. omega. omega. - + omega. + + rewrite bits_shl; auto. rewrite zlt_true. auto. lia. + + rewrite bits_shl; auto. rewrite zlt_false. f_equal. lia. lia. + + lia. Qed. Theorem sub_ltu: @@ -1734,12 +1734,12 @@ Theorem sub_ltu: Proof. intros. generalize (ltu_inv x y H). intros . - split. omega. omega. + split. lia. lia. Qed. Theorem shru_zero: forall x, shru x zero = x. Proof. - bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; omega. omega. + bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; lia. lia. Qed. Lemma bitwise_binop_shru: @@ -1751,7 +1751,7 @@ Proof. intros. apply same_bits_eq; intros. rewrite H; auto. rewrite !bits_shru; auto. destruct (zlt (i + unsigned n) zwordsize); auto. - rewrite H; auto. generalize (unsigned_range n); omega. + rewrite H; auto. generalize (unsigned_range n); lia. Qed. Theorem and_shru: @@ -1786,20 +1786,20 @@ Proof. generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros. assert (unsigned (add y z) = unsigned y + unsigned z). unfold add. apply unsigned_repr. - generalize two_wordsize_max_unsigned; omega. + generalize two_wordsize_max_unsigned; lia. apply same_bits_eq; intros. rewrite bits_shru; auto. destruct (zlt (i + unsigned z) zwordsize). - rewrite bits_shru. destruct (zlt (i + unsigned z + unsigned y) zwordsize). - + rewrite bits_shru; auto. rewrite zlt_true. f_equal. omega. omega. - + rewrite bits_shru; auto. rewrite zlt_false. auto. omega. - + omega. - - rewrite bits_shru; auto. rewrite zlt_false. auto. omega. + + rewrite bits_shru; auto. rewrite zlt_true. f_equal. lia. lia. + + rewrite bits_shru; auto. rewrite zlt_false. auto. lia. + + lia. + - rewrite bits_shru; auto. rewrite zlt_false. auto. lia. Qed. Theorem shr_zero: forall x, shr x zero = x. Proof. - bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; omega. omega. + bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; lia. lia. Qed. Lemma bitwise_binop_shr: @@ -1811,8 +1811,8 @@ Proof. rewrite H; auto. rewrite !bits_shr; auto. rewrite H; auto. destruct (zlt (i + unsigned n) zwordsize). - generalize (unsigned_range n); omega. - omega. + generalize (unsigned_range n); lia. + lia. Qed. Theorem and_shr: @@ -1847,15 +1847,15 @@ Proof. generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros. assert (unsigned (add y z) = unsigned y + unsigned z). unfold add. apply unsigned_repr. - generalize two_wordsize_max_unsigned; omega. + generalize two_wordsize_max_unsigned; lia. apply same_bits_eq; intros. rewrite !bits_shr; auto. f_equal. destruct (zlt (i + unsigned z) zwordsize). - rewrite H4. replace (i + (unsigned y + unsigned z)) with (i + unsigned z + unsigned y) by omega. auto. + rewrite H4. replace (i + (unsigned y + unsigned z)) with (i + unsigned z + unsigned y) by lia. auto. rewrite (zlt_false _ (i + unsigned (add y z))). - destruct (zlt (zwordsize - 1 + unsigned y) zwordsize); omega. - omega. - destruct (zlt (i + unsigned z) zwordsize); omega. + destruct (zlt (zwordsize - 1 + unsigned y) zwordsize); lia. + lia. + destruct (zlt (i + unsigned z) zwordsize); lia. Qed. Theorem and_shr_shru: @@ -1865,7 +1865,7 @@ Proof. intros. apply same_bits_eq; intros. rewrite bits_and; auto. rewrite bits_shr; auto. rewrite !bits_shru; auto. destruct (zlt (i + unsigned z) zwordsize). - - rewrite bits_and; auto. generalize (unsigned_range z); omega. + - rewrite bits_and; auto. generalize (unsigned_range z); lia. - apply andb_false_r. Qed. @@ -1891,17 +1891,17 @@ Proof. rewrite sign_bit_of_unsigned. unfold lt. rewrite signed_zero. unfold signed. destruct (zlt (unsigned x) half_modulus). - rewrite zlt_false. auto. generalize (unsigned_range x); omega. + rewrite zlt_false. auto. generalize (unsigned_range x); lia. rewrite zlt_true. unfold one; rewrite testbit_repr; auto. - generalize (unsigned_range x); omega. - omega. + generalize (unsigned_range x); lia. + lia. rewrite zlt_false. unfold testbit. rewrite Ztestbit_eq. rewrite zeq_false. destruct (lt x zero). rewrite unsigned_one. simpl Z.div2. rewrite Z.testbit_0_l; auto. rewrite unsigned_zero. simpl Z.div2. rewrite Z.testbit_0_l; auto. - auto. omega. omega. - generalize wordsize_max_unsigned; omega. + auto. lia. lia. + generalize wordsize_max_unsigned; lia. Qed. Theorem shr_lt_zero: @@ -1912,13 +1912,13 @@ Proof. rewrite bits_shr; auto. rewrite unsigned_repr. transitivity (testbit x (zwordsize - 1)). - f_equal. destruct (zlt (i + (zwordsize - 1)) zwordsize); omega. + f_equal. destruct (zlt (i + (zwordsize - 1)) zwordsize); lia. rewrite sign_bit_of_unsigned. unfold lt. rewrite signed_zero. unfold signed. destruct (zlt (unsigned x) half_modulus). - rewrite zlt_false. rewrite bits_zero; auto. generalize (unsigned_range x); omega. - rewrite zlt_true. rewrite bits_mone; auto. generalize (unsigned_range x); omega. - generalize wordsize_max_unsigned; omega. + rewrite zlt_false. rewrite bits_zero; auto. generalize (unsigned_range x); lia. + rewrite zlt_true. rewrite bits_mone; auto. generalize (unsigned_range x); lia. + generalize wordsize_max_unsigned; lia. Qed. (** ** Properties of rotations *) @@ -1935,20 +1935,20 @@ Proof. exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos. fold j. intros RANGE. rewrite testbit_repr; auto. - rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: omega. + rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: lia. destruct (zlt i j). - rewrite Z.shiftl_spec_low; auto. simpl. unfold testbit. f_equal. symmetry. apply Zmod_unique with (-k - 1). rewrite EQ. ring. - omega. + lia. - rewrite Z.shiftl_spec_high. fold (testbit x (i + (zwordsize - j))). rewrite bits_above. rewrite orb_false_r. fold (testbit x (i - j)). f_equal. symmetry. apply Zmod_unique with (-k). rewrite EQ. ring. - omega. omega. omega. omega. + lia. lia. lia. lia. Qed. Lemma bits_ror: @@ -1963,20 +1963,20 @@ Proof. exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos. fold j. intros RANGE. rewrite testbit_repr; auto. - rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: omega. + rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: lia. destruct (zlt (i + j) zwordsize). - rewrite Z.shiftl_spec_low; auto. rewrite orb_false_r. unfold testbit. f_equal. symmetry. apply Zmod_unique with k. rewrite EQ. ring. - omega. omega. + lia. lia. - rewrite Z.shiftl_spec_high. fold (testbit x (i + j)). rewrite bits_above. simpl. unfold testbit. f_equal. symmetry. apply Zmod_unique with (k + 1). rewrite EQ. ring. - omega. omega. omega. omega. + lia. lia. lia. lia. Qed. Hint Rewrite bits_rol bits_ror: ints. @@ -1993,8 +1993,8 @@ Proof. - rewrite andb_false_r; auto. - generalize (unsigned_range n); intros. rewrite bits_mone. rewrite andb_true_r. f_equal. - symmetry. apply Z.mod_small. omega. - omega. + symmetry. apply Z.mod_small. lia. + lia. Qed. Theorem shru_rolm: @@ -2009,9 +2009,9 @@ Proof. - generalize (unsigned_range n); intros. rewrite bits_mone. rewrite andb_true_r. f_equal. unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize. - symmetry. apply Zmod_unique with (-1). ring. omega. - rewrite unsigned_repr_wordsize. generalize wordsize_max_unsigned. omega. - omega. + symmetry. apply Zmod_unique with (-1). ring. lia. + rewrite unsigned_repr_wordsize. generalize wordsize_max_unsigned. lia. + lia. - rewrite andb_false_r; auto. Qed. @@ -2065,11 +2065,11 @@ Proof. apply eqmod_sub. apply eqmod_sym. apply eqmod_mod. apply wordsize_pos. apply eqmod_refl. - replace (i - M - N) with (i - (M + N)) by omega. + replace (i - M - N) with (i - (M + N)) by lia. apply eqmod_sub. apply eqmod_refl. apply eqmod_trans with (Z.modulo (unsigned n + unsigned m) zwordsize). - replace (M + N) with (N + M) by omega. apply eqmod_mod. apply wordsize_pos. + replace (M + N) with (N + M) by lia. apply eqmod_mod. apply wordsize_pos. unfold modu, add. fold M; fold N. rewrite unsigned_repr_wordsize. assert (forall a, eqmod zwordsize a (unsigned (repr a))). intros. eapply eqmod_divides. apply eqm_unsigned_repr. assumption. @@ -2116,7 +2116,7 @@ Proof. unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize. apply eqmod_mod_eq. apply wordsize_pos. exists 1. ring. rewrite unsigned_repr_wordsize. - generalize wordsize_pos; generalize wordsize_max_unsigned; omega. + generalize wordsize_pos; generalize wordsize_max_unsigned; lia. Qed. Theorem ror_rol_neg: @@ -2124,9 +2124,9 @@ Theorem ror_rol_neg: Proof. intros. apply same_bits_eq; intros. rewrite bits_ror by auto. rewrite bits_rol by auto. - f_equal. apply eqmod_mod_eq. omega. + f_equal. apply eqmod_mod_eq. lia. apply eqmod_trans with (i - (- unsigned y)). - apply eqmod_refl2; omega. + apply eqmod_refl2; lia. apply eqmod_sub. apply eqmod_refl. apply eqmod_divides with modulus. apply eqm_unsigned_repr. auto. @@ -2149,8 +2149,8 @@ Proof. assert (unsigned (add y z) = zwordsize). rewrite H1. apply unsigned_repr_wordsize. unfold add in H5. rewrite unsigned_repr in H5. - omega. - generalize two_wordsize_max_unsigned; omega. + lia. + generalize two_wordsize_max_unsigned; lia. - apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal. apply Z.mod_small; auto. Qed. @@ -2166,10 +2166,10 @@ Proof. destruct (Z_is_power2 (unsigned n)) as [i|] eqn:E; inv H. assert (0 <= i < zwordsize). { apply Z_is_power2_range with (unsigned n). - generalize wordsize_pos; omega. + generalize wordsize_pos; lia. rewrite <- modulus_power. apply unsigned_range. auto. } - rewrite unsigned_repr; auto. generalize wordsize_max_unsigned; omega. + rewrite unsigned_repr; auto. generalize wordsize_max_unsigned; lia. Qed. Lemma is_power2_rng: @@ -2203,10 +2203,10 @@ Remark two_p_range: 0 <= two_p n <= max_unsigned. Proof. intros. split. - assert (two_p n > 0). apply two_p_gt_ZERO. omega. omega. + assert (two_p n > 0). apply two_p_gt_ZERO. lia. lia. generalize (two_p_monotone_strict _ _ H). unfold zwordsize; rewrite <- two_power_nat_two_p. - unfold max_unsigned, modulus. omega. + unfold max_unsigned, modulus. lia. Qed. Lemma is_power2_two_p: @@ -2214,7 +2214,7 @@ Lemma is_power2_two_p: is_power2 (repr (two_p n)) = Some (repr n). Proof. intros. unfold is_power2. rewrite unsigned_repr. - rewrite Z_is_power2_complete by omega; auto. + rewrite Z_is_power2_complete by lia; auto. apply two_p_range. auto. Qed. @@ -2228,7 +2228,7 @@ Lemma shl_mul_two_p: Proof. intros. unfold shl, mul. apply eqm_samerepr. rewrite Zshiftl_mul_two_p. auto with ints. - generalize (unsigned_range y); omega. + generalize (unsigned_range y); lia. Qed. Theorem shl_mul: @@ -2264,19 +2264,19 @@ Proof. rewrite shl_mul_two_p. unfold mul. apply eqm_unsigned_repr_l. apply eqm_mult; auto with ints. apply eqm_unsigned_repr_l. apply eqm_refl2. rewrite unsigned_repr. auto. - generalize wordsize_max_unsigned; omega. + generalize wordsize_max_unsigned; lia. - bit_solve. rewrite unsigned_repr. destruct (zlt i n). + auto. + replace (testbit y i) with false. apply andb_false_r. symmetry. unfold testbit. - assert (EQ: Z.of_nat (Z.to_nat n) = n) by (apply Z2Nat.id; omega). + assert (EQ: Z.of_nat (Z.to_nat n) = n) by (apply Z2Nat.id; lia). apply Ztestbit_above with (Z.to_nat n). rewrite <- EQ in H0. rewrite <- two_power_nat_two_p in H0. - generalize (unsigned_range y); omega. + generalize (unsigned_range y); lia. rewrite EQ; auto. - + generalize wordsize_max_unsigned; omega. + + generalize wordsize_max_unsigned; lia. Qed. (** Unsigned right shifts and unsigned divisions by powers of 2. *) @@ -2287,7 +2287,7 @@ Lemma shru_div_two_p: Proof. intros. unfold shru. rewrite Zshiftr_div_two_p. auto. - generalize (unsigned_range y); omega. + generalize (unsigned_range y); lia. Qed. Theorem divu_pow2: @@ -2307,7 +2307,7 @@ Lemma shr_div_two_p: Proof. intros. unfold shr. rewrite Zshiftr_div_two_p. auto. - generalize (unsigned_range y); omega. + generalize (unsigned_range y); lia. Qed. Theorem divs_pow2: @@ -2360,24 +2360,24 @@ Proof. set (uy := unsigned y). assert (0 <= uy < zwordsize - 1). generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto. - generalize wordsize_pos wordsize_max_unsigned; omega. + generalize wordsize_pos wordsize_max_unsigned; lia. rewrite shr_div_two_p. unfold shrx. unfold divs. assert (shl one y = repr (two_p uy)). transitivity (mul one (repr (two_p uy))). symmetry. apply mul_pow2. replace y with (repr uy). - apply is_power2_two_p. omega. apply repr_unsigned. + apply is_power2_two_p. lia. apply repr_unsigned. rewrite mul_commut. apply mul_one. - assert (two_p uy > 0). apply two_p_gt_ZERO. omega. + assert (two_p uy > 0). apply two_p_gt_ZERO. lia. assert (two_p uy < half_modulus). rewrite half_modulus_power. apply two_p_monotone_strict. auto. assert (two_p uy < modulus). - rewrite modulus_power. apply two_p_monotone_strict. omega. + rewrite modulus_power. apply two_p_monotone_strict. lia. assert (unsigned (shl one y) = two_p uy). - rewrite H1. apply unsigned_repr. unfold max_unsigned. omega. + rewrite H1. apply unsigned_repr. unfold max_unsigned. lia. assert (signed (shl one y) = two_p uy). rewrite H1. apply signed_repr. - unfold max_signed. generalize min_signed_neg. omega. + unfold max_signed. generalize min_signed_neg. lia. rewrite H6. rewrite Zquot_Zdiv; auto. unfold lt. rewrite signed_zero. @@ -2386,10 +2386,10 @@ Proof. assert (signed (sub (shl one y) one) = two_p uy - 1). unfold sub. rewrite H5. rewrite unsigned_one. apply signed_repr. - generalize min_signed_neg. unfold max_signed. omega. - rewrite H7. rewrite signed_repr. f_equal. f_equal. omega. + generalize min_signed_neg. unfold max_signed. lia. + rewrite H7. rewrite signed_repr. f_equal. f_equal. lia. generalize (signed_range x). intros. - assert (two_p uy - 1 <= max_signed). unfold max_signed. omega. omega. + assert (two_p uy - 1 <= max_signed). unfold max_signed. lia. lia. Qed. Theorem shrx_shr_2: @@ -2404,19 +2404,19 @@ Proof. generalize (unsigned_range y); fold uy; intros. assert (0 <= uy < zwordsize - 1). generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto. - generalize wordsize_pos wordsize_max_unsigned; omega. + generalize wordsize_pos wordsize_max_unsigned; lia. assert (two_p uy < modulus). - rewrite modulus_power. apply two_p_monotone_strict. omega. + rewrite modulus_power. apply two_p_monotone_strict. lia. f_equal. rewrite shl_mul_two_p. fold uy. rewrite mul_commut. rewrite mul_one. unfold sub. rewrite unsigned_one. rewrite unsigned_repr. rewrite unsigned_repr_wordsize. fold uy. apply same_bits_eq; intros. rewrite bits_shru by auto. - rewrite testbit_repr by auto. rewrite Ztestbit_two_p_m1 by omega. - rewrite unsigned_repr by (generalize wordsize_max_unsigned; omega). + rewrite testbit_repr by auto. rewrite Ztestbit_two_p_m1 by lia. + rewrite unsigned_repr by (generalize wordsize_max_unsigned; lia). destruct (zlt i uy). - rewrite zlt_true by omega. rewrite bits_mone by omega. auto. - rewrite zlt_false by omega. auto. - assert (two_p uy > 0) by (apply two_p_gt_ZERO; omega). unfold max_unsigned; omega. + rewrite zlt_true by lia. rewrite bits_mone by lia. auto. + rewrite zlt_false by lia. auto. + assert (two_p uy > 0) by (apply two_p_gt_ZERO; lia). unfold max_unsigned; lia. - replace (shru zero (sub iwordsize y)) with zero. rewrite add_zero; auto. bit_solve. destruct (zlt (i + unsigned (sub iwordsize y)) zwordsize); auto. @@ -2434,23 +2434,23 @@ Proof. set (uy := unsigned y). assert (0 <= uy < zwordsize - 1). generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto. - generalize wordsize_pos wordsize_max_unsigned; omega. + generalize wordsize_pos wordsize_max_unsigned; lia. assert (shl one y = repr (two_p uy)). rewrite shl_mul_two_p. rewrite mul_commut. apply mul_one. assert (and x (sub (shl one y) one) = modu x (repr (two_p uy))). symmetry. rewrite H1. apply modu_and with (logn := y). rewrite is_power2_two_p. unfold uy. rewrite repr_unsigned. auto. - omega. + lia. rewrite H2. rewrite H1. repeat rewrite shr_div_two_p. fold sx. fold uy. - assert (two_p uy > 0). apply two_p_gt_ZERO. omega. + assert (two_p uy > 0). apply two_p_gt_ZERO. lia. assert (two_p uy < modulus). - rewrite modulus_power. apply two_p_monotone_strict. omega. + rewrite modulus_power. apply two_p_monotone_strict. lia. assert (two_p uy < half_modulus). rewrite half_modulus_power. apply two_p_monotone_strict. auto. assert (two_p uy < modulus). - rewrite modulus_power. apply two_p_monotone_strict. omega. + rewrite modulus_power. apply two_p_monotone_strict. lia. assert (sub (repr (two_p uy)) one = repr (two_p uy - 1)). unfold sub. apply eqm_samerepr. apply eqm_sub. apply eqm_sym; apply eqm_unsigned_repr. rewrite unsigned_one. apply eqm_refl. @@ -2463,17 +2463,17 @@ Proof. fold eqm. unfold sx. apply eqm_sym. apply eqm_signed_unsigned. unfold modulus. rewrite two_power_nat_two_p. exists (two_p (zwordsize - uy)). rewrite <- two_p_is_exp. - f_equal. fold zwordsize; omega. omega. omega. + f_equal. fold zwordsize; lia. lia. lia. rewrite H8. rewrite Zdiv_shift; auto. unfold add. apply eqm_samerepr. apply eqm_add. apply eqm_unsigned_repr. destruct (zeq (sx mod two_p uy) 0); simpl. rewrite unsigned_zero. apply eqm_refl. rewrite unsigned_one. apply eqm_refl. - generalize (Z_mod_lt (unsigned x) (two_p uy) H3). unfold max_unsigned. omega. - unfold max_unsigned; omega. - generalize (signed_range x). fold sx. intros. split. omega. unfold max_signed. omega. - generalize min_signed_neg. unfold max_signed. omega. + generalize (Z_mod_lt (unsigned x) (two_p uy) H3). unfold max_unsigned. lia. + unfold max_unsigned; lia. + generalize (signed_range x). fold sx. intros. split. lia. unfold max_signed. lia. + generalize min_signed_neg. unfold max_signed. lia. Qed. (** Connections between [shr] and [shru]. *) @@ -2492,14 +2492,14 @@ Lemma and_positive: forall x y, signed y >= 0 -> signed (and x y) >= 0. Proof. intros. - assert (unsigned y < half_modulus). rewrite signed_positive in H. unfold max_signed in H; omega. + assert (unsigned y < half_modulus). rewrite signed_positive in H. unfold max_signed in H; lia. generalize (sign_bit_of_unsigned y). rewrite zlt_true; auto. intros A. generalize (sign_bit_of_unsigned (and x y)). rewrite bits_and. rewrite A. rewrite andb_false_r. unfold signed. destruct (zlt (unsigned (and x y)) half_modulus). - intros. generalize (unsigned_range (and x y)); omega. + intros. generalize (unsigned_range (and x y)); lia. congruence. - generalize wordsize_pos; omega. + generalize wordsize_pos; lia. Qed. Theorem shr_and_is_shru_and: @@ -2526,7 +2526,7 @@ Lemma bits_sign_ext: testbit (sign_ext n x) i = testbit x (if zlt i n then i else n - 1). Proof. intros. unfold sign_ext. - rewrite testbit_repr; auto. apply Zsign_ext_spec. omega. + rewrite testbit_repr; auto. apply Zsign_ext_spec. lia. Qed. Hint Rewrite bits_zero_ext bits_sign_ext: ints. @@ -2535,13 +2535,13 @@ Theorem zero_ext_above: forall n x, n >= zwordsize -> zero_ext n x = x. Proof. intros. apply same_bits_eq; intros. - rewrite bits_zero_ext. apply zlt_true. omega. omega. + rewrite bits_zero_ext. apply zlt_true. lia. lia. Qed. Theorem zero_ext_below: forall n x, n <= 0 -> zero_ext n x = zero. Proof. - intros. bit_solve. destruct (zlt i n); auto. apply bits_below; omega. omega. + intros. bit_solve. destruct (zlt i n); auto. apply bits_below; lia. lia. Qed. Theorem sign_ext_above: @@ -2549,13 +2549,13 @@ Theorem sign_ext_above: Proof. intros. apply same_bits_eq; intros. unfold sign_ext; rewrite testbit_repr; auto. - rewrite Zsign_ext_spec. rewrite zlt_true. auto. omega. omega. + rewrite Zsign_ext_spec. rewrite zlt_true. auto. lia. lia. Qed. Theorem sign_ext_below: forall n x, n <= 0 -> sign_ext n x = zero. Proof. - intros. bit_solve. apply bits_below. destruct (zlt i n); omega. + intros. bit_solve. apply bits_below. destruct (zlt i n); lia. Qed. Theorem zero_ext_and: @@ -2577,8 +2577,8 @@ Proof. fold (testbit (zero_ext n x) i). destruct (zlt i zwordsize). rewrite bits_zero_ext; auto. - rewrite bits_above. rewrite zlt_false; auto. omega. omega. - omega. + rewrite bits_above. rewrite zlt_false; auto. lia. lia. + lia. Qed. Theorem zero_ext_widen: @@ -2586,7 +2586,7 @@ Theorem zero_ext_widen: zero_ext n' (zero_ext n x) = zero_ext n x. Proof. bit_solve. destruct (zlt i n). - apply zlt_true. omega. + apply zlt_true. lia. destruct (zlt i n'); auto. tauto. tauto. Qed. @@ -2599,9 +2599,9 @@ Proof. bit_solve. destruct (zlt i n'). auto. rewrite (zlt_false _ i n). - destruct (zlt (n' - 1) n); f_equal; omega. - omega. - destruct (zlt i n'); omega. + destruct (zlt (n' - 1) n); f_equal; lia. + lia. + destruct (zlt i n'); lia. apply sign_ext_above; auto. Qed. @@ -2613,8 +2613,8 @@ Proof. bit_solve. destruct (zlt i n'). auto. - rewrite !zlt_false. auto. omega. omega. omega. - destruct (zlt i n'); omega. + rewrite !zlt_false. auto. lia. lia. lia. + destruct (zlt i n'); lia. apply sign_ext_above; auto. Qed. @@ -2623,9 +2623,9 @@ Theorem zero_ext_narrow: zero_ext n (zero_ext n' x) = zero_ext n x. Proof. bit_solve. destruct (zlt i n). - apply zlt_true. omega. + apply zlt_true. lia. auto. - omega. omega. omega. + lia. lia. lia. Qed. Theorem sign_ext_narrow: @@ -2633,9 +2633,9 @@ Theorem sign_ext_narrow: sign_ext n (sign_ext n' x) = sign_ext n x. Proof. intros. destruct (zlt n zwordsize). - bit_solve. destruct (zlt i n); f_equal; apply zlt_true; omega. - destruct (zlt i n); omega. - rewrite (sign_ext_above n'). auto. omega. + bit_solve. destruct (zlt i n); f_equal; apply zlt_true; lia. + destruct (zlt i n); lia. + rewrite (sign_ext_above n'). auto. lia. Qed. Theorem zero_sign_ext_narrow: @@ -2645,21 +2645,21 @@ Proof. intros. destruct (zlt n' zwordsize). bit_solve. destruct (zlt i n); auto. - rewrite zlt_true; auto. omega. - omega. omega. + rewrite zlt_true; auto. lia. + lia. lia. rewrite sign_ext_above; auto. Qed. Theorem zero_ext_idem: forall n x, 0 <= n -> zero_ext n (zero_ext n x) = zero_ext n x. Proof. - intros. apply zero_ext_widen. omega. + intros. apply zero_ext_widen. lia. Qed. Theorem sign_ext_idem: forall n x, 0 < n -> sign_ext n (sign_ext n x) = sign_ext n x. Proof. - intros. apply sign_ext_widen. omega. + intros. apply sign_ext_widen. lia. Qed. Theorem sign_ext_zero_ext: @@ -2669,15 +2669,15 @@ Proof. bit_solve. destruct (zlt i n). rewrite zlt_true; auto. - rewrite zlt_true; auto. omega. - destruct (zlt i n); omega. + rewrite zlt_true; auto. lia. + destruct (zlt i n); lia. rewrite zero_ext_above; auto. Qed. Theorem zero_ext_sign_ext: forall n x, 0 < n -> zero_ext n (sign_ext n x) = zero_ext n x. Proof. - intros. apply zero_sign_ext_narrow. omega. + intros. apply zero_sign_ext_narrow. lia. Qed. Theorem sign_ext_equal_if_zero_equal: @@ -2700,21 +2700,21 @@ Proof. apply same_bits_eq; intros. rewrite bits_shru by auto. fold Z. destruct (zlt Z Y). - assert (A: unsigned (sub y z) = Y - Z). - { apply unsigned_repr. generalize wordsize_max_unsigned; omega. } - symmetry; rewrite bits_shl, A by omega. + { apply unsigned_repr. generalize wordsize_max_unsigned; lia. } + symmetry; rewrite bits_shl, A by lia. destruct (zlt (i + Z) zwordsize). -+ rewrite bits_shl by omega. fold Y. - destruct (zlt i (Y - Z)); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. - rewrite bits_zero_ext by omega. rewrite zlt_true by omega. f_equal; omega. -+ rewrite bits_zero_ext by omega. rewrite ! zlt_false by omega. auto. ++ rewrite bits_shl by lia. fold Y. + destruct (zlt i (Y - Z)); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. + rewrite bits_zero_ext by lia. rewrite zlt_true by lia. f_equal; lia. ++ rewrite bits_zero_ext by lia. rewrite ! zlt_false by lia. auto. - assert (A: unsigned (sub z y) = Z - Y). - { apply unsigned_repr. generalize wordsize_max_unsigned; omega. } - rewrite bits_zero_ext, bits_shru, A by omega. - destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. - rewrite bits_shl by omega. fold Y. + { apply unsigned_repr. generalize wordsize_max_unsigned; lia. } + rewrite bits_zero_ext, bits_shru, A by lia. + destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. + rewrite bits_shl by lia. fold Y. destruct (zlt (i + Z) Y). -+ rewrite zlt_false by omega. auto. -+ rewrite zlt_true by omega. f_equal; omega. ++ rewrite zlt_false by lia. auto. ++ rewrite zlt_true by lia. f_equal; lia. Qed. Corollary zero_ext_shru_shl: @@ -2725,11 +2725,11 @@ Corollary zero_ext_shru_shl: Proof. intros. assert (A: unsigned y = zwordsize - n). - { unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega. } + { unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. lia. } assert (B: ltu y iwordsize = true). - { unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; omega. } - rewrite shru_shl by auto. unfold ltu; rewrite zlt_false by omega. - rewrite sub_idem, shru_zero. f_equal. rewrite A; omega. + { unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; lia. } + rewrite shru_shl by auto. unfold ltu; rewrite zlt_false by lia. + rewrite sub_idem, shru_zero. f_equal. rewrite A; lia. Qed. Theorem shr_shl: @@ -2741,26 +2741,26 @@ Proof. intros. apply ltu_iwordsize_inv in H; apply ltu_iwordsize_inv in H0. unfold ltu. set (Y := unsigned y) in *; set (Z := unsigned z) in *. apply same_bits_eq; intros. rewrite bits_shr by auto. fold Z. - rewrite bits_shl by (destruct (zlt (i + Z) zwordsize); omega). fold Y. + rewrite bits_shl by (destruct (zlt (i + Z) zwordsize); lia). fold Y. destruct (zlt Z Y). - assert (A: unsigned (sub y z) = Y - Z). - { apply unsigned_repr. generalize wordsize_max_unsigned; omega. } - rewrite bits_shl, A by omega. + { apply unsigned_repr. generalize wordsize_max_unsigned; lia. } + rewrite bits_shl, A by lia. destruct (zlt i (Y - Z)). -+ apply zlt_true. destruct (zlt (i + Z) zwordsize); omega. -+ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); omega). - rewrite bits_sign_ext by omega. f_equal. ++ apply zlt_true. destruct (zlt (i + Z) zwordsize); lia. ++ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia). + rewrite bits_sign_ext by lia. f_equal. destruct (zlt (i + Z) zwordsize). - rewrite zlt_true by omega. omega. - rewrite zlt_false by omega. omega. + rewrite zlt_true by lia. lia. + rewrite zlt_false by lia. lia. - assert (A: unsigned (sub z y) = Z - Y). - { apply unsigned_repr. generalize wordsize_max_unsigned; omega. } - rewrite bits_sign_ext by omega. - rewrite bits_shr by (destruct (zlt i (zwordsize - Z)); omega). - rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); omega). + { apply unsigned_repr. generalize wordsize_max_unsigned; lia. } + rewrite bits_sign_ext by lia. + rewrite bits_shr by (destruct (zlt i (zwordsize - Z)); lia). + rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia). f_equal. destruct (zlt i (zwordsize - Z)). -+ rewrite ! zlt_true by omega. omega. -+ rewrite ! zlt_false by omega. rewrite zlt_true by omega. omega. ++ rewrite ! zlt_true by lia. lia. ++ rewrite ! zlt_false by lia. rewrite zlt_true by lia. lia. Qed. Corollary sign_ext_shr_shl: @@ -2771,11 +2771,11 @@ Corollary sign_ext_shr_shl: Proof. intros. assert (A: unsigned y = zwordsize - n). - { unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega. } + { unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. lia. } assert (B: ltu y iwordsize = true). - { unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; omega. } - rewrite shr_shl by auto. unfold ltu; rewrite zlt_false by omega. - rewrite sub_idem, shr_zero. f_equal. rewrite A; omega. + { unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; lia. } + rewrite shr_shl by auto. unfold ltu; rewrite zlt_false by lia. + rewrite sub_idem, shr_zero. f_equal. rewrite A; lia. Qed. (** [zero_ext n x] is the unique integer congruent to [x] modulo [2^n] @@ -2784,14 +2784,14 @@ Qed. Lemma zero_ext_range: forall n x, 0 <= n < zwordsize -> 0 <= unsigned (zero_ext n x) < two_p n. Proof. - intros. rewrite zero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. omega. + intros. rewrite zero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. lia. Qed. Lemma eqmod_zero_ext: forall n x, 0 <= n < zwordsize -> eqmod (two_p n) (unsigned (zero_ext n x)) (unsigned x). Proof. intros. rewrite zero_ext_mod; auto. apply eqmod_sym. apply eqmod_mod. - apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. Qed. (** [sign_ext n x] is the unique integer congruent to [x] modulo [2^n] @@ -2802,26 +2802,26 @@ Lemma sign_ext_range: Proof. intros. rewrite sign_ext_shr_shl; auto. set (X := shl x (repr (zwordsize - n))). - assert (two_p (n - 1) > 0) by (apply two_p_gt_ZERO; omega). + assert (two_p (n - 1) > 0) by (apply two_p_gt_ZERO; lia). assert (unsigned (repr (zwordsize - n)) = zwordsize - n). apply unsigned_repr. - split. omega. generalize wordsize_max_unsigned; omega. + split. lia. generalize wordsize_max_unsigned; lia. rewrite shr_div_two_p. rewrite signed_repr. rewrite H1. apply Zdiv_interval_1. - omega. omega. apply two_p_gt_ZERO; omega. + lia. lia. apply two_p_gt_ZERO; lia. replace (- two_p (n - 1) * two_p (zwordsize - n)) with (- (two_p (n - 1) * two_p (zwordsize - n))) by ring. rewrite <- two_p_is_exp. - replace (n - 1 + (zwordsize - n)) with (zwordsize - 1) by omega. + replace (n - 1 + (zwordsize - n)) with (zwordsize - 1) by lia. rewrite <- half_modulus_power. - generalize (signed_range X). unfold min_signed, max_signed. omega. - omega. omega. + generalize (signed_range X). unfold min_signed, max_signed. lia. + lia. lia. apply Zdiv_interval_2. apply signed_range. - generalize min_signed_neg; omega. - generalize max_signed_pos; omega. - rewrite H1. apply two_p_gt_ZERO. omega. + generalize min_signed_neg; lia. + generalize max_signed_pos; lia. + rewrite H1. apply two_p_gt_ZERO. lia. Qed. Lemma eqmod_sign_ext': @@ -2830,12 +2830,12 @@ Lemma eqmod_sign_ext': Proof. intros. set (N := Z.to_nat n). - assert (Z.of_nat N = n) by (apply Z2Nat.id; omega). + assert (Z.of_nat N = n) by (apply Z2Nat.id; lia). rewrite <- H0. rewrite <- two_power_nat_two_p. apply eqmod_same_bits; intros. rewrite H0 in H1. rewrite H0. fold (testbit (sign_ext n x) i). rewrite bits_sign_ext. - rewrite zlt_true. auto. omega. omega. + rewrite zlt_true. auto. lia. lia. Qed. Lemma eqmod_sign_ext: @@ -2846,7 +2846,7 @@ Proof. apply eqmod_divides with modulus. apply eqm_signed_unsigned. exists (two_p (zwordsize - n)). unfold modulus. rewrite two_power_nat_two_p. fold zwordsize. - rewrite <- two_p_is_exp. f_equal. omega. omega. omega. + rewrite <- two_p_is_exp. f_equal. lia. lia. lia. apply eqmod_sign_ext'; auto. Qed. @@ -2857,11 +2857,11 @@ Lemma shl_zero_ext: shl (zero_ext n x) m = zero_ext (n + unsigned m) (shl x m). Proof. intros. apply same_bits_eq; intros. - rewrite bits_zero_ext, ! bits_shl by omega. + rewrite bits_zero_ext, ! bits_shl by lia. destruct (zlt i (unsigned m)). -- rewrite zlt_true by omega; auto. -- rewrite bits_zero_ext by omega. - destruct (zlt (i - unsigned m) n); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. +- rewrite zlt_true by lia; auto. +- rewrite bits_zero_ext by lia. + destruct (zlt (i - unsigned m) n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. Qed. Lemma shl_sign_ext: @@ -2870,12 +2870,12 @@ Lemma shl_sign_ext: Proof. intros. generalize (unsigned_range m); intros. apply same_bits_eq; intros. - rewrite bits_sign_ext, ! bits_shl by omega. + rewrite bits_sign_ext, ! bits_shl by lia. destruct (zlt i (n + unsigned m)). - rewrite bits_shl by auto. destruct (zlt i (unsigned m)); auto. - rewrite bits_sign_ext by omega. f_equal. apply zlt_true. omega. -- rewrite zlt_false by omega. rewrite bits_shl by omega. rewrite zlt_false by omega. - rewrite bits_sign_ext by omega. f_equal. rewrite zlt_false by omega. omega. + rewrite bits_sign_ext by lia. f_equal. apply zlt_true. lia. +- rewrite zlt_false by lia. rewrite bits_shl by lia. rewrite zlt_false by lia. + rewrite bits_sign_ext by lia. f_equal. rewrite zlt_false by lia. lia. Qed. Lemma shru_zero_ext: @@ -2884,10 +2884,10 @@ Lemma shru_zero_ext: Proof. intros. bit_solve. - destruct (zlt (i + unsigned m) zwordsize). -* destruct (zlt i n); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. +* destruct (zlt i n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. * destruct (zlt i n); auto. -- generalize (unsigned_range m); omega. -- omega. +- generalize (unsigned_range m); lia. +- lia. Qed. Lemma shru_zero_ext_0: @@ -2896,8 +2896,8 @@ Lemma shru_zero_ext_0: Proof. intros. bit_solve. - destruct (zlt (i + unsigned m) zwordsize); auto. - apply zlt_false. omega. -- generalize (unsigned_range m); omega. + apply zlt_false. lia. +- generalize (unsigned_range m); lia. Qed. Lemma shr_sign_ext: @@ -2910,12 +2910,12 @@ Proof. rewrite bits_sign_ext, bits_shr. - f_equal. destruct (zlt i n), (zlt (i + unsigned m) zwordsize). -+ apply zlt_true; omega. -+ apply zlt_true; omega. -+ rewrite zlt_false by omega. rewrite zlt_true by omega. omega. -+ rewrite zlt_false by omega. rewrite zlt_true by omega. omega. -- destruct (zlt i n); omega. -- destruct (zlt (i + unsigned m) zwordsize); omega. ++ apply zlt_true; lia. ++ apply zlt_true; lia. ++ rewrite zlt_false by lia. rewrite zlt_true by lia. lia. ++ rewrite zlt_false by lia. rewrite zlt_true by lia. lia. +- destruct (zlt i n); lia. +- destruct (zlt (i + unsigned m) zwordsize); lia. Qed. Lemma zero_ext_shru_min: @@ -2924,10 +2924,10 @@ Lemma zero_ext_shru_min: Proof. intros. apply ltu_iwordsize_inv in H. apply Z.min_case_strong; intros; auto. - bit_solve; try omega. + bit_solve; try lia. destruct (zlt i (zwordsize - unsigned n)). - rewrite zlt_true by omega. auto. - destruct (zlt i s); auto. rewrite zlt_false by omega; auto. + rewrite zlt_true by lia. auto. + destruct (zlt i s); auto. rewrite zlt_false by lia; auto. Qed. Lemma sign_ext_shr_min: @@ -2939,12 +2939,12 @@ Proof. destruct (Z.min_spec (zwordsize - unsigned n) s) as [[A B] | [A B]]; rewrite B; auto. apply same_bits_eq; intros. rewrite ! bits_sign_ext by auto. destruct (zlt i (zwordsize - unsigned n)). - rewrite zlt_true by omega. auto. + rewrite zlt_true by lia. auto. assert (C: testbit (shr x n) (zwordsize - unsigned n - 1) = testbit x (zwordsize - 1)). - { rewrite bits_shr by omega. rewrite zlt_true by omega. f_equal; omega. } - rewrite C. destruct (zlt i s); rewrite bits_shr by omega. - rewrite zlt_false by omega. auto. - rewrite zlt_false by omega. auto. + { rewrite bits_shr by lia. rewrite zlt_true by lia. f_equal; lia. } + rewrite C. destruct (zlt i s); rewrite bits_shr by lia. + rewrite zlt_false by lia. auto. + rewrite zlt_false by lia. auto. Qed. Lemma shl_zero_ext_min: @@ -2955,10 +2955,10 @@ Proof. apply Z.min_case_strong; intros; auto. apply same_bits_eq; intros. rewrite ! bits_shl by auto. destruct (zlt i (unsigned n)); auto. - rewrite ! bits_zero_ext by omega. + rewrite ! bits_zero_ext by lia. destruct (zlt (i - unsigned n) s). - rewrite zlt_true by omega; auto. - rewrite zlt_false by omega; auto. + rewrite zlt_true by lia; auto. + rewrite zlt_false by lia; auto. Qed. Lemma shl_sign_ext_min: @@ -2970,10 +2970,10 @@ Proof. destruct (Z.min_spec (zwordsize - unsigned n) s) as [[A B] | [A B]]; rewrite B; auto. apply same_bits_eq; intros. rewrite ! bits_shl by auto. destruct (zlt i (unsigned n)); auto. - rewrite ! bits_sign_ext by omega. f_equal. + rewrite ! bits_sign_ext by lia. f_equal. destruct (zlt (i - unsigned n) s). - rewrite zlt_true by omega; auto. - omegaContradiction. + rewrite zlt_true by lia; auto. + extlia. Qed. (** ** Properties of [one_bits] (decomposition in sum of powers of two) *) @@ -2984,8 +2984,8 @@ Proof. assert (A: forall p, 0 <= p < zwordsize -> ltu (repr p) iwordsize = true). intros. unfold ltu, iwordsize. apply zlt_true. repeat rewrite unsigned_repr. tauto. - generalize wordsize_max_unsigned; omega. - generalize wordsize_max_unsigned; omega. + generalize wordsize_max_unsigned; lia. + generalize wordsize_max_unsigned; lia. unfold one_bits. intros. destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]]. subst i. apply A. apply Z_one_bits_range with (unsigned x); auto. @@ -3015,7 +3015,7 @@ Proof. rewrite mul_one. apply eqm_unsigned_repr_r. rewrite unsigned_repr. auto with ints. generalize (H a (in_eq _ _)). change (Z.of_nat wordsize) with zwordsize. - generalize wordsize_max_unsigned. omega. + generalize wordsize_max_unsigned. lia. auto with ints. intros; apply H; auto with coqlib. Qed. @@ -3059,7 +3059,7 @@ Proof. apply eqm_sub. apply eqm_trans with (unsigned (repr (unsigned x + unsigned d))). eauto with ints. apply eqm_trans with (unsigned (repr (unsigned y + unsigned d))). eauto with ints. eauto with ints. eauto with ints. - omega. omega. + lia. lia. Qed. Lemma translate_ltu: @@ -3070,8 +3070,8 @@ Lemma translate_ltu: Proof. intros. unfold add. unfold ltu. repeat rewrite unsigned_repr; auto. case (zlt (unsigned x) (unsigned y)); intro. - apply zlt_true. omega. - apply zlt_false. omega. + apply zlt_true. lia. + apply zlt_false. lia. Qed. Theorem translate_cmpu: @@ -3092,8 +3092,8 @@ Lemma translate_lt: Proof. intros. repeat rewrite add_signed. unfold lt. repeat rewrite signed_repr; auto. case (zlt (signed x) (signed y)); intro. - apply zlt_true. omega. - apply zlt_false. omega. + apply zlt_true. lia. + apply zlt_false. lia. Qed. Theorem translate_cmp: @@ -3129,7 +3129,7 @@ Proof. intros. unfold ltu in H. destruct (zlt (unsigned x) (unsigned y)); try discriminate. rewrite signed_eq_unsigned. - generalize (unsigned_range x). omega. omega. + generalize (unsigned_range x). lia. lia. Qed. Theorem lt_sub_overflow: @@ -3143,30 +3143,30 @@ Proof. unfold min_signed, max_signed in *. generalize half_modulus_pos half_modulus_modulus; intros HM MM. destruct (zle 0 (X - Y)). -- unfold proj_sumbool at 1; rewrite zle_true at 1 by omega. simpl. - rewrite (zlt_false _ X) by omega. +- unfold proj_sumbool at 1; rewrite zle_true at 1 by lia. simpl. + rewrite (zlt_false _ X) by lia. destruct (zlt (X - Y) half_modulus). - + unfold proj_sumbool; rewrite zle_true by omega. - rewrite signed_repr. rewrite zlt_false by omega. apply xor_idem. - unfold min_signed, max_signed; omega. - + unfold proj_sumbool; rewrite zle_false by omega. + + unfold proj_sumbool; rewrite zle_true by lia. + rewrite signed_repr. rewrite zlt_false by lia. apply xor_idem. + unfold min_signed, max_signed; lia. + + unfold proj_sumbool; rewrite zle_false by lia. replace (signed (repr (X - Y))) with (X - Y - modulus). - rewrite zlt_true by omega. apply xor_idem. + rewrite zlt_true by lia. apply xor_idem. rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y). rewrite zlt_false; auto. - symmetry. apply Zmod_unique with 0; omega. -- unfold proj_sumbool at 2. rewrite zle_true at 1 by omega. rewrite andb_true_r. - rewrite (zlt_true _ X) by omega. + symmetry. apply Zmod_unique with 0; lia. +- unfold proj_sumbool at 2. rewrite zle_true at 1 by lia. rewrite andb_true_r. + rewrite (zlt_true _ X) by lia. destruct (zlt (X - Y) (-half_modulus)). - + unfold proj_sumbool; rewrite zle_false by omega. + + unfold proj_sumbool; rewrite zle_false by lia. replace (signed (repr (X - Y))) with (X - Y + modulus). - rewrite zlt_false by omega. apply xor_zero. + rewrite zlt_false by lia. apply xor_zero. rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y + modulus). - rewrite zlt_true by omega; auto. - symmetry. apply Zmod_unique with (-1); omega. - + unfold proj_sumbool; rewrite zle_true by omega. - rewrite signed_repr. rewrite zlt_true by omega. apply xor_zero_l. - unfold min_signed, max_signed; omega. + rewrite zlt_true by lia; auto. + symmetry. apply Zmod_unique with (-1); lia. + + unfold proj_sumbool; rewrite zle_true by lia. + rewrite signed_repr. rewrite zlt_true by lia. apply xor_zero_l. + unfold min_signed, max_signed; lia. Qed. Lemma signed_eq: @@ -3186,10 +3186,10 @@ Lemma not_lt: Proof. intros. unfold lt. rewrite signed_eq. unfold proj_sumbool. destruct (zlt (signed y) (signed x)). - rewrite zlt_false. rewrite zeq_false. auto. omega. omega. + rewrite zlt_false. rewrite zeq_false. auto. lia. lia. destruct (zeq (signed x) (signed y)). - rewrite zlt_false. auto. omega. - rewrite zlt_true. auto. omega. + rewrite zlt_false. auto. lia. + rewrite zlt_true. auto. lia. Qed. Lemma lt_not: @@ -3203,10 +3203,10 @@ Lemma not_ltu: Proof. intros. unfold ltu, eq. destruct (zlt (unsigned y) (unsigned x)). - rewrite zlt_false. rewrite zeq_false. auto. omega. omega. + rewrite zlt_false. rewrite zeq_false. auto. lia. lia. destruct (zeq (unsigned x) (unsigned y)). - rewrite zlt_false. auto. omega. - rewrite zlt_true. auto. omega. + rewrite zlt_false. auto. lia. + rewrite zlt_true. auto. lia. Qed. Lemma ltu_not: @@ -3238,7 +3238,7 @@ Proof. clear H3. generalize (unsigned_range ofs1) (unsigned_range ofs2). intros P Q. generalize (unsigned_add_either base ofs1) (unsigned_add_either base ofs2). - intros [C|C] [D|D]; omega. + intros [C|C] [D|D]; lia. Qed. (** ** Size of integers, in bits. *) @@ -3255,14 +3255,14 @@ Theorem bits_size_1: Proof. intros. destruct (zeq (unsigned x) 0). left. rewrite <- (repr_unsigned x). rewrite e; auto. - right. apply Ztestbit_size_1. generalize (unsigned_range x); omega. + right. apply Ztestbit_size_1. generalize (unsigned_range x); lia. Qed. Theorem bits_size_2: forall x i, size x <= i -> testbit x i = false. Proof. - intros. apply Ztestbit_size_2. generalize (unsigned_range x); omega. - fold (size x); omega. + intros. apply Ztestbit_size_2. generalize (unsigned_range x); lia. + fold (size x); lia. Qed. Theorem size_range: @@ -3270,9 +3270,9 @@ Theorem size_range: Proof. intros; split. apply Zsize_pos. destruct (bits_size_1 x). - subst x; unfold size; rewrite unsigned_zero; simpl. generalize wordsize_pos; omega. + subst x; unfold size; rewrite unsigned_zero; simpl. generalize wordsize_pos; lia. destruct (zle (size x) zwordsize); auto. - rewrite bits_above in H. congruence. omega. + rewrite bits_above in H. congruence. lia. Qed. Theorem bits_size_3: @@ -3285,7 +3285,7 @@ Proof. destruct (bits_size_1 x). subst x. unfold size; rewrite unsigned_zero; assumption. rewrite (H0 (Z.pred (size x))) in H1. congruence. - generalize (size_range x); omega. + generalize (size_range x); lia. Qed. Theorem bits_size_4: @@ -3299,14 +3299,14 @@ Proof. assert (size x <= n). apply bits_size_3; auto. destruct (zlt (size x) n). - rewrite bits_size_2 in H0. congruence. omega. - omega. + rewrite bits_size_2 in H0. congruence. lia. + lia. Qed. Theorem size_interval_1: forall x, 0 <= unsigned x < two_p (size x). Proof. - intros; apply Zsize_interval_1. generalize (unsigned_range x); omega. + intros; apply Zsize_interval_1. generalize (unsigned_range x); lia. Qed. Theorem size_interval_2: @@ -3320,9 +3320,9 @@ Theorem size_and: Proof. intros. assert (0 <= Z.min (size a) (size b)). - generalize (size_range a) (size_range b). zify; omega. + generalize (size_range a) (size_range b). zify; lia. apply bits_size_3. auto. intros. - rewrite bits_and by omega. + rewrite bits_and by lia. rewrite andb_false_iff. generalize (bits_size_2 a i). generalize (bits_size_2 b i). @@ -3335,9 +3335,9 @@ Proof. intros. generalize (size_interval_1 (and a b)); intros. assert (two_p (size (and a b)) <= two_p (Z.min (size a) (size b))). - apply two_p_monotone. split. generalize (size_range (and a b)); omega. + apply two_p_monotone. split. generalize (size_range (and a b)); lia. apply size_and. - omega. + lia. Qed. Theorem size_or: @@ -3345,17 +3345,17 @@ Theorem size_or: Proof. intros. generalize (size_range a) (size_range b); intros. destruct (bits_size_1 a). - subst a. rewrite size_zero. rewrite or_zero_l. zify; omega. + subst a. rewrite size_zero. rewrite or_zero_l. zify; lia. destruct (bits_size_1 b). - subst b. rewrite size_zero. rewrite or_zero. zify; omega. + subst b. rewrite size_zero. rewrite or_zero. zify; lia. zify. destruct H3 as [[P Q] | [P Q]]; subst. apply bits_size_4. tauto. rewrite bits_or. rewrite H2. apply orb_true_r. - omega. - intros. rewrite bits_or. rewrite !bits_size_2. auto. omega. omega. omega. + lia. + intros. rewrite bits_or. rewrite !bits_size_2. auto. lia. lia. lia. apply bits_size_4. tauto. rewrite bits_or. rewrite H1. apply orb_true_l. destruct (zeq (size a) 0). unfold testbit in H1. rewrite Z.testbit_neg_r in H1. - congruence. omega. omega. - intros. rewrite bits_or. rewrite !bits_size_2. auto. omega. omega. omega. + congruence. lia. lia. + intros. rewrite bits_or. rewrite !bits_size_2. auto. lia. lia. lia. Qed. Corollary or_interval: @@ -3369,12 +3369,12 @@ Theorem size_xor: Proof. intros. assert (0 <= Z.max (size a) (size b)). - generalize (size_range a) (size_range b). zify; omega. + generalize (size_range a) (size_range b). zify; lia. apply bits_size_3. auto. intros. rewrite bits_xor. rewrite !bits_size_2. auto. - zify; omega. - zify; omega. - omega. + zify; lia. + zify; lia. + lia. Qed. Corollary xor_interval: @@ -3383,9 +3383,9 @@ Proof. intros. generalize (size_interval_1 (xor a b)); intros. assert (two_p (size (xor a b)) <= two_p (Z.max (size a) (size b))). - apply two_p_monotone. split. generalize (size_range (xor a b)); omega. + apply two_p_monotone. split. generalize (size_range (xor a b)); lia. apply size_xor. - omega. + lia. Qed. End Make. @@ -3465,7 +3465,7 @@ Proof. intros. unfold shl'. rewrite testbit_repr; auto. destruct (zlt i (Int.unsigned y)). apply Z.shiftl_spec_low. auto. - apply Z.shiftl_spec_high. omega. omega. + apply Z.shiftl_spec_high. lia. lia. Qed. Lemma bits_shru': @@ -3479,7 +3479,7 @@ Proof. destruct (zlt (i + Int.unsigned y) zwordsize). auto. apply bits_above; auto. - omega. + lia. Qed. Lemma bits_shr': @@ -3490,8 +3490,8 @@ Lemma bits_shr': Proof. intros. unfold shr'. rewrite testbit_repr; auto. rewrite Z.shiftr_spec. apply bits_signed. - generalize (Int.unsigned_range y); omega. - omega. + generalize (Int.unsigned_range y); lia. + lia. Qed. Lemma shl'_mul_two_p: @@ -3500,7 +3500,7 @@ Lemma shl'_mul_two_p: Proof. intros. unfold shl', mul. apply eqm_samerepr. rewrite Zshiftl_mul_two_p. apply eqm_mult. apply eqm_refl. apply eqm_unsigned_repr. - generalize (Int.unsigned_range y); omega. + generalize (Int.unsigned_range y); lia. Qed. Lemma shl'_one_two_p: @@ -3551,7 +3551,7 @@ Proof. intros. apply Int.ltu_inv in H. change (Int.unsigned (Int.repr 63)) with 63 in H. set (y1 := Int64.repr (Int.unsigned y)). assert (U: unsigned y1 = Int.unsigned y). - { apply unsigned_repr. assert (63 < max_unsigned) by reflexivity. omega. } + { apply unsigned_repr. assert (63 < max_unsigned) by reflexivity. lia. } transitivity (shrx x y1). - unfold shrx', shrx, shl', shl. rewrite U; auto. - rewrite shrx_carry. @@ -3572,20 +3572,20 @@ Proof. assert (N1: 63 < max_unsigned) by reflexivity. assert (N2: 63 < Int.max_unsigned) by reflexivity. assert (A: unsigned z = Int.unsigned y). - { unfold z; apply unsigned_repr; omega. } + { unfold z; apply unsigned_repr; lia. } assert (B: unsigned (sub (repr 64) z) = Int.unsigned (Int.sub (Int.repr 64) y)). { unfold z. unfold sub, Int.sub. change (unsigned (repr 64)) with 64. change (Int.unsigned (Int.repr 64)) with 64. - rewrite (unsigned_repr (Int.unsigned y)) by omega. - rewrite unsigned_repr, Int.unsigned_repr by omega. + rewrite (unsigned_repr (Int.unsigned y)) by lia. + rewrite unsigned_repr, Int.unsigned_repr by lia. auto. } unfold shrx', shr', shru', shl'. rewrite <- A. change (Int.unsigned (Int.repr 63)) with (unsigned (repr 63)). rewrite <- B. apply shrx_shr_2. - unfold ltu. apply zlt_true. change (unsigned z < 63). rewrite A; omega. + unfold ltu. apply zlt_true. change (unsigned z < 63). rewrite A; lia. Qed. Remark int_ltu_2_inv: @@ -3606,11 +3606,11 @@ Proof. change (Int.unsigned iwordsize') with 64 in *. assert (128 < max_unsigned) by reflexivity. assert (128 < Int.max_unsigned) by reflexivity. - assert (Y: unsigned y' = Int.unsigned y) by (apply unsigned_repr; omega). - assert (Z: unsigned z' = Int.unsigned z) by (apply unsigned_repr; omega). + assert (Y: unsigned y' = Int.unsigned y) by (apply unsigned_repr; lia). + assert (Z: unsigned z' = Int.unsigned z) by (apply unsigned_repr; lia). assert (P: Int.unsigned (Int.add y z) = unsigned (add y' z')). - { unfold Int.add. rewrite Int.unsigned_repr by omega. - unfold add. rewrite unsigned_repr by omega. congruence. } + { unfold Int.add. rewrite Int.unsigned_repr by lia. + unfold add. rewrite unsigned_repr by lia. congruence. } intuition auto. apply zlt_true. rewrite Y; auto. apply zlt_true. rewrite Z; auto. @@ -3624,7 +3624,7 @@ Theorem or_ror': Int.add y z = iwordsize' -> ror x (repr (Int.unsigned z)) = or (shl' x y) (shru' x z). Proof. - intros. destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. rewrite H1; omega. + intros. destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. rewrite H1; lia. replace (shl' x y) with (shl x (repr (Int.unsigned y))). replace (shru' x z) with (shru x (repr (Int.unsigned z))). apply or_ror; auto. rewrite F, H1. reflexivity. @@ -3640,7 +3640,7 @@ Theorem shl'_shl': shl' (shl' x y) z = shl' x (Int.add y z). Proof. intros. apply Int.ltu_inv in H1. - destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega. + destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia. set (y' := repr (Int.unsigned y)) in *. set (z' := repr (Int.unsigned z)) in *. replace (shl' x y) with (shl x y'). @@ -3661,7 +3661,7 @@ Theorem shru'_shru': shru' (shru' x y) z = shru' x (Int.add y z). Proof. intros. apply Int.ltu_inv in H1. - destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega. + destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia. set (y' := repr (Int.unsigned y)) in *. set (z' := repr (Int.unsigned z)) in *. replace (shru' x y) with (shru x y'). @@ -3682,7 +3682,7 @@ Theorem shr'_shr': shr' (shr' x y) z = shr' x (Int.add y z). Proof. intros. apply Int.ltu_inv in H1. - destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega. + destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia. set (y' := repr (Int.unsigned y)) in *. set (z' := repr (Int.unsigned z)) in *. replace (shr' x y) with (shr x y'). @@ -3707,21 +3707,21 @@ Proof. apply same_bits_eq; intros. rewrite bits_shru' by auto. fold Z. destruct (zlt Z Y). - assert (A: Int.unsigned (Int.sub y z) = Y - Z). - { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. } - symmetry; rewrite bits_shl', A by omega. + { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. } + symmetry; rewrite bits_shl', A by lia. destruct (zlt (i + Z) zwordsize). -+ rewrite bits_shl' by omega. fold Y. - destruct (zlt i (Y - Z)); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. - rewrite bits_zero_ext by omega. rewrite zlt_true by omega. f_equal; omega. -+ rewrite bits_zero_ext by omega. rewrite ! zlt_false by omega. auto. ++ rewrite bits_shl' by lia. fold Y. + destruct (zlt i (Y - Z)); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. + rewrite bits_zero_ext by lia. rewrite zlt_true by lia. f_equal; lia. ++ rewrite bits_zero_ext by lia. rewrite ! zlt_false by lia. auto. - assert (A: Int.unsigned (Int.sub z y) = Z - Y). - { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. } - rewrite bits_zero_ext, bits_shru', A by omega. - destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. - rewrite bits_shl' by omega. fold Y. + { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. } + rewrite bits_zero_ext, bits_shru', A by lia. + destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. + rewrite bits_shl' by lia. fold Y. destruct (zlt (i + Z) Y). -+ rewrite zlt_false by omega. auto. -+ rewrite zlt_true by omega. f_equal; omega. ++ rewrite zlt_false by lia. auto. ++ rewrite zlt_true by lia. f_equal; lia. Qed. Theorem shr'_shl': @@ -3734,26 +3734,26 @@ Proof. change (Int.unsigned iwordsize') with zwordsize in *. unfold Int.ltu. set (Y := Int.unsigned y) in *; set (Z := Int.unsigned z) in *. apply same_bits_eq; intros. rewrite bits_shr' by auto. fold Z. - rewrite bits_shl' by (destruct (zlt (i + Z) zwordsize); omega). fold Y. + rewrite bits_shl' by (destruct (zlt (i + Z) zwordsize); lia). fold Y. destruct (zlt Z Y). - assert (A: Int.unsigned (Int.sub y z) = Y - Z). - { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. } - rewrite bits_shl', A by omega. + { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. } + rewrite bits_shl', A by lia. destruct (zlt i (Y - Z)). -+ apply zlt_true. destruct (zlt (i + Z) zwordsize); omega. -+ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); omega). - rewrite bits_sign_ext by omega. f_equal. ++ apply zlt_true. destruct (zlt (i + Z) zwordsize); lia. ++ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia). + rewrite bits_sign_ext by lia. f_equal. destruct (zlt (i + Z) zwordsize). - rewrite zlt_true by omega. omega. - rewrite zlt_false by omega. omega. + rewrite zlt_true by lia. lia. + rewrite zlt_false by lia. lia. - assert (A: Int.unsigned (Int.sub z y) = Z - Y). - { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. } - rewrite bits_sign_ext by omega. - rewrite bits_shr' by (destruct (zlt i (zwordsize - Z)); omega). - rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); omega). + { apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. } + rewrite bits_sign_ext by lia. + rewrite bits_shr' by (destruct (zlt i (zwordsize - Z)); lia). + rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia). f_equal. destruct (zlt i (zwordsize - Z)). -+ rewrite ! zlt_true by omega. omega. -+ rewrite ! zlt_false by omega. rewrite zlt_true by omega. omega. ++ rewrite ! zlt_true by lia. lia. ++ rewrite ! zlt_false by lia. rewrite zlt_true by lia. lia. Qed. Lemma shl'_zero_ext: @@ -3761,11 +3761,11 @@ Lemma shl'_zero_ext: shl' (zero_ext n x) m = zero_ext (n + Int.unsigned m) (shl' x m). Proof. intros. apply same_bits_eq; intros. - rewrite bits_zero_ext, ! bits_shl' by omega. + rewrite bits_zero_ext, ! bits_shl' by lia. destruct (zlt i (Int.unsigned m)). -- rewrite zlt_true by omega; auto. -- rewrite bits_zero_ext by omega. - destruct (zlt (i - Int.unsigned m) n); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. +- rewrite zlt_true by lia; auto. +- rewrite bits_zero_ext by lia. + destruct (zlt (i - Int.unsigned m) n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. Qed. Lemma shl'_sign_ext: @@ -3774,12 +3774,12 @@ Lemma shl'_sign_ext: Proof. intros. generalize (Int.unsigned_range m); intros. apply same_bits_eq; intros. - rewrite bits_sign_ext, ! bits_shl' by omega. + rewrite bits_sign_ext, ! bits_shl' by lia. destruct (zlt i (n + Int.unsigned m)). - rewrite bits_shl' by auto. destruct (zlt i (Int.unsigned m)); auto. - rewrite bits_sign_ext by omega. f_equal. apply zlt_true. omega. -- rewrite zlt_false by omega. rewrite bits_shl' by omega. rewrite zlt_false by omega. - rewrite bits_sign_ext by omega. f_equal. rewrite zlt_false by omega. omega. + rewrite bits_sign_ext by lia. f_equal. apply zlt_true. lia. +- rewrite zlt_false by lia. rewrite bits_shl' by lia. rewrite zlt_false by lia. + rewrite bits_sign_ext by lia. f_equal. rewrite zlt_false by lia. lia. Qed. Lemma shru'_zero_ext: @@ -3787,9 +3787,9 @@ Lemma shru'_zero_ext: shru' (zero_ext (n + Int.unsigned m) x) m = zero_ext n (shru' x m). Proof. intros. generalize (Int.unsigned_range m); intros. - bit_solve; [|omega]. rewrite bits_shru', bits_zero_ext, bits_shru' by omega. + bit_solve; [|lia]. rewrite bits_shru', bits_zero_ext, bits_shru' by lia. destruct (zlt (i + Int.unsigned m) zwordsize). -* destruct (zlt i n); [rewrite zlt_true by omega|rewrite zlt_false by omega]; auto. +* destruct (zlt i n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto. * destruct (zlt i n); auto. Qed. @@ -3798,9 +3798,9 @@ Lemma shru'_zero_ext_0: shru' (zero_ext n x) m = zero. Proof. intros. generalize (Int.unsigned_range m); intros. - bit_solve. rewrite bits_shru', bits_zero_ext by omega. + bit_solve. rewrite bits_shru', bits_zero_ext by lia. destruct (zlt (i + Int.unsigned m) zwordsize); auto. - apply zlt_false. omega. + apply zlt_false. lia. Qed. Lemma shr'_sign_ext: @@ -3813,12 +3813,12 @@ Proof. rewrite bits_sign_ext, bits_shr'. - f_equal. destruct (zlt i n), (zlt (i + Int.unsigned m) zwordsize). -+ apply zlt_true; omega. -+ apply zlt_true; omega. -+ rewrite zlt_false by omega. rewrite zlt_true by omega. omega. -+ rewrite zlt_false by omega. rewrite zlt_true by omega. omega. -- destruct (zlt i n); omega. -- destruct (zlt (i + Int.unsigned m) zwordsize); omega. ++ apply zlt_true; lia. ++ apply zlt_true; lia. ++ rewrite zlt_false by lia. rewrite zlt_true by lia. lia. ++ rewrite zlt_false by lia. rewrite zlt_true by lia. lia. +- destruct (zlt i n); lia. +- destruct (zlt (i + Int.unsigned m) zwordsize); lia. Qed. Lemma zero_ext_shru'_min: @@ -3827,10 +3827,10 @@ Lemma zero_ext_shru'_min: Proof. intros. apply Int.ltu_inv in H. change (Int.unsigned iwordsize') with zwordsize in H. apply Z.min_case_strong; intros; auto. - bit_solve; try omega. rewrite ! bits_shru' by omega. + bit_solve; try lia. rewrite ! bits_shru' by lia. destruct (zlt i (zwordsize - Int.unsigned n)). - rewrite zlt_true by omega. auto. - destruct (zlt i s); auto. rewrite zlt_false by omega; auto. + rewrite zlt_true by lia. auto. + destruct (zlt i s); auto. rewrite zlt_false by lia; auto. Qed. Lemma sign_ext_shr'_min: @@ -3842,12 +3842,12 @@ Proof. destruct (Z.min_spec (zwordsize - Int.unsigned n) s) as [[A B] | [A B]]; rewrite B; auto. apply same_bits_eq; intros. rewrite ! bits_sign_ext by auto. destruct (zlt i (zwordsize - Int.unsigned n)). - rewrite zlt_true by omega. auto. + rewrite zlt_true by lia. auto. assert (C: testbit (shr' x n) (zwordsize - Int.unsigned n - 1) = testbit x (zwordsize - 1)). - { rewrite bits_shr' by omega. rewrite zlt_true by omega. f_equal; omega. } - rewrite C. destruct (zlt i s); rewrite bits_shr' by omega. - rewrite zlt_false by omega. auto. - rewrite zlt_false by omega. auto. + { rewrite bits_shr' by lia. rewrite zlt_true by lia. f_equal; lia. } + rewrite C. destruct (zlt i s); rewrite bits_shr' by lia. + rewrite zlt_false by lia. auto. + rewrite zlt_false by lia. auto. Qed. Lemma shl'_zero_ext_min: @@ -3858,10 +3858,10 @@ Proof. apply Z.min_case_strong; intros; auto. apply same_bits_eq; intros. rewrite ! bits_shl' by auto. destruct (zlt i (Int.unsigned n)); auto. - rewrite ! bits_zero_ext by omega. + rewrite ! bits_zero_ext by lia. destruct (zlt (i - Int.unsigned n) s). - rewrite zlt_true by omega; auto. - rewrite zlt_false by omega; auto. + rewrite zlt_true by lia; auto. + rewrite zlt_false by lia; auto. Qed. Lemma shl'_sign_ext_min: @@ -3873,10 +3873,10 @@ Proof. destruct (Z.min_spec (zwordsize - Int.unsigned n) s) as [[A B] | [A B]]; rewrite B; auto. apply same_bits_eq; intros. rewrite ! bits_shl' by auto. destruct (zlt i (Int.unsigned n)); auto. - rewrite ! bits_sign_ext by omega. f_equal. + rewrite ! bits_sign_ext by lia. f_equal. destruct (zlt (i - Int.unsigned n) s). - rewrite zlt_true by omega; auto. - omegaContradiction. + rewrite zlt_true by lia; auto. + extlia. Qed. (** Powers of two with exponents given as 32-bit ints *) @@ -3897,8 +3897,8 @@ Proof. destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]]. exploit Z_one_bits_range; eauto. fold zwordsize. intros R. unfold Int.ltu. rewrite EQ. rewrite Int.unsigned_repr. - change (Int.unsigned iwordsize') with zwordsize. apply zlt_true. omega. - assert (zwordsize < Int.max_unsigned) by reflexivity. omega. + change (Int.unsigned iwordsize') with zwordsize. apply zlt_true. lia. + assert (zwordsize < Int.max_unsigned) by reflexivity. lia. Qed. Fixpoint int_of_one_bits' (l: list Int.int) : int := @@ -3917,7 +3917,7 @@ Proof. - auto. - rewrite IHl by eauto. apply eqm_samerepr; apply eqm_add. + rewrite shl'_one_two_p. rewrite Int.unsigned_repr. apply eqm_sym; apply eqm_unsigned_repr. - exploit (H a). auto. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. + exploit (H a). auto. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. + apply eqm_sym; apply eqm_unsigned_repr. } intros. rewrite <- (repr_unsigned x) at 1. unfold one_bits'. rewrite REC. @@ -3936,7 +3936,7 @@ Proof. { apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. } rewrite Int.unsigned_repr. auto. assert (zwordsize < Int.max_unsigned) by reflexivity. - omega. + lia. Qed. Theorem is_power2'_range: @@ -3955,11 +3955,11 @@ Proof. unfold is_power2'; intros. destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv H. rewrite (Z_one_bits_powerserie wordsize (unsigned n)) by (apply unsigned_range). - rewrite Int.unsigned_repr. rewrite B; simpl. omega. + rewrite Int.unsigned_repr. rewrite B; simpl. lia. assert (0 <= i < zwordsize). { apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. } assert (zwordsize < Int.max_unsigned) by reflexivity. - omega. + lia. Qed. Theorem mul_pow2': @@ -4003,7 +4003,7 @@ Proof. assert (zwordsize = 2 * Int.zwordsize) by reflexivity. fold (testbit (shru n (repr Int.zwordsize)) i). rewrite bits_shru. change (unsigned (repr Int.zwordsize)) with Int.zwordsize. - apply zlt_true. omega. omega. + apply zlt_true. lia. lia. Qed. Lemma bits_ofwords: @@ -4018,15 +4018,15 @@ Proof. rewrite testbit_repr; auto. rewrite !testbit_repr; auto. fold (Int.testbit lo i). rewrite Int.bits_above. apply orb_false_r. auto. - omega. + lia. Qed. Lemma lo_ofwords: forall hi lo, loword (ofwords hi lo) = lo. Proof. intros. apply Int.same_bits_eq; intros. - rewrite bits_loword; auto. rewrite bits_ofwords. apply zlt_true. omega. - assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega. + rewrite bits_loword; auto. rewrite bits_ofwords. apply zlt_true. lia. + assert (zwordsize = 2 * Int.zwordsize) by reflexivity. lia. Qed. Lemma hi_ofwords: @@ -4034,8 +4034,8 @@ Lemma hi_ofwords: Proof. intros. apply Int.same_bits_eq; intros. rewrite bits_hiword; auto. rewrite bits_ofwords. - rewrite zlt_false. f_equal. omega. omega. - assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega. + rewrite zlt_false. f_equal. lia. lia. + assert (zwordsize = 2 * Int.zwordsize) by reflexivity. lia. Qed. Lemma ofwords_recompose: @@ -4043,9 +4043,9 @@ Lemma ofwords_recompose: Proof. intros. apply same_bits_eq; intros. rewrite bits_ofwords; auto. destruct (zlt i Int.zwordsize). - apply bits_loword. omega. - rewrite bits_hiword. f_equal. omega. - assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega. + apply bits_loword. lia. + rewrite bits_hiword. f_equal. lia. + assert (zwordsize = 2 * Int.zwordsize) by reflexivity. lia. Qed. Lemma ofwords_add: @@ -4056,10 +4056,10 @@ Proof. apply eqm_sym; apply eqm_unsigned_repr. apply eqm_refl. apply eqm_sym; apply eqm_unsigned_repr. - change Int.zwordsize with 32; change zwordsize with 64; omega. + change Int.zwordsize with 32; change zwordsize with 64; lia. rewrite unsigned_repr. generalize (Int.unsigned_range lo). intros [A B]. exact B. assert (Int.max_unsigned < max_unsigned) by (compute; auto). - generalize (Int.unsigned_range_2 lo); omega. + generalize (Int.unsigned_range_2 lo); lia. Qed. Lemma ofwords_add': @@ -4070,7 +4070,7 @@ Proof. change (two_p 32) with Int.modulus. change Int.modulus with 4294967296. change max_unsigned with 18446744073709551615. - omega. + lia. Qed. Remark eqm_mul_2p32: @@ -4094,7 +4094,7 @@ Proof. change min_signed with (Int.min_signed * Int.modulus). change max_signed with (Int.max_signed * Int.modulus + Int.modulus - 1). change Int.modulus with 4294967296. - omega. + lia. apply eqm_samerepr. apply eqm_add. apply eqm_mul_2p32. apply Int.eqm_signed_unsigned. apply eqm_refl. Qed. @@ -4109,7 +4109,7 @@ Proof. intros. apply Int64.same_bits_eq; intros. rewrite H by auto. rewrite ! bits_ofwords by auto. assert (zwordsize = 2 * Int.zwordsize) by reflexivity. - destruct (zlt i Int.zwordsize); rewrite H0 by omega; auto. + destruct (zlt i Int.zwordsize); rewrite H0 by lia; auto. Qed. Lemma decompose_and: @@ -4154,21 +4154,21 @@ Proof. intros. assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. } + rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. } assert (zwordsize = 2 * Int.zwordsize) by reflexivity. apply Int64.same_bits_eq; intros. rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto. - destruct (zlt i Int.zwordsize). rewrite Int.bits_shl by omega. + destruct (zlt i Int.zwordsize). rewrite Int.bits_shl by lia. destruct (zlt i (Int.unsigned y)). auto. - rewrite bits_ofwords by omega. rewrite zlt_true by omega. auto. - rewrite zlt_false by omega. rewrite bits_ofwords by omega. - rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega. - rewrite Int.bits_shru by omega. rewrite H0. + rewrite bits_ofwords by lia. rewrite zlt_true by lia. auto. + rewrite zlt_false by lia. rewrite bits_ofwords by lia. + rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia. + rewrite Int.bits_shru by lia. rewrite H0. destruct (zlt (i - Int.unsigned y) (Int.zwordsize)). - rewrite zlt_true by omega. rewrite zlt_true by omega. - rewrite orb_false_l. f_equal. omega. - rewrite zlt_false by omega. rewrite zlt_false by omega. - rewrite orb_false_r. f_equal. omega. + rewrite zlt_true by lia. rewrite zlt_true by lia. + rewrite orb_false_l. f_equal. lia. + rewrite zlt_false by lia. rewrite zlt_false by lia. + rewrite orb_false_r. f_equal. lia. Qed. Lemma decompose_shl_2: @@ -4181,15 +4181,15 @@ Proof. assert (zwordsize = 2 * Int.zwordsize) by reflexivity. assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. } + rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. } apply Int64.same_bits_eq; intros. rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto. - destruct (zlt i Int.zwordsize). rewrite zlt_true by omega. apply Int.bits_zero. - rewrite Int.bits_shl by omega. + destruct (zlt i Int.zwordsize). rewrite zlt_true by lia. apply Int.bits_zero. + rewrite Int.bits_shl by lia. destruct (zlt i (Int.unsigned y)). - rewrite zlt_true by omega. auto. - rewrite zlt_false by omega. - rewrite bits_ofwords by omega. rewrite zlt_true by omega. f_equal. omega. + rewrite zlt_true by lia. auto. + rewrite zlt_false by lia. + rewrite bits_ofwords by lia. rewrite zlt_true by lia. f_equal. lia. Qed. Lemma decompose_shru_1: @@ -4202,25 +4202,25 @@ Proof. intros. assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. } + rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. } assert (zwordsize = 2 * Int.zwordsize) by reflexivity. apply Int64.same_bits_eq; intros. rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto. destruct (zlt i Int.zwordsize). - rewrite zlt_true by omega. - rewrite bits_ofwords by omega. - rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega. - rewrite Int.bits_shru by omega. rewrite H0. + rewrite zlt_true by lia. + rewrite bits_ofwords by lia. + rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia. + rewrite Int.bits_shru by lia. rewrite H0. destruct (zlt (i + Int.unsigned y) (Int.zwordsize)). - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite orb_false_r. auto. - rewrite zlt_false by omega. - rewrite orb_false_l. f_equal. omega. - rewrite Int.bits_shru by omega. + rewrite zlt_false by lia. + rewrite orb_false_l. f_equal. lia. + rewrite Int.bits_shru by lia. destruct (zlt (i + Int.unsigned y) zwordsize). - rewrite bits_ofwords by omega. - rewrite zlt_true by omega. rewrite zlt_false by omega. f_equal. omega. - rewrite zlt_false by omega. auto. + rewrite bits_ofwords by lia. + rewrite zlt_true by lia. rewrite zlt_false by lia. f_equal. lia. + rewrite zlt_false by lia. auto. Qed. Lemma decompose_shru_2: @@ -4233,16 +4233,16 @@ Proof. assert (zwordsize = 2 * Int.zwordsize) by reflexivity. assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. } + rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. } apply Int64.same_bits_eq; intros. rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto. destruct (zlt i Int.zwordsize). - rewrite Int.bits_shru by omega. rewrite H1. + rewrite Int.bits_shru by lia. rewrite H1. destruct (zlt (i + Int.unsigned y) zwordsize). - rewrite zlt_true by omega. rewrite bits_ofwords by omega. - rewrite zlt_false by omega. f_equal; omega. - rewrite zlt_false by omega. auto. - rewrite zlt_false by omega. apply Int.bits_zero. + rewrite zlt_true by lia. rewrite bits_ofwords by lia. + rewrite zlt_false by lia. f_equal; lia. + rewrite zlt_false by lia. auto. + rewrite zlt_false by lia. apply Int.bits_zero. Qed. Lemma decompose_shr_1: @@ -4255,26 +4255,26 @@ Proof. intros. assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. } + rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. } assert (zwordsize = 2 * Int.zwordsize) by reflexivity. apply Int64.same_bits_eq; intros. rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto. destruct (zlt i Int.zwordsize). - rewrite zlt_true by omega. - rewrite bits_ofwords by omega. - rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega. - rewrite Int.bits_shru by omega. rewrite H0. + rewrite zlt_true by lia. + rewrite bits_ofwords by lia. + rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia. + rewrite Int.bits_shru by lia. rewrite H0. destruct (zlt (i + Int.unsigned y) (Int.zwordsize)). - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite orb_false_r. auto. - rewrite zlt_false by omega. - rewrite orb_false_l. f_equal. omega. - rewrite Int.bits_shr by omega. + rewrite zlt_false by lia. + rewrite orb_false_l. f_equal. lia. + rewrite Int.bits_shr by lia. destruct (zlt (i + Int.unsigned y) zwordsize). - rewrite bits_ofwords by omega. - rewrite zlt_true by omega. rewrite zlt_false by omega. f_equal. omega. - rewrite zlt_false by omega. rewrite bits_ofwords by omega. - rewrite zlt_false by omega. f_equal. + rewrite bits_ofwords by lia. + rewrite zlt_true by lia. rewrite zlt_false by lia. f_equal. lia. + rewrite zlt_false by lia. rewrite bits_ofwords by lia. + rewrite zlt_false by lia. f_equal. Qed. Lemma decompose_shr_2: @@ -4288,24 +4288,24 @@ Proof. assert (zwordsize = 2 * Int.zwordsize) by reflexivity. assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize). { unfold Int.sub. rewrite Int.unsigned_repr. auto. - rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. } + rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. } apply Int64.same_bits_eq; intros. rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto. destruct (zlt i Int.zwordsize). - rewrite Int.bits_shr by omega. rewrite H1. + rewrite Int.bits_shr by lia. rewrite H1. destruct (zlt (i + Int.unsigned y) zwordsize). - rewrite zlt_true by omega. rewrite bits_ofwords by omega. - rewrite zlt_false by omega. f_equal; omega. - rewrite zlt_false by omega. rewrite bits_ofwords by omega. - rewrite zlt_false by omega. auto. - rewrite Int.bits_shr by omega. + rewrite zlt_true by lia. rewrite bits_ofwords by lia. + rewrite zlt_false by lia. f_equal; lia. + rewrite zlt_false by lia. rewrite bits_ofwords by lia. + rewrite zlt_false by lia. auto. + rewrite Int.bits_shr by lia. change (Int.unsigned (Int.sub Int.iwordsize Int.one)) with (Int.zwordsize - 1). destruct (zlt (i + Int.unsigned y) zwordsize); - rewrite bits_ofwords by omega. - symmetry. rewrite zlt_false by omega. f_equal. - destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); omega. - symmetry. rewrite zlt_false by omega. f_equal. - destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); omega. + rewrite bits_ofwords by lia. + symmetry. rewrite zlt_false by lia. f_equal. + destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); lia. + symmetry. rewrite zlt_false by lia. f_equal. + destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); lia. Qed. Lemma decompose_add: @@ -4442,14 +4442,14 @@ Proof. intros. unfold ltu. rewrite ! ofwords_add'. unfold Int.ltu, Int.eq. destruct (zeq (Int.unsigned xh) (Int.unsigned yh)). rewrite e. destruct (zlt (Int.unsigned xl) (Int.unsigned yl)). - apply zlt_true; omega. - apply zlt_false; omega. + apply zlt_true; lia. + apply zlt_false; lia. change (two_p 32) with Int.modulus. generalize (Int.unsigned_range xl) (Int.unsigned_range yl). change Int.modulus with 4294967296. intros. destruct (zlt (Int.unsigned xh) (Int.unsigned yh)). - apply zlt_true; omega. - apply zlt_false; omega. + apply zlt_true; lia. + apply zlt_false; lia. Qed. Lemma decompose_leu: @@ -4461,8 +4461,8 @@ Proof. unfold Int.eq. destruct (zeq (Int.unsigned xh) (Int.unsigned yh)). auto. unfold Int.ltu. destruct (zlt (Int.unsigned xh) (Int.unsigned yh)). - rewrite zlt_false by omega; auto. - rewrite zlt_true by omega; auto. + rewrite zlt_false by lia; auto. + rewrite zlt_true by lia; auto. Qed. Lemma decompose_lt: @@ -4472,14 +4472,14 @@ Proof. intros. unfold lt. rewrite ! ofwords_add''. rewrite Int.eq_signed. destruct (zeq (Int.signed xh) (Int.signed yh)). rewrite e. unfold Int.ltu. destruct (zlt (Int.unsigned xl) (Int.unsigned yl)). - apply zlt_true; omega. - apply zlt_false; omega. + apply zlt_true; lia. + apply zlt_false; lia. change (two_p 32) with Int.modulus. generalize (Int.unsigned_range xl) (Int.unsigned_range yl). change Int.modulus with 4294967296. intros. unfold Int.lt. destruct (zlt (Int.signed xh) (Int.signed yh)). - apply zlt_true; omega. - apply zlt_false; omega. + apply zlt_true; lia. + apply zlt_false; lia. Qed. Lemma decompose_le: @@ -4491,8 +4491,8 @@ Proof. rewrite Int.eq_signed. destruct (zeq (Int.signed xh) (Int.signed yh)). auto. unfold Int.lt. destruct (zlt (Int.signed xh) (Int.signed yh)). - rewrite zlt_false by omega; auto. - rewrite zlt_true by omega; auto. + rewrite zlt_false by lia; auto. + rewrite zlt_true by lia; auto. Qed. (** Utility proofs for mixed 32bit and 64bit arithmetic *) @@ -4507,7 +4507,7 @@ Proof. change (wordsize) with 64%nat in *. change (Int.wordsize) with 32%nat in *. unfold two_power_nat. simpl. - omega. + lia. Qed. Remark int_unsigned_repr: @@ -4527,9 +4527,9 @@ Proof. rewrite unsigned_repr by apply int_unsigned_range. rewrite int_unsigned_repr. reflexivity. rewrite unsigned_repr by apply int_unsigned_range. rewrite int_unsigned_repr. generalize (int_unsigned_range y). - omega. + lia. generalize (Int.sub_ltu x y H). intros. - generalize (Int.unsigned_range_2 y). intros. omega. + generalize (Int.unsigned_range_2 y). intros. lia. Qed. End Int64. @@ -4687,7 +4687,7 @@ Lemma to_int_of_int: forall n, to_int (of_int n) = n. Proof. intros; unfold of_int, to_int. rewrite unsigned_repr. apply Int.repr_unsigned. - unfold max_unsigned. rewrite modulus_eq32. destruct (Int.unsigned_range n); omega. + unfold max_unsigned. rewrite modulus_eq32. destruct (Int.unsigned_range n); lia. Qed. End AGREE32. @@ -4797,7 +4797,7 @@ Lemma to_int64_of_int64: forall n, to_int64 (of_int64 n) = n. Proof. intros; unfold of_int64, to_int64. rewrite unsigned_repr. apply Int64.repr_unsigned. - unfold max_unsigned. rewrite modulus_eq64. destruct (Int64.unsigned_range n); omega. + unfold max_unsigned. rewrite modulus_eq64. destruct (Int64.unsigned_range n); lia. Qed. End AGREE64. @@ -41,14 +41,14 @@ Lemma notin_range: forall x i, x < fst i \/ x >= snd i -> ~In x i. Proof. - unfold In; intros; omega. + unfold In; intros; lia. Qed. Lemma range_notin: forall x i, ~In x i -> fst i < snd i -> x < fst i \/ x >= snd i. Proof. - unfold In; intros; omega. + unfold In; intros; lia. Qed. (** * Emptyness *) @@ -60,26 +60,26 @@ Lemma empty_dec: Proof. unfold empty; intros. case (zle (snd i) (fst i)); intros. - left; omega. - right; omega. + left; lia. + right; lia. Qed. Lemma is_notempty: forall i, fst i < snd i -> ~empty i. Proof. - unfold empty; intros; omega. + unfold empty; intros; lia. Qed. Lemma empty_notin: forall x i, empty i -> ~In x i. Proof. - unfold empty, In; intros. omega. + unfold empty, In; intros. lia. Qed. Lemma in_notempty: forall x i, In x i -> ~empty i. Proof. - unfold empty, In; intros. omega. + unfold empty, In; intros. lia. Qed. (** * Disjointness *) @@ -109,7 +109,7 @@ Lemma disjoint_range: forall i j, snd i <= fst j \/ snd j <= fst i -> disjoint i j. Proof. - unfold disjoint, In; intros. omega. + unfold disjoint, In; intros. lia. Qed. Lemma range_disjoint: @@ -127,13 +127,13 @@ Proof. (* Case 1.1: i ends to the left of j, OK *) auto. (* Case 1.2: i ends to the right of j's start, not disjoint. *) - elim (H (fst j)). red; omega. red; omega. + elim (H (fst j)). red; lia. red; lia. (* Case 2: j starts to the left of i *) destruct (zle (snd j) (fst i)). (* Case 2.1: j ends to the left of i, OK *) auto. (* Case 2.2: j ends to the right of i's start, not disjoint. *) - elim (H (fst i)). red; omega. red; omega. + elim (H (fst i)). red; lia. red; lia. Qed. Lemma range_disjoint': @@ -141,7 +141,7 @@ Lemma range_disjoint': disjoint i j -> fst i < snd i -> fst j < snd j -> snd i <= fst j \/ snd j <= fst i. Proof. - intros. exploit range_disjoint; eauto. unfold empty; intuition omega. + intros. exploit range_disjoint; eauto. unfold empty; intuition lia. Qed. Lemma disjoint_dec: @@ -163,14 +163,14 @@ Lemma in_shift: forall x i delta, In x i -> In (x + delta) (shift i delta). Proof. - unfold shift, In; intros. simpl. omega. + unfold shift, In; intros. simpl. lia. Qed. Lemma in_shift_inv: forall x i delta, In x (shift i delta) -> In (x - delta) i. Proof. - unfold shift, In; simpl; intros. omega. + unfold shift, In; simpl; intros. lia. Qed. (** * Enumerating the elements of an interval *) @@ -182,7 +182,7 @@ Variable lo: Z. Function elements_rec (hi: Z) {wf (Zwf lo) hi} : list Z := if zlt lo hi then (hi-1) :: elements_rec (hi-1) else nil. Proof. - intros. red. omega. + intros. red. lia. apply Zwf_well_founded. Qed. @@ -192,8 +192,8 @@ Lemma In_elements_rec: Proof. intros. functional induction (elements_rec hi). simpl; split; intros. - destruct H. clear IHl. omega. rewrite IHl in H. clear IHl. omega. - destruct (zeq (hi - 1) x); auto. right. rewrite IHl. clear IHl. omega. + destruct H. clear IHl. lia. rewrite IHl in H. clear IHl. lia. + destruct (zeq (hi - 1) x); auto. right. rewrite IHl. clear IHl. lia. simpl; intuition. Qed. @@ -241,20 +241,20 @@ Program Fixpoint forall_rec (hi: Z) {wf (Zwf lo) hi}: left _ _ . Next Obligation. - red. omega. + red. lia. Qed. Next Obligation. - assert (x = hi - 1 \/ x < hi - 1) by omega. + assert (x = hi - 1 \/ x < hi - 1) by lia. destruct H2. congruence. auto. Qed. Next Obligation. - exists wildcard'; split; auto. omega. + exists wildcard'; split; auto. lia. Qed. Next Obligation. - exists (hi - 1); split; auto. omega. + exists (hi - 1); split; auto. lia. Qed. Next Obligation. - omegaContradiction. + extlia. Defined. End FORALL. @@ -276,7 +276,7 @@ Variable a: A. Function fold_rec (hi: Z) {wf (Zwf lo) hi} : A := if zlt lo hi then f (hi - 1) (fold_rec (hi - 1)) else a. Proof. - intros. red. omega. + intros. red. lia. apply Zwf_well_founded. Qed. diff --git a/lib/IntvSets.v b/lib/IntvSets.v index b97d9882..7250a9f6 100644 --- a/lib/IntvSets.v +++ b/lib/IntvSets.v @@ -59,7 +59,7 @@ Proof. + destruct (zle l x); simpl. * tauto. * split; intros. congruence. - exfalso. destruct H0. omega. exploit BELOW; eauto. omega. + exfalso. destruct H0. lia. exploit BELOW; eauto. lia. + rewrite IHok. intuition. Qed. @@ -74,14 +74,14 @@ Lemma contains_In: (contains l0 h0 s = true <-> (forall x, l0 <= x < h0 -> In x s)). Proof. induction 2; simpl. -- intuition. elim (H0 l0); omega. +- intuition. elim (H0 l0); lia. - destruct (zle h0 h); simpl. destruct (zle l l0); simpl. intuition. rewrite IHok. intuition. destruct (H3 x); auto. exfalso. - destruct (H3 l0). omega. omega. exploit BELOW; eauto. omega. + destruct (H3 l0). lia. lia. exploit BELOW; eauto. lia. rewrite IHok. intuition. destruct (H3 x); auto. exfalso. - destruct (H3 h). omega. omega. exploit BELOW; eauto. omega. + destruct (H3 h). lia. lia. exploit BELOW; eauto. lia. Qed. Fixpoint add (L H: Z) (s: t) {struct s} : t := @@ -103,9 +103,9 @@ Proof. destruct (zlt h0 l). simpl. tauto. rewrite IHok. intuition idtac. - assert (l0 <= x < h0 \/ l <= x < h) by xomega. tauto. - left; xomega. - left; xomega. + assert (l0 <= x < h0 \/ l <= x < h) by extlia. tauto. + left; extlia. + left; extlia. Qed. Lemma add_ok: @@ -115,11 +115,11 @@ Proof. constructor. auto. intros. inv H0. constructor. destruct (zlt h l0). constructor; auto. intros. rewrite In_add in H1; auto. - destruct H1. omega. auto. + destruct H1. lia. auto. destruct (zlt h0 l). - constructor. auto. simpl; intros. destruct H1. omega. exploit BELOW; eauto. omega. - constructor. omega. auto. auto. - apply IHok. xomega. + constructor. auto. simpl; intros. destruct H1. lia. exploit BELOW; eauto. lia. + constructor. lia. auto. auto. + apply IHok. extlia. Qed. Fixpoint remove (L H: Z) (s: t) {struct s} : t := @@ -141,22 +141,22 @@ Proof. induction 1; simpl. tauto. destruct (zlt h l0). - simpl. rewrite IHok. intuition omega. + simpl. rewrite IHok. intuition lia. destruct (zlt h0 l). - simpl. intuition. exploit BELOW; eauto. omega. + simpl. intuition. exploit BELOW; eauto. lia. destruct (zlt l l0). destruct (zlt h0 h); simpl. clear IHok. split. intros [A | [A | A]]. - split. omega. left; omega. - split. omega. left; omega. - split. exploit BELOW; eauto. omega. auto. + split. lia. left; lia. + split. lia. left; lia. + split. exploit BELOW; eauto. lia. auto. intros [A [B | B]]. - destruct (zlt x l0). left; omega. right; left; omega. + destruct (zlt x l0). left; lia. right; left; lia. auto. - intuition omega. + intuition lia. destruct (zlt h0 h); simpl. - intuition. exploit BELOW; eauto. omega. - rewrite IHok. intuition. omegaContradiction. + intuition. exploit BELOW; eauto. lia. + rewrite IHok. intuition. extlia. Qed. Lemma remove_ok: @@ -170,9 +170,9 @@ Proof. constructor; auto. destruct (zlt l l0). destruct (zlt h0 h). - constructor. omega. intros. inv H1. omega. exploit BELOW; eauto. omega. - constructor. omega. auto. auto. - constructor; auto. intros. rewrite In_remove in H1 by auto. destruct H1. exploit BELOW; eauto. omega. + constructor. lia. intros. inv H1. lia. exploit BELOW; eauto. lia. + constructor. lia. auto. auto. + constructor; auto. intros. rewrite In_remove in H1 by auto. destruct H1. exploit BELOW; eauto. lia. destruct (zlt h0 h). constructor; auto. auto. @@ -204,19 +204,19 @@ Proof. tauto. assert (ok (Cons l0 h0 s0)) by (constructor; auto). destruct (zle h l0). - rewrite IHok; auto. simpl. intuition. omegaContradiction. - exploit BELOW0; eauto. intros. omegaContradiction. + rewrite IHok; auto. simpl. intuition. extlia. + exploit BELOW0; eauto. intros. extlia. destruct (zle h0 l). - simpl in IHok0; rewrite IHok0. intuition. omegaContradiction. - exploit BELOW; eauto. intros; omegaContradiction. + simpl in IHok0; rewrite IHok0. intuition. extlia. + exploit BELOW; eauto. intros; extlia. destruct (zle l l0). destruct (zle h0 h). simpl. simpl in IHok0; rewrite IHok0. intuition. - simpl. rewrite IHok; auto. simpl. intuition. exploit BELOW0; eauto. intros; omegaContradiction. + simpl. rewrite IHok; auto. simpl. intuition. exploit BELOW0; eauto. intros; extlia. destruct (zle h h0). simpl. rewrite IHok; auto. simpl. intuition. simpl. simpl in IHok0; rewrite IHok0. intuition. - exploit BELOW; eauto. intros; omegaContradiction. + exploit BELOW; eauto. intros; extlia. Qed. Lemma inter_ok: @@ -237,12 +237,12 @@ Proof. constructor; auto. intros. assert (In x (inter (Cons l h s) s0)) by exact H3. rewrite In_inter in H4; auto. apply BELOW0. tauto. - constructor. omega. intros. rewrite In_inter in H3; auto. apply BELOW. tauto. + constructor. lia. intros. rewrite In_inter in H3; auto. apply BELOW. tauto. auto. destruct (zle h h0). - constructor. omega. intros. rewrite In_inter in H3; auto. apply BELOW. tauto. + constructor. lia. intros. rewrite In_inter in H3; auto. apply BELOW. tauto. auto. - constructor. omega. intros. + constructor. lia. intros. assert (In x (inter (Cons l h s) s0)) by exact H3. rewrite In_inter in H4; auto. apply BELOW0. tauto. auto. @@ -281,20 +281,20 @@ Lemma beq_spec: Proof. induction 1; destruct 1; simpl. - tauto. -- split; intros. discriminate. exfalso. apply (H0 l). left; omega. -- split; intros. discriminate. exfalso. apply (H0 l). left; omega. +- split; intros. discriminate. exfalso. apply (H0 l). left; lia. +- split; intros. discriminate. exfalso. apply (H0 l). left; lia. - split; intros. + InvBooleans. subst. rewrite IHok in H3 by auto. rewrite H3. tauto. + destruct (zeq l l0). destruct (zeq h h0). simpl. subst. apply IHok. auto. intros; split; intros. - destruct (proj1 (H1 x)); auto. exfalso. exploit BELOW; eauto. omega. - destruct (proj2 (H1 x)); auto. exfalso. exploit BELOW0; eauto. omega. + destruct (proj1 (H1 x)); auto. exfalso. exploit BELOW; eauto. lia. + destruct (proj2 (H1 x)); auto. exfalso. exploit BELOW0; eauto. lia. exfalso. subst l0. destruct (zlt h h0). - destruct (proj2 (H1 h)). left; omega. omega. exploit BELOW; eauto. omega. - destruct (proj1 (H1 h0)). left; omega. omega. exploit BELOW0; eauto. omega. + destruct (proj2 (H1 h)). left; lia. lia. exploit BELOW; eauto. lia. + destruct (proj1 (H1 h0)). left; lia. lia. exploit BELOW0; eauto. lia. exfalso. destruct (zlt l l0). - destruct (proj1 (H1 l)). left; omega. omega. exploit BELOW0; eauto. omega. - destruct (proj2 (H1 l0)). left; omega. omega. exploit BELOW; eauto. omega. + destruct (proj1 (H1 l)). left; lia. lia. exploit BELOW0; eauto. lia. + destruct (proj2 (H1 l0)). left; lia. lia. exploit BELOW; eauto. lia. Qed. End R. @@ -340,7 +340,7 @@ Proof. unfold add, In; intros. destruct (zlt l h). simpl. apply R.In_add. apply proj2_sig. - intuition. omegaContradiction. + intuition. extlia. Qed. Program Definition remove (l h: Z) (s: t) : t := @@ -392,7 +392,7 @@ Theorem contains_spec: Proof. unfold contains, In; intros. destruct (zlt l h). apply R.contains_In. auto. apply proj2_sig. - split; intros. omegaContradiction. auto. + split; intros. extlia. auto. Qed. Program Definition beq (s1 s2: t) : bool := R.beq s1 s2. diff --git a/lib/Iteration.v b/lib/Iteration.v index 6a9d3253..0cca7fb7 100644 --- a/lib/Iteration.v +++ b/lib/Iteration.v @@ -237,8 +237,8 @@ Lemma iter_monot: Proof. induction p; intros. simpl. red; intros; red; auto. - destruct q. elimtype False; omega. - simpl. apply F_iter_monot. apply IHp. omega. + destruct q. elimtype False; lia. + simpl. apply F_iter_monot. apply IHp. lia. Qed. Lemma iter_either: @@ -1395,7 +1395,7 @@ Theorem cardinal_remove: Proof. unfold cardinal; intros. exploit T.elements_remove; eauto. intros (l1 & l2 & P & Q). - rewrite P, Q. rewrite ! app_length. simpl. omega. + rewrite P, Q. rewrite ! app_length. simpl. lia. Qed. Theorem cardinal_set: diff --git a/lib/Ordered.v b/lib/Ordered.v index 1adbd330..69dc1c69 100644 --- a/lib/Ordered.v +++ b/lib/Ordered.v @@ -70,7 +70,7 @@ Proof (@eq_trans t). Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z. Proof Z.lt_trans. Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. -Proof. unfold lt, eq, t; intros. omega. Qed. +Proof. unfold lt, eq, t; intros. lia. Qed. Lemma compare : forall x y : t, Compare lt eq x y. Proof. intros. destruct (Z.compare x y) as [] eqn:E. @@ -99,11 +99,11 @@ Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. Proof (@eq_trans t). Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z. Proof. - unfold lt; intros. omega. + unfold lt; intros. lia. Qed. Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. Proof. - unfold lt,eq; intros; red; intros. subst. omega. + unfold lt,eq; intros; red; intros. subst. lia. Qed. Lemma compare : forall x y : t, Compare lt eq x y. Proof. @@ -114,7 +114,7 @@ Proof. apply GT. assert (Int.unsigned x <> Int.unsigned y). red; intros. rewrite <- (Int.repr_unsigned x) in n. rewrite <- (Int.repr_unsigned y) in n. congruence. - red. omega. + red. lia. Defined. Definition eq_dec : forall x y, { eq x y } + { ~ eq x y } := Int.eq_dec. diff --git a/lib/Parmov.v b/lib/Parmov.v index db27e83f..f602bd60 100644 --- a/lib/Parmov.v +++ b/lib/Parmov.v @@ -1106,7 +1106,7 @@ Lemma measure_decreasing_1: forall st st', dtransition st st' -> measure st' < measure st. Proof. - induction 1; repeat (simpl; rewrite List.app_length); simpl; omega. + induction 1; repeat (simpl; rewrite List.app_length); simpl; lia. Qed. Lemma measure_decreasing_2: diff --git a/lib/Postorder.v b/lib/Postorder.v index 3181c4cc..eaeaea37 100644 --- a/lib/Postorder.v +++ b/lib/Postorder.v @@ -314,10 +314,10 @@ Proof. destruct (wrk s) as [ | [x succs] l]. discriminate. destruct succs as [ | y succs ]. - inv H. simpl. apply lex_ord_right. omega. + inv H. simpl. apply lex_ord_right. lia. destruct ((gr s)!y) as [succs'|] eqn:?. inv H. simpl. apply lex_ord_left. eapply PTree_Properties.cardinal_remove; eauto. - inv H. simpl. apply lex_ord_right. omega. + inv H. simpl. apply lex_ord_right. lia. Qed. End POSTORDER. diff --git a/lib/UnionFind.v b/lib/UnionFind.v index 20bb91cd..c4a2f5b1 100644 --- a/lib/UnionFind.v +++ b/lib/UnionFind.v @@ -552,10 +552,10 @@ Proof. rewrite H; auto. simpl in G. rewrite M.gsspec in G. destruct (M.elt_eq x0 (repr uf a)). rewrite e. rewrite repr_canonical. rewrite dec_eq_true. inversion G. subst x'. rewrite dec_eq_false; auto. - replace (pathlen uf (repr uf a)) with 0. omega. + replace (pathlen uf (repr uf a)) with 0. lia. symmetry. apply pathlen_none. apply repr_res_none. rewrite (repr_unroll uf x0), (pathlen_unroll uf x0); rewrite G. - destruct (M.elt_eq (repr uf x') (repr uf a)); omega. + destruct (M.elt_eq (repr uf x') (repr uf a)); lia. simpl in G. rewrite M.gsspec in G. destruct (M.elt_eq x0 (repr uf a)); try discriminate. rewrite (repr_none uf x0) by auto. rewrite dec_eq_false; auto. symmetry. apply pathlen_zero; auto. apply repr_none; auto. @@ -570,7 +570,7 @@ Proof. intros. repeat rewrite pathlen_merge. destruct (M.elt_eq (repr uf a) (repr uf b)). auto. rewrite H. destruct (M.elt_eq (repr uf y) (repr uf a)). - omega. auto. + lia. auto. Qed. (* Path compression *) diff --git a/lib/Zbits.v b/lib/Zbits.v index 6f3acaab..0539d04b 100644 --- a/lib/Zbits.v +++ b/lib/Zbits.v @@ -33,7 +33,7 @@ Definition eqmod (x y: Z) : Prop := exists k, x = k * modul + y. Lemma eqmod_refl: forall x, eqmod x x. Proof. - intros; red. exists 0. omega. + intros; red. exists 0. lia. Qed. Lemma eqmod_refl2: forall x y, x = y -> eqmod x y. @@ -57,7 +57,7 @@ Lemma eqmod_small_eq: Proof. intros x y [k EQ] I1 I2. generalize (Zdiv_unique _ _ _ _ EQ I2). intro. - rewrite (Z.div_small x modul I1) in H. subst k. omega. + rewrite (Z.div_small x modul I1) in H. subst k. lia. Qed. Lemma eqmod_mod_eq: @@ -136,11 +136,11 @@ Lemma P_mod_two_p_range: forall n p, 0 <= P_mod_two_p p n < two_power_nat n. Proof. induction n; simpl; intros. - - rewrite two_power_nat_O. omega. + - rewrite two_power_nat_O. lia. - rewrite two_power_nat_S. destruct p. - + generalize (IHn p). rewrite Z.succ_double_spec. omega. - + generalize (IHn p). rewrite Z.double_spec. omega. - + generalize (two_power_nat_pos n). omega. + + generalize (IHn p). rewrite Z.succ_double_spec. lia. + + generalize (IHn p). rewrite Z.double_spec. lia. + + generalize (two_power_nat_pos n). lia. Qed. Lemma P_mod_two_p_eq: @@ -157,7 +157,7 @@ Proof. + destruct (IHn p) as [y EQ]. exists y. change (Zpos p~0) with (2 * Zpos p). rewrite EQ. rewrite (Z.double_spec (P_mod_two_p p n)). ring. - + exists 0; omega. + + exists 0; lia. } intros. destruct (H n p) as [y EQ]. @@ -221,8 +221,8 @@ Remark Zshiftin_spec: forall b x, Zshiftin b x = 2 * x + (if b then 1 else 0). Proof. unfold Zshiftin; intros. destruct b. - - rewrite Z.succ_double_spec. omega. - - rewrite Z.double_spec. omega. + - rewrite Z.succ_double_spec. lia. + - rewrite Z.double_spec. lia. Qed. Remark Zshiftin_inj: @@ -231,10 +231,10 @@ Remark Zshiftin_inj: Proof. intros. rewrite !Zshiftin_spec in H. destruct b1; destruct b2. - split; [auto|omega]. - omegaContradiction. - omegaContradiction. - split; [auto|omega]. + split; [auto|lia]. + extlia. + extlia. + split; [auto|lia]. Qed. Remark Zdecomp: @@ -255,9 +255,9 @@ Proof. - subst n. destruct b. + apply Z.testbit_odd_0. + rewrite Z.add_0_r. apply Z.testbit_even_0. - - assert (0 <= Z.pred n) by omega. + - assert (0 <= Z.pred n) by lia. set (n' := Z.pred n) in *. - replace n with (Z.succ n') by (unfold n'; omega). + replace n with (Z.succ n') by (unfold n'; lia). destruct b. + apply Z.testbit_odd_succ; auto. + rewrite Z.add_0_r. apply Z.testbit_even_succ; auto. @@ -273,7 +273,7 @@ Remark Ztestbit_shiftin_succ: forall b x n, 0 <= n -> Z.testbit (Zshiftin b x) (Z.succ n) = Z.testbit x n. Proof. intros. rewrite Ztestbit_shiftin. rewrite zeq_false. rewrite Z.pred_succ. auto. - omega. omega. + lia. lia. Qed. Lemma Zshiftin_ind: @@ -287,7 +287,7 @@ Proof. - induction p. + change (P (Zshiftin true (Z.pos p))). auto. + change (P (Zshiftin false (Z.pos p))). auto. - + change (P (Zshiftin true 0)). apply H0. omega. auto. + + change (P (Zshiftin true 0)). apply H0. lia. auto. - compute in H1. intuition congruence. Qed. @@ -323,7 +323,7 @@ Remark Ztestbit_succ: forall n x, 0 <= n -> Z.testbit x (Z.succ n) = Z.testbit (Z.div2 x) n. Proof. intros. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ. auto. - omega. omega. + lia. lia. Qed. Lemma eqmod_same_bits: @@ -335,13 +335,13 @@ Proof. - change (two_power_nat 0) with 1. exists (x-y); ring. - rewrite two_power_nat_S. assert (eqmod (two_power_nat n) (Z.div2 x) (Z.div2 y)). - apply IHn. intros. rewrite <- !Ztestbit_succ. apply H. rewrite Nat2Z.inj_succ; omega. - omega. omega. + apply IHn. intros. rewrite <- !Ztestbit_succ. apply H. rewrite Nat2Z.inj_succ; lia. + lia. lia. destruct H0 as [k EQ]. exists k. rewrite (Zdecomp x). rewrite (Zdecomp y). replace (Z.odd y) with (Z.odd x). rewrite EQ. rewrite !Zshiftin_spec. ring. - exploit (H 0). rewrite Nat2Z.inj_succ; omega. + exploit (H 0). rewrite Nat2Z.inj_succ; lia. rewrite !Ztestbit_base. auto. Qed. @@ -351,7 +351,7 @@ Lemma same_bits_eqmod: Z.testbit x i = Z.testbit y i. Proof. induction n; intros. - - simpl in H0. omegaContradiction. + - simpl in H0. extlia. - rewrite Nat2Z.inj_succ in H0. rewrite two_power_nat_S in H. rewrite !(Ztestbit_eq i); intuition. destruct H as [k EQ]. @@ -364,7 +364,7 @@ Proof. exploit Zshiftin_inj; eauto. intros [A B]. destruct (zeq i 0). + auto. - + apply IHn. exists k; auto. omega. + + apply IHn. exists k; auto. lia. Qed. Lemma equal_same_bits: @@ -383,7 +383,7 @@ Proof. replace (- Zshiftin (Z.odd x) y - 1) with (Zshiftin (negb (Z.odd x)) (- y - 1)). rewrite !Ztestbit_shiftin; auto. - destruct (zeq i 0). auto. apply IND. omega. + destruct (zeq i 0). auto. apply IND. lia. rewrite !Zshiftin_spec. destruct (Z.odd x); simpl negb; ring. Qed. @@ -395,12 +395,12 @@ Lemma Ztestbit_above: Proof. induction n; intros. - change (two_power_nat 0) with 1 in H. - replace x with 0 by omega. + replace x with 0 by lia. apply Z.testbit_0_l. - rewrite Nat2Z.inj_succ in H0. rewrite Ztestbit_eq. rewrite zeq_false. apply IHn. rewrite two_power_nat_S in H. rewrite (Zdecomp x) in H. - rewrite Zshiftin_spec in H. destruct (Z.odd x); omega. - omega. omega. omega. + rewrite Zshiftin_spec in H. destruct (Z.odd x); lia. + lia. lia. lia. Qed. Lemma Ztestbit_above_neg: @@ -412,10 +412,10 @@ Proof. intros. set (y := -x-1). assert (Z.testbit y i = false). apply Ztestbit_above with n. - unfold y; omega. auto. + unfold y; lia. auto. unfold y in H1. rewrite Z_one_complement in H1. change true with (negb false). rewrite <- H1. rewrite negb_involutive; auto. - omega. + lia. Qed. Lemma Zsign_bit: @@ -425,16 +425,16 @@ Lemma Zsign_bit: Proof. induction n; intros. - change (two_power_nat 1) with 2 in H. - assert (x = 0 \/ x = 1) by omega. + assert (x = 0 \/ x = 1) by lia. destruct H0; subst x; reflexivity. - rewrite Nat2Z.inj_succ. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ. rewrite IHn. rewrite two_power_nat_S. destruct (zlt (Z.div2 x) (two_power_nat n)); rewrite (Zdecomp x); rewrite Zshiftin_spec. - rewrite zlt_true. auto. destruct (Z.odd x); omega. - rewrite zlt_false. auto. destruct (Z.odd x); omega. + rewrite zlt_true. auto. destruct (Z.odd x); lia. + rewrite zlt_false. auto. destruct (Z.odd x); lia. rewrite (Zdecomp x) in H; rewrite Zshiftin_spec in H. - rewrite two_power_nat_S in H. destruct (Z.odd x); omega. - omega. omega. + rewrite two_power_nat_S in H. destruct (Z.odd x); lia. + lia. lia. Qed. Lemma Ztestbit_le: @@ -444,16 +444,16 @@ Lemma Ztestbit_le: x <= y. Proof. intros x y0 POS0; revert x; pattern y0; apply Zshiftin_ind; auto; intros. - - replace x with 0. omega. apply equal_same_bits; intros. + - replace x with 0. lia. apply equal_same_bits; intros. rewrite Ztestbit_0. destruct (Z.testbit x i) as [] eqn:E; auto. exploit H; eauto. rewrite Ztestbit_0. auto. - assert (Z.div2 x0 <= x). { apply H0. intros. exploit (H1 (Z.succ i)). - omega. rewrite Ztestbit_succ; auto. rewrite Ztestbit_shiftin_succ; auto. + lia. rewrite Ztestbit_succ; auto. rewrite Ztestbit_shiftin_succ; auto. } rewrite (Zdecomp x0). rewrite !Zshiftin_spec. - destruct (Z.odd x0) as [] eqn:E1; destruct b as [] eqn:E2; try omega. - exploit (H1 0). omega. rewrite Ztestbit_base; auto. + destruct (Z.odd x0) as [] eqn:E1; destruct b as [] eqn:E2; try lia. + exploit (H1 0). lia. rewrite Ztestbit_base; auto. rewrite Ztestbit_shiftin_base. congruence. Qed. @@ -464,16 +464,16 @@ Lemma Ztestbit_mod_two_p: Proof. intros n0 x i N0POS. revert x i; pattern n0; apply natlike_ind; auto. - intros. change (two_p 0) with 1. rewrite Zmod_1_r. rewrite Z.testbit_0_l. - rewrite zlt_false; auto. omega. + rewrite zlt_false; auto. lia. - intros. rewrite two_p_S; auto. replace (x0 mod (2 * two_p x)) with (Zshiftin (Z.odd x0) (Z.div2 x0 mod two_p x)). rewrite Ztestbit_shiftin; auto. rewrite (Ztestbit_eq i x0); auto. destruct (zeq i 0). - + rewrite zlt_true; auto. omega. + + rewrite zlt_true; auto. lia. + rewrite H0. destruct (zlt (Z.pred i) x). - * rewrite zlt_true; auto. omega. - * rewrite zlt_false; auto. omega. - * omega. + * rewrite zlt_true; auto. lia. + * rewrite zlt_false; auto. lia. + * lia. + rewrite (Zdecomp x0) at 3. set (x1 := Z.div2 x0). symmetry. apply Zmod_unique with (x1 / two_p x). rewrite !Zshiftin_spec. rewrite Z.add_assoc. f_equal. @@ -481,7 +481,7 @@ Proof. f_equal. apply Z_div_mod_eq. apply two_p_gt_ZERO; auto. ring. rewrite Zshiftin_spec. exploit (Z_mod_lt x1 (two_p x)). apply two_p_gt_ZERO; auto. - destruct (Z.odd x0); omega. + destruct (Z.odd x0); lia. Qed. Corollary Ztestbit_two_p_m1: @@ -491,7 +491,7 @@ Proof. intros. replace (two_p n - 1) with ((-1) mod (two_p n)). rewrite Ztestbit_mod_two_p; auto. destruct (zlt i n); auto. apply Ztestbit_m1; auto. apply Zmod_unique with (-1). ring. - exploit (two_p_gt_ZERO n). auto. omega. + exploit (two_p_gt_ZERO n). auto. lia. Qed. Corollary Ztestbit_neg_two_p: @@ -499,7 +499,7 @@ Corollary Ztestbit_neg_two_p: Z.testbit (- (two_p n)) i = if zlt i n then false else true. Proof. intros. - replace (- two_p n) with (- (two_p n - 1) - 1) by omega. + replace (- two_p n) with (- (two_p n - 1) - 1) by lia. rewrite Z_one_complement by auto. rewrite Ztestbit_two_p_m1 by auto. destruct (zlt i n); auto. @@ -516,16 +516,16 @@ Proof. rewrite (Zdecomp x) in *. rewrite (Zdecomp y) in *. transitivity (Z.testbit (Zshiftin (Z.odd x || Z.odd y) (Z.div2 x + Z.div2 y)) i). - f_equal. rewrite !Zshiftin_spec. - exploit (EXCL 0). omega. rewrite !Ztestbit_shiftin_base. intros. + exploit (EXCL 0). lia. rewrite !Ztestbit_shiftin_base. intros. Opaque Z.mul. destruct (Z.odd x); destruct (Z.odd y); simpl in *; discriminate || ring. - rewrite !Ztestbit_shiftin; auto. destruct (zeq i 0). + auto. - + apply IND. omega. intros. - exploit (EXCL (Z.succ j)). omega. + + apply IND. lia. intros. + exploit (EXCL (Z.succ j)). lia. rewrite !Ztestbit_shiftin_succ. auto. - omega. omega. + lia. lia. Qed. (** ** Zero and sign extensions *) @@ -583,8 +583,8 @@ Lemma Znatlike_ind: forall n, P n. Proof. intros. destruct (zle 0 n). - apply natlike_ind; auto. apply H; omega. - apply H. omega. + apply natlike_ind; auto. apply H; lia. + apply H. lia. Qed. Lemma Zzero_ext_spec: @@ -593,16 +593,16 @@ Lemma Zzero_ext_spec: Proof. unfold Zzero_ext. induction n using Znatlike_ind. - intros. rewrite Ziter_base; auto. - rewrite zlt_false. rewrite Ztestbit_0; auto. omega. + rewrite zlt_false. rewrite Ztestbit_0; auto. lia. - intros. rewrite Ziter_succ; auto. rewrite Ztestbit_shiftin; auto. rewrite (Ztestbit_eq i x); auto. destruct (zeq i 0). - + subst i. rewrite zlt_true; auto. omega. + + subst i. rewrite zlt_true; auto. lia. + rewrite IHn. destruct (zlt (Z.pred i) n). - rewrite zlt_true; auto. omega. - rewrite zlt_false; auto. omega. - omega. + rewrite zlt_true; auto. lia. + rewrite zlt_false; auto. lia. + lia. Qed. Lemma Zsign_ext_spec: @@ -611,29 +611,29 @@ Lemma Zsign_ext_spec: Proof. intros n0 x i I0. unfold Zsign_ext. unfold proj_sumbool; destruct (zlt 0 n0) as [N0|N0]; simpl. -- revert x i I0. pattern n0. apply Zlt_lower_bound_ind with (z := 1); [ | omega ]. +- revert x i I0. pattern n0. apply Zlt_lower_bound_ind with (z := 1); [ | lia ]. unfold Zsign_ext. intros. destruct (zeq x 1). + subst x; simpl. replace (if zlt i 1 then i else 0) with 0. rewrite Ztestbit_base. destruct (Z.odd x0); [ apply Ztestbit_m1; auto | apply Ztestbit_0 ]. - destruct (zlt i 1); omega. - + set (x1 := Z.pred x). replace x1 with (Z.succ (Z.pred x1)) by omega. - rewrite Ziter_succ by (unfold x1; omega). rewrite Ztestbit_shiftin by auto. + destruct (zlt i 1); lia. + + set (x1 := Z.pred x). replace x1 with (Z.succ (Z.pred x1)) by lia. + rewrite Ziter_succ by (unfold x1; lia). rewrite Ztestbit_shiftin by auto. destruct (zeq i 0). - * subst i. rewrite zlt_true. rewrite Ztestbit_base; auto. omega. - * rewrite H by (unfold x1; omega). + * subst i. rewrite zlt_true. rewrite Ztestbit_base; auto. lia. + * rewrite H by (unfold x1; lia). unfold x1; destruct (zlt (Z.pred i) (Z.pred x)). - ** rewrite zlt_true by omega. - rewrite (Ztestbit_eq i x0) by omega. - rewrite zeq_false by omega. auto. - ** rewrite zlt_false by omega. - rewrite (Ztestbit_eq (x - 1) x0) by omega. - rewrite zeq_false by omega. auto. -- rewrite Ziter_base by omega. rewrite andb_false_r. + ** rewrite zlt_true by lia. + rewrite (Ztestbit_eq i x0) by lia. + rewrite zeq_false by lia. auto. + ** rewrite zlt_false by lia. + rewrite (Ztestbit_eq (x - 1) x0) by lia. + rewrite zeq_false by lia. auto. +- rewrite Ziter_base by lia. rewrite andb_false_r. rewrite Z.testbit_0_l, Z.testbit_neg_r. auto. - destruct (zlt i n0); omega. + destruct (zlt i n0); lia. Qed. (** [Zzero_ext n x] is [x modulo 2^n] *) @@ -650,14 +650,14 @@ Qed. Lemma Zzero_ext_range: forall n x, 0 <= n -> 0 <= Zzero_ext n x < two_p n. Proof. - intros. rewrite Zzero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. omega. + intros. rewrite Zzero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. lia. Qed. Lemma eqmod_Zzero_ext: forall n x, 0 <= n -> eqmod (two_p n) (Zzero_ext n x) x. Proof. intros. rewrite Zzero_ext_mod; auto. apply eqmod_sym. apply eqmod_mod. - apply two_p_gt_ZERO. omega. + apply two_p_gt_ZERO. lia. Qed. (** Relation between [Zsign_ext n x] and (Zzero_ext n x] *) @@ -670,13 +670,13 @@ Proof. rewrite Zsign_ext_spec by auto. destruct (Z.testbit x (n - 1)) eqn:SIGNBIT. - set (n' := - two_p n). - replace (Zzero_ext n x - two_p n) with (Zzero_ext n x + n') by (unfold n'; omega). + replace (Zzero_ext n x - two_p n) with (Zzero_ext n x + n') by (unfold n'; lia). rewrite Z_add_is_or; auto. - rewrite Zzero_ext_spec by auto. unfold n'; rewrite Ztestbit_neg_two_p by omega. + rewrite Zzero_ext_spec by auto. unfold n'; rewrite Ztestbit_neg_two_p by lia. destruct (zlt i n). rewrite orb_false_r; auto. auto. - intros. rewrite Zzero_ext_spec by omega. unfold n'; rewrite Ztestbit_neg_two_p by omega. + intros. rewrite Zzero_ext_spec by lia. unfold n'; rewrite Ztestbit_neg_two_p by lia. destruct (zlt j n); auto using andb_false_r. -- replace (Zzero_ext n x - 0) with (Zzero_ext n x) by omega. +- replace (Zzero_ext n x - 0) with (Zzero_ext n x) by lia. rewrite Zzero_ext_spec by auto. destruct (zlt i n); auto. Qed. @@ -688,20 +688,20 @@ Lemma Zsign_ext_range: forall n x, 0 < n -> -two_p (n-1) <= Zsign_ext n x < two_p (n-1). Proof. intros. - assert (A: 0 <= Zzero_ext n x < two_p n) by (apply Zzero_ext_range; omega). + assert (A: 0 <= Zzero_ext n x < two_p n) by (apply Zzero_ext_range; lia). assert (B: Z.testbit (Zzero_ext n x) (n - 1) = if zlt (Zzero_ext n x) (two_p (n - 1)) then false else true). { set (N := Z.to_nat (n - 1)). generalize (Zsign_bit N (Zzero_ext n x)). rewrite ! two_power_nat_two_p. - rewrite inj_S. unfold N; rewrite Z2Nat.id by omega. - intros X; apply X. replace (Z.succ (n - 1)) with n by omega. exact A. + rewrite inj_S. unfold N; rewrite Z2Nat.id by lia. + intros X; apply X. replace (Z.succ (n - 1)) with n by lia. exact A. } assert (C: two_p n = 2 * two_p (n - 1)). - { rewrite <- two_p_S by omega. f_equal; omega. } - rewrite Zzero_ext_spec, zlt_true in B by omega. - rewrite Zsign_ext_zero_ext by omega. rewrite B. - destruct (zlt (Zzero_ext n x) (two_p (n - 1))); omega. + { rewrite <- two_p_S by lia. f_equal; lia. } + rewrite Zzero_ext_spec, zlt_true in B by lia. + rewrite Zsign_ext_zero_ext by lia. rewrite B. + destruct (zlt (Zzero_ext n x) (two_p (n - 1))); lia. Qed. Lemma eqmod_Zsign_ext: @@ -711,9 +711,9 @@ Proof. intros. rewrite Zsign_ext_zero_ext by auto. apply eqmod_trans with (x - 0). apply eqmod_sub. - apply eqmod_Zzero_ext; omega. + apply eqmod_Zzero_ext; lia. exists (if Z.testbit x (n - 1) then 1 else 0). destruct (Z.testbit x (n - 1)); ring. - apply eqmod_refl2; omega. + apply eqmod_refl2; lia. Qed. (** ** Decomposition of a number as a sum of powers of two. *) @@ -743,19 +743,19 @@ Proof. { induction n; intros. simpl. rewrite two_power_nat_O in H0. - assert (x = 0) by omega. subst x. omega. + assert (x = 0) by lia. subst x. lia. rewrite two_power_nat_S in H0. simpl Z_one_bits. rewrite (Zdecomp x) in H0. rewrite Zshiftin_spec in H0. assert (EQ: Z.div2 x * two_p (i + 1) = powerserie (Z_one_bits n (Z.div2 x) (i + 1))). - apply IHn. omega. - destruct (Z.odd x); omega. + apply IHn. lia. + destruct (Z.odd x); lia. rewrite two_p_is_exp in EQ. change (two_p 1) with 2 in EQ. rewrite (Zdecomp x) at 1. rewrite Zshiftin_spec. destruct (Z.odd x); simpl powerserie; rewrite <- EQ; ring. - omega. omega. + lia. lia. } - intros. rewrite <- H. change (two_p 0) with 1. omega. - omega. exact H0. + intros. rewrite <- H. change (two_p 0) with 1. lia. + lia. exact H0. Qed. Lemma Z_one_bits_range: @@ -768,12 +768,12 @@ Proof. tauto. intros x i j. rewrite Nat2Z.inj_succ. assert (In j (Z_one_bits n (Z.div2 x) (i + 1)) -> i <= j < i + Z.succ (Z.of_nat n)). - intros. exploit IHn; eauto. omega. + intros. exploit IHn; eauto. lia. destruct (Z.odd x); simpl. - intros [A|B]. subst j. omega. auto. + intros [A|B]. subst j. lia. auto. auto. } - intros. generalize (H n x 0 i H0). omega. + intros. generalize (H n x 0 i H0). lia. Qed. Remark Z_one_bits_zero: @@ -787,15 +787,15 @@ Remark Z_one_bits_two_p: 0 <= x < Z.of_nat n -> Z_one_bits n (two_p x) i = (i + x) :: nil. Proof. - induction n; intros; simpl. simpl in H. omegaContradiction. + induction n; intros; simpl. simpl in H. extlia. rewrite Nat2Z.inj_succ in H. - assert (x = 0 \/ 0 < x) by omega. destruct H0. - subst x; simpl. decEq. omega. apply Z_one_bits_zero. + assert (x = 0 \/ 0 < x) by lia. destruct H0. + subst x; simpl. decEq. lia. apply Z_one_bits_zero. assert (Z.odd (two_p x) = false /\ Z.div2 (two_p x) = two_p (x-1)). apply Zshiftin_inj. rewrite <- Zdecomp. rewrite !Zshiftin_spec. - rewrite <- two_p_S. rewrite Z.add_0_r. f_equal; omega. omega. + rewrite <- two_p_S. rewrite Z.add_0_r. f_equal; lia. lia. destruct H1 as [A B]; rewrite A; rewrite B. - rewrite IHn. f_equal; omega. omega. + rewrite IHn. f_equal; lia. lia. Qed. (** ** Recognition of powers of two *) @@ -820,7 +820,7 @@ Proof. induction p; simpl P_is_power2; intros. - discriminate. - change (Z.pos p~0) with (2 * Z.pos p). apply IHp in H. - rewrite Z.log2_double by xomega. rewrite two_p_S. congruence. + rewrite Z.log2_double by extlia. rewrite two_p_S. congruence. apply Z.log2_nonneg. - reflexivity. Qed. @@ -848,7 +848,7 @@ Proof. intros. assert (x <> 0) by (red; intros; subst x; discriminate). apply Z_is_power2_sound in H1. destruct H1 as [P Q]. subst i. - split. apply Z.log2_nonneg. apply Z.log2_lt_pow2. omega. rewrite <- two_p_equiv; tauto. + split. apply Z.log2_nonneg. apply Z.log2_lt_pow2. lia. rewrite <- two_p_equiv; tauto. Qed. Lemma Z_is_power2_complete: @@ -858,11 +858,11 @@ Opaque Z.log2. assert (A: forall x i, Z_is_power2 x = Some i -> Z_is_power2 (2 * x) = Some (Z.succ i)). { destruct x; simpl; intros; try discriminate. change (2 * Z.pos p) with (Z.pos (xO p)); simpl. - destruct (P_is_power2 p); inv H. rewrite <- Z.log2_double by xomega. auto. + destruct (P_is_power2 p); inv H. rewrite <- Z.log2_double by extlia. auto. } induction i using Znatlike_ind; intros. -- replace i with 0 by omega. reflexivity. -- rewrite two_p_S by omega. apply A. apply IHi; omega. +- replace i with 0 by lia. reflexivity. +- rewrite two_p_S by lia. apply A. apply IHi; lia. Qed. Definition Z_is_power2m1 (x: Z) : option Z := Z_is_power2 (Z.succ x). @@ -876,13 +876,13 @@ Qed. Lemma Z_is_power2m1_sound: forall x i, Z_is_power2m1 x = Some i -> x = two_p i - 1. Proof. - unfold Z_is_power2m1; intros. apply Z_is_power2_sound in H. omega. + unfold Z_is_power2m1; intros. apply Z_is_power2_sound in H. lia. Qed. Lemma Z_is_power2m1_complete: forall i, 0 <= i -> Z_is_power2m1 (two_p i - 1) = Some i. Proof. - intros. unfold Z_is_power2m1. replace (Z.succ (two_p i - 1)) with (two_p i) by omega. + intros. unfold Z_is_power2m1. replace (Z.succ (two_p i - 1)) with (two_p i) by lia. apply Z_is_power2_complete; auto. Qed. @@ -891,8 +891,8 @@ Lemma Z_is_power2m1_range: 0 <= n -> 0 <= x < two_p n -> Z_is_power2m1 x = Some i -> 0 <= i <= n. Proof. intros. destruct (zeq x (two_p n - 1)). -- subst x. rewrite Z_is_power2m1_complete in H1 by auto. inv H1; omega. -- unfold Z_is_power2m1 in H1. apply (Z_is_power2_range n (Z.succ x) i) in H1; omega. +- subst x. rewrite Z_is_power2m1_complete in H1 by auto. inv H1; lia. +- unfold Z_is_power2m1 in H1. apply (Z_is_power2_range n (Z.succ x) i) in H1; lia. Qed. (** ** Relation between bitwise operations and multiplications / divisions by powers of 2 *) @@ -903,7 +903,7 @@ Lemma Zshiftl_mul_two_p: forall x n, 0 <= n -> Z.shiftl x n = x * two_p n. Proof. intros. destruct n; simpl. - - omega. + - lia. - pattern p. apply Pos.peano_ind. + change (two_power_pos 1) with 2. simpl. ring. + intros. rewrite Pos.iter_succ. rewrite H0. @@ -925,7 +925,7 @@ Proof. rewrite Pplus_one_succ_l. rewrite two_power_pos_is_exp. change (two_power_pos 1) with 2. rewrite Zdiv2_div. rewrite Z.mul_comm. apply Zdiv_Zdiv. - rewrite two_power_pos_nat. apply two_power_nat_pos. omega. + rewrite two_power_pos_nat. apply two_power_nat_pos. lia. - compute in H. congruence. Qed. @@ -938,12 +938,12 @@ Lemma Zquot_Zdiv: Proof. intros. destruct (zlt x 0). - symmetry. apply Zquot_unique_full with ((x + y - 1) mod y - (y - 1)). - + red. right; split. omega. + + red. right; split. lia. exploit (Z_mod_lt (x + y - 1) y); auto. - rewrite Z.abs_eq. omega. omega. + rewrite Z.abs_eq. lia. lia. + transitivity ((y * ((x + y - 1) / y) + (x + y - 1) mod y) - (y-1)). rewrite <- Z_div_mod_eq. ring. auto. ring. - - apply Zquot_Zdiv_pos; omega. + - apply Zquot_Zdiv_pos; lia. Qed. Lemma Zdiv_shift: @@ -953,8 +953,8 @@ Proof. intros. generalize (Z_div_mod_eq x y H). generalize (Z_mod_lt x y H). set (q := x / y). set (r := x mod y). intros. destruct (zeq r 0). - apply Zdiv_unique with (y - 1). rewrite H1. rewrite e. ring. omega. - apply Zdiv_unique with (r - 1). rewrite H1. ring. omega. + apply Zdiv_unique with (y - 1). rewrite H1. rewrite e. ring. lia. + apply Zdiv_unique with (r - 1). rewrite H1. ring. lia. Qed. (** ** Size of integers, in bits. *) @@ -967,7 +967,7 @@ Definition Zsize (x: Z) : Z := Remark Zsize_pos: forall x, 0 <= Zsize x. Proof. - destruct x; simpl. omega. compute; intuition congruence. omega. + destruct x; simpl. lia. compute; intuition congruence. lia. Qed. Remark Zsize_pos': forall x, 0 < x -> 0 < Zsize x. @@ -991,8 +991,8 @@ Lemma Ztestbit_size_1: Proof. intros x0 POS0; pattern x0; apply Zshiftin_pos_ind; auto. intros. rewrite Zsize_shiftin; auto. - replace (Z.pred (Z.succ (Zsize x))) with (Z.succ (Z.pred (Zsize x))) by omega. - rewrite Ztestbit_shiftin_succ. auto. generalize (Zsize_pos' x H); omega. + replace (Z.pred (Z.succ (Zsize x))) with (Z.succ (Z.pred (Zsize x))) by lia. + rewrite Ztestbit_shiftin_succ. auto. generalize (Zsize_pos' x H); lia. Qed. Lemma Ztestbit_size_2: @@ -1002,12 +1002,12 @@ Proof. - subst x0; intros. apply Ztestbit_0. - pattern x0; apply Zshiftin_pos_ind. + simpl. intros. change 1 with (Zshiftin true 0). rewrite Ztestbit_shiftin. - rewrite zeq_false. apply Ztestbit_0. omega. omega. + rewrite zeq_false. apply Ztestbit_0. lia. lia. + intros. rewrite Zsize_shiftin in H1; auto. generalize (Zsize_pos' _ H); intros. - rewrite Ztestbit_shiftin. rewrite zeq_false. apply H0. omega. - omega. omega. - + omega. + rewrite Ztestbit_shiftin. rewrite zeq_false. apply H0. lia. + lia. lia. + + lia. Qed. Lemma Zsize_interval_1: @@ -1029,18 +1029,18 @@ Proof. assert (Z.of_nat N = n) by (apply Z2Nat.id; auto). rewrite <- H1 in H0. rewrite <- two_power_nat_two_p in H0. destruct (zeq x 0). - subst x; simpl; omega. + subst x; simpl; lia. destruct (zlt n (Zsize x)); auto. - exploit (Ztestbit_above N x (Z.pred (Zsize x))). auto. omega. - rewrite Ztestbit_size_1. congruence. omega. + exploit (Ztestbit_above N x (Z.pred (Zsize x))). auto. lia. + rewrite Ztestbit_size_1. congruence. lia. Qed. Lemma Zsize_monotone: forall x y, 0 <= x <= y -> Zsize x <= Zsize y. Proof. intros. apply Z.ge_le. apply Zsize_interval_2. apply Zsize_pos. - exploit (Zsize_interval_1 y). omega. - omega. + exploit (Zsize_interval_1 y). lia. + lia. Qed. (** ** Bit insertion, bit extraction *) @@ -1070,7 +1070,7 @@ Lemma Zextract_s_spec: Proof. unfold Zextract_s; intros. rewrite Zsign_ext_spec by auto. rewrite Z.shiftr_spec. rewrite Z.add_comm. auto. - destruct (zlt i len); omega. + destruct (zlt i len); lia. Qed. (** Insert bits [0...len-1] of [y] into bits [to...to+len-1] of [x] *) @@ -1092,10 +1092,10 @@ Proof. { intros; apply Ztestbit_two_p_m1; auto. } rewrite Z.lor_spec, Z.land_spec, Z.ldiff_spec by auto. destruct (zle to i). -- rewrite ! Z.shiftl_spec by auto. rewrite ! M by omega. +- rewrite ! Z.shiftl_spec by auto. rewrite ! M by lia. unfold proj_sumbool; destruct (zlt (i - to) len); simpl; rewrite andb_true_r, andb_false_r. -+ rewrite zlt_true by omega. apply orb_false_r. -+ rewrite zlt_false by omega; auto. -- rewrite ! Z.shiftl_spec_low by omega. simpl. apply andb_true_r. ++ rewrite zlt_true by lia. apply orb_false_r. ++ rewrite zlt_false by lia; auto. +- rewrite ! Z.shiftl_spec_low by lia. simpl. apply andb_true_r. Qed. diff --git a/powerpc/Asm.v b/powerpc/Asm.v index d9901960..93bc31b8 100644 --- a/powerpc/Asm.v +++ b/powerpc/Asm.v @@ -1276,7 +1276,7 @@ Ltac Equalities := split. auto. intros. destruct B; auto. subst. auto. (* trace length *) red; intros. inv H; simpl. - omega. + lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. (* initial states *) diff --git a/powerpc/Asmgenproof.v b/powerpc/Asmgenproof.v index a1ae5855..23071756 100644 --- a/powerpc/Asmgenproof.v +++ b/powerpc/Asmgenproof.v @@ -69,7 +69,7 @@ Lemma transf_function_no_overflow: transf_function f = OK tf -> list_length_z tf.(fn_code) <= Ptrofs.max_unsigned. Proof. intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z x.(fn_code))); inv EQ0. - omega. + lia. Qed. Lemma exec_straight_exec: @@ -401,8 +401,8 @@ Proof. split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Ptrofs.unsigned_repr. replace (pos' - 0) with pos' in Q. - auto. omega. - generalize (transf_function_no_overflow _ _ H0). omega. + auto. lia. + generalize (transf_function_no_overflow _ _ H0). lia. intros. apply Pregmap.gso; auto. Qed. @@ -926,14 +926,14 @@ Local Transparent destroyed_by_jumptable. simpl const_low. rewrite ATLR. erewrite storev_offset_ptr by eexact P. auto. congruence. auto. auto. auto. left; exists (State rs5 m3'); split. - eapply exec_straight_steps_1; eauto. omega. constructor. + eapply exec_straight_steps_1; eauto. lia. constructor. econstructor; eauto. change (rs5 PC) with (Val.offset_ptr (Val.offset_ptr (Val.offset_ptr (Val.offset_ptr (rs0 PC) Ptrofs.one) Ptrofs.one) Ptrofs.one) Ptrofs.one). rewrite ATPC. simpl. constructor; eauto. - eapply code_tail_next_int. omega. - eapply code_tail_next_int. omega. - eapply code_tail_next_int. omega. - eapply code_tail_next_int. omega. + eapply code_tail_next_int. lia. + eapply code_tail_next_int. lia. + eapply code_tail_next_int. lia. + eapply code_tail_next_int. lia. constructor. unfold rs5, rs4, rs3, rs2. apply agree_nextinstr. apply agree_nextinstr. @@ -958,7 +958,7 @@ Local Transparent destroyed_by_jumptable. - (* return *) inv STACKS. simpl in *. - right. split. omega. split. auto. + right. split. lia. split. auto. rewrite <- ATPC in H5. econstructor; eauto. congruence. diff --git a/powerpc/Asmgenproof1.v b/powerpc/Asmgenproof1.v index 0442f7e8..14ca22f9 100644 --- a/powerpc/Asmgenproof1.v +++ b/powerpc/Asmgenproof1.v @@ -81,12 +81,12 @@ Proof. unfold Int.modu, Int.zero. decEq. change 0 with (0 mod 65536). change (Int.unsigned (Int.repr 65536)) with 65536. - apply eqmod_mod_eq. omega. + apply eqmod_mod_eq. lia. unfold x, low_s. eapply eqmod_trans. apply eqmod_divides with Int.modulus. unfold Int.sub. apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl. exists 65536. compute; auto. - replace 0 with (Int.unsigned n - Int.unsigned n) by omega. + replace 0 with (Int.unsigned n - Int.unsigned n) by lia. apply eqmod_sub. apply eqmod_refl. apply Int.eqmod_sign_ext'. compute; auto. rewrite H0 in H. rewrite Int.add_zero in H. @@ -543,7 +543,7 @@ Proof. - econstructor; split; [|split]. + apply exec_straight_one. simpl; eauto. auto. + Simpl. rewrite Int64.add_zero_l. rewrite H. unfold low64_s. - rewrite Int64.sign_ext_widen by omega. auto. + rewrite Int64.sign_ext_widen by lia. auto. + intros; Simpl. - econstructor; split; [|split]. + eapply exec_straight_two. simpl; eauto. simpl; eauto. auto. auto. @@ -551,16 +551,16 @@ Proof. apply Int64.same_bits_eq; intros. assert (Int64.zwordsize = 64) by auto. rewrite Int64.bits_or, Int64.bits_shl by auto. unfold low64_s, low64_u. - rewrite Int64.bits_zero_ext by omega. + rewrite Int64.bits_zero_ext by lia. change (Int64.unsigned (Int64.repr 16)) with 16. destruct (zlt i 16). - * rewrite Int64.bits_sign_ext by omega. rewrite zlt_true by omega. auto. - * rewrite ! Int64.bits_sign_ext by omega. rewrite orb_false_r. + * rewrite Int64.bits_sign_ext by lia. rewrite zlt_true by lia. auto. + * rewrite ! Int64.bits_sign_ext by lia. rewrite orb_false_r. destruct (zlt i 32). - ** rewrite zlt_true by omega. rewrite Int64.bits_shr by omega. + ** rewrite zlt_true by lia. rewrite Int64.bits_shr by lia. change (Int64.unsigned (Int64.repr 16)) with 16. - rewrite zlt_true by omega. f_equal; omega. - ** rewrite zlt_false by omega. rewrite Int64.bits_shr by omega. + rewrite zlt_true by lia. f_equal; lia. + ** rewrite zlt_false by lia. rewrite Int64.bits_shr by lia. change (Int64.unsigned (Int64.repr 16)) with 16. reflexivity. + intros; Simpl. @@ -605,11 +605,11 @@ Proof. rewrite Int64.bits_shl by auto. change (Int64.unsigned (Int64.repr 32)) with 32. destruct (zlt i 32); auto. - rewrite Int64.bits_sign_ext by omega. - rewrite zlt_true by omega. - unfold n2. rewrite Int64.bits_shru by omega. + rewrite Int64.bits_sign_ext by lia. + rewrite zlt_true by lia. + unfold n2. rewrite Int64.bits_shru by lia. change (Int64.unsigned (Int64.repr 32)) with 32. - rewrite zlt_true by omega. f_equal; omega. + rewrite zlt_true by lia. f_equal; lia. } assert (MI: forall i, 0 <= i < Int64.zwordsize -> Int64.testbit mi i = @@ -619,21 +619,21 @@ Proof. rewrite Int64.bits_shl by auto. change (Int64.unsigned (Int64.repr 16)) with 16. destruct (zlt i 16); auto. - unfold n1. rewrite Int64.bits_zero_ext by omega. - rewrite Int64.bits_shru by omega. + unfold n1. rewrite Int64.bits_zero_ext by lia. + rewrite Int64.bits_shru by lia. destruct (zlt i 32). - rewrite zlt_true by omega. + rewrite zlt_true by lia. change (Int64.unsigned (Int64.repr 16)) with 16. - rewrite zlt_true by omega. f_equal; omega. - rewrite zlt_false by omega. auto. + rewrite zlt_true by lia. f_equal; lia. + rewrite zlt_false by lia. auto. } assert (EQ: Int64.or (Int64.or hi mi) n0 = n). { apply Int64.same_bits_eq; intros. rewrite ! Int64.bits_or by auto. - unfold n0; rewrite Int64.bits_zero_ext by omega. + unfold n0; rewrite Int64.bits_zero_ext by lia. rewrite HI, MI by auto. destruct (zlt i 16). - rewrite zlt_true by omega. auto. + rewrite zlt_true by lia. auto. destruct (zlt i 32); rewrite ! orb_false_r; auto. } edestruct (loadimm64_32s_correct r n2) as (rs' & A & B & C). @@ -1180,7 +1180,7 @@ Local Transparent Int.repr. rewrite H2. apply Int.mkint_eq; reflexivity. rewrite Int.not_involutive in H3. congruence. - omega. + lia. Qed. Remark add_carry_ne0: @@ -1198,8 +1198,8 @@ Transparent Int.eq. rewrite Int.unsigned_zero. rewrite Int.unsigned_mone. unfold negb, Val.of_bool, Vtrue, Vfalse. destruct (zeq (Int.unsigned i) 0); decEq. - apply zlt_true. omega. - apply zlt_false. generalize (Int.unsigned_range i). omega. + apply zlt_true. lia. + apply zlt_false. generalize (Int.unsigned_range i). lia. Qed. Lemma transl_cond_op_correct: diff --git a/powerpc/ConstpropOpproof.v b/powerpc/ConstpropOpproof.v index 8687b056..1dd2e0e4 100644 --- a/powerpc/ConstpropOpproof.v +++ b/powerpc/ConstpropOpproof.v @@ -374,7 +374,7 @@ Proof. Int.bit_solve. destruct (zlt i0 n0). replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)). rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto. - rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto. + rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto. rewrite Int.bits_not by auto. apply negb_involutive. rewrite H6 by auto. auto. econstructor; split; eauto. auto. diff --git a/powerpc/Conventions1.v b/powerpc/Conventions1.v index 5c9cbd4f..045eb471 100644 --- a/powerpc/Conventions1.v +++ b/powerpc/Conventions1.v @@ -268,7 +268,7 @@ Remark loc_arguments_rec_charact: forall_rpair (loc_argument_charact ofs) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_charact ofs1 l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition lia. } assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_charact ofs2) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_charact ofs1) p). { destruct p; simpl; intuition eauto. } Opaque list_nth_z. @@ -279,52 +279,52 @@ Opaque list_nth_z. destruct (list_nth_z int_param_regs ir) as [r|] eqn:E; destruct H. subst. left. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. split. lia. apply Z.divide_1_l. + eapply Y; eauto. lia. - (* float *) - assert (ofs <= align ofs 2) by (apply align_le; omega). + assert (ofs <= align ofs 2) by (apply align_le; lia). destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H. subst. right. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. split. lia. apply Z.divide_1_l. + eapply Y; eauto. lia. - (* long *) - assert (ofs <= align ofs 2) by (apply align_le; omega). + assert (ofs <= align ofs 2) by (apply align_le; lia). set (ir' := align ir 2) in *. destruct (list_nth_z int_param_regs ir') as [r1|] eqn:E1. destruct (list_nth_z int_param_regs (ir' + 1)) as [r2|] eqn:E2. destruct H. subst; split; left; eapply list_nth_z_in; eauto. eapply IHtyl; eauto. destruct H. - subst. destruct Archi.ptr64; [split|split;split]; try omega. - apply align_divides; omega. apply Z.divide_1_l. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. destruct Archi.ptr64; [split|split;split]; try lia. + apply align_divides; lia. apply Z.divide_1_l. apply Z.divide_1_l. + eapply Y; eauto. lia. destruct H. - subst. destruct Archi.ptr64; [split|split;split]; try omega. - apply align_divides; omega. apply Z.divide_1_l. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. destruct Archi.ptr64; [split|split;split]; try lia. + apply align_divides; lia. apply Z.divide_1_l. apply Z.divide_1_l. + eapply Y; eauto. lia. - (* single *) - assert (ofs <= align ofs 1) by (apply align_le; omega). - assert (ofs <= align ofs 2) by (apply align_le; omega). + assert (ofs <= align ofs 1) by (apply align_le; lia). + assert (ofs <= align ofs 2) by (apply align_le; lia). destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H. subst. right. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. destruct Archi.single_passed_as_single; simpl; omega. + subst. split. destruct Archi.single_passed_as_single; simpl; lia. destruct Archi.single_passed_as_single; simpl; apply Z.divide_1_l. - eapply Y; eauto. destruct Archi.single_passed_as_single; simpl; omega. + eapply Y; eauto. destruct Archi.single_passed_as_single; simpl; lia. - (* any32 *) destruct (list_nth_z int_param_regs ir) as [r|] eqn:E; destruct H. subst. left. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. split. lia. apply Z.divide_1_l. + eapply Y; eauto. lia. - (* float *) - assert (ofs <= align ofs 2) by (apply align_le; omega). + assert (ofs <= align ofs 2) by (apply align_le; lia). destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H. subst. right. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. apply Z.divide_1_l. - eapply Y; eauto. omega. + subst. split. lia. apply Z.divide_1_l. + eapply Y; eauto. lia. Qed. Lemma loc_arguments_acceptable: diff --git a/powerpc/NeedOp.v b/powerpc/NeedOp.v index 74ee6b85..85dd9b2e 100644 --- a/powerpc/NeedOp.v +++ b/powerpc/NeedOp.v @@ -162,8 +162,8 @@ Lemma operation_is_redundant_sound: vagree v arg1' nv. Proof. intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst. -- apply sign_ext_redundant_sound; auto. omega. -- apply sign_ext_redundant_sound; auto. omega. +- apply sign_ext_redundant_sound; auto. lia. +- apply sign_ext_redundant_sound; auto. lia. - apply andimm_redundant_sound; auto. - apply orimm_redundant_sound; auto. - apply rolm_redundant_sound; auto. diff --git a/powerpc/SelectLongproof.v b/powerpc/SelectLongproof.v index f16c967e..ea14668f 100644 --- a/powerpc/SelectLongproof.v +++ b/powerpc/SelectLongproof.v @@ -221,15 +221,15 @@ Proof. change (Int64.unsigned Int64.iwordsize) with 64. f_equal. rewrite Int.unsigned_repr. - apply eqmod_mod_eq. omega. + apply eqmod_mod_eq. lia. apply eqmod_trans with a. apply eqmod_divides with Int.modulus. apply Int.eqm_sym. apply Int.eqm_unsigned_repr. exists (two_p (32-6)); auto. apply eqmod_divides with Int64.modulus. apply Int64.eqm_unsigned_repr. exists (two_p (64-6)); auto. - assert (0 <= Int.unsigned (Int.repr a) mod 64 < 64) by (apply Z_mod_lt; omega). + assert (0 <= Int.unsigned (Int.repr a) mod 64 < 64) by (apply Z_mod_lt; lia). assert (64 < Int.max_unsigned) by (compute; auto). - omega. + lia. - InvEval. TrivialExists. simpl. rewrite <- H. unfold Val.rolml; destruct v1; simpl; auto. unfold Int64.rolm. rewrite Int64.rol_and. rewrite Int64.and_assoc. auto. diff --git a/powerpc/SelectOpproof.v b/powerpc/SelectOpproof.v index 7b34ea89..73fadc46 100644 --- a/powerpc/SelectOpproof.v +++ b/powerpc/SelectOpproof.v @@ -805,7 +805,7 @@ Qed. Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8). Proof. red; intros. unfold cast8unsigned. - rewrite Val.zero_ext_and. apply eval_andimm; auto. omega. + rewrite Val.zero_ext_and. apply eval_andimm; auto. lia. Qed. Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16). @@ -818,7 +818,7 @@ Qed. Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16). Proof. red; intros. unfold cast16unsigned. - rewrite Val.zero_ext_and. apply eval_andimm; auto. omega. + rewrite Val.zero_ext_and. apply eval_andimm; auto. lia. Qed. Theorem eval_singleoffloat: unary_constructor_sound singleoffloat Val.singleoffloat. diff --git a/powerpc/Stacklayout.v b/powerpc/Stacklayout.v index cb3806bd..32b11ad5 100644 --- a/powerpc/Stacklayout.v +++ b/powerpc/Stacklayout.v @@ -77,11 +77,11 @@ Local Opaque Z.add Z.mul sepconj range. set (ostkdata := align oendcs 8). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. unfold fe_ofs_arg. - assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; omega). - assert (ol <= ora) by (unfold ora; omega). - assert (ora <= ocs) by (unfold ocs; omega). + assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; lia). + assert (ol <= ora) by (unfold ora; lia). + assert (ora <= ocs) by (unfold ocs; lia). assert (ocs <= oendcs) by (apply size_callee_save_area_incr). - assert (oendcs <= ostkdata) by (apply align_le; omega). + assert (oendcs <= ostkdata) by (apply align_le; lia). (* Reorder as: back link outgoing @@ -90,12 +90,12 @@ Local Opaque Z.add Z.mul sepconj range. callee-save *) rewrite sep_swap3. (* Apply range_split and range_split2 repeatedly *) - apply range_drop_right with 8. omega. - apply range_split. omega. - apply range_split_2. fold ol; omega. omega. - apply range_split. omega. - apply range_split. omega. - apply range_drop_right with ostkdata. omega. + apply range_drop_right with 8. lia. + apply range_split. lia. + apply range_split_2. fold ol; lia. lia. + apply range_split. lia. + apply range_split. lia. + apply range_drop_right with ostkdata. lia. eapply sep_drop2. eexact H. Qed. @@ -112,12 +112,12 @@ Proof. set (ostkdata := align oendcs 8). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. unfold fe_ofs_arg. - assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; omega). - assert (ol <= ora) by (unfold ora; omega). - assert (ora <= ocs) by (unfold ocs; omega). + assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; lia). + assert (ol <= ora) by (unfold ora; lia). + assert (ora <= ocs) by (unfold ocs; lia). assert (ocs <= oendcs) by (apply size_callee_save_area_incr). - assert (oendcs <= ostkdata) by (apply align_le; omega). - split. omega. apply align_le. omega. + assert (oendcs <= ostkdata) by (apply align_le; lia). + split. lia. apply align_le. lia. Qed. Lemma frame_env_aligned: @@ -136,10 +136,10 @@ Proof. set (oendcs := size_callee_save_area b ocs). set (ostkdata := align oendcs 8). split. exists (fe_ofs_arg / 8); reflexivity. - split. apply align_divides; omega. - split. apply align_divides; omega. + split. apply align_divides; lia. + split. apply align_divides; lia. split. apply Z.divide_0_r. apply Z.divide_add_r. - apply Z.divide_trans with 8. exists 2; auto. apply align_divides; omega. + apply Z.divide_trans with 8. exists 2; auto. apply align_divides; lia. apply Z.divide_factor_l. Qed. diff --git a/riscV/Asm.v b/riscV/Asm.v index dc410a3b..30b128ec 100644 --- a/riscV/Asm.v +++ b/riscV/Asm.v @@ -1157,7 +1157,7 @@ Ltac Equalities := split. auto. intros. destruct B; auto. subst. auto. - (* trace length *) red; intros. inv H; simpl. - omega. + lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. - (* initial states *) diff --git a/riscV/Asmgenproof.v b/riscV/Asmgenproof.v index 5ec57886..ab07d071 100644 --- a/riscV/Asmgenproof.v +++ b/riscV/Asmgenproof.v @@ -67,7 +67,7 @@ Lemma transf_function_no_overflow: transf_function f = OK tf -> list_length_z tf.(fn_code) <= Ptrofs.max_unsigned. Proof. intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z x.(fn_code))); inv EQ0. - omega. + lia. Qed. Lemma exec_straight_exec: @@ -432,8 +432,8 @@ Proof. split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Ptrofs.unsigned_repr. replace (pos' - 0) with pos' in Q. - auto. omega. - generalize (transf_function_no_overflow _ _ H0). omega. + auto. lia. + generalize (transf_function_no_overflow _ _ H0). lia. intros. apply Pregmap.gso; auto. Qed. @@ -948,10 +948,10 @@ Local Transparent destroyed_by_op. rewrite <- (sp_val _ _ _ AG). rewrite chunk_of_Tptr in F. rewrite F. reflexivity. reflexivity. eexact U. } - exploit exec_straight_steps_2; eauto using functions_transl. omega. constructor. + exploit exec_straight_steps_2; eauto using functions_transl. lia. constructor. intros (ofs' & X & Y). left; exists (State rs3 m3'); split. - eapply exec_straight_steps_1; eauto. omega. constructor. + eapply exec_straight_steps_1; eauto. lia. constructor. econstructor; eauto. rewrite X; econstructor; eauto. apply agree_exten with rs2; eauto with asmgen. @@ -980,7 +980,7 @@ Local Transparent destroyed_at_function_entry. - (* return *) inv STACKS. simpl in *. - right. split. omega. split. auto. + right. split. lia. split. auto. rewrite <- ATPC in H5. econstructor; eauto. congruence. Qed. diff --git a/riscV/Asmgenproof1.v b/riscV/Asmgenproof1.v index c20c4e49..253e769f 100644 --- a/riscV/Asmgenproof1.v +++ b/riscV/Asmgenproof1.v @@ -35,7 +35,7 @@ Proof. - set (m := Int.sub n lo). assert (A: eqmod (two_p 12) (Int.unsigned lo) (Int.unsigned n)) by (apply Int.eqmod_sign_ext'; compute; auto). assert (B: eqmod (two_p 12) (Int.unsigned n - Int.unsigned lo) 0). - { replace 0 with (Int.unsigned n - Int.unsigned n) by omega. + { replace 0 with (Int.unsigned n - Int.unsigned n) by lia. auto using eqmod_sub, eqmod_refl. } assert (C: eqmod (two_p 12) (Int.unsigned m) 0). { apply eqmod_trans with (Int.unsigned n - Int.unsigned lo); auto. @@ -45,7 +45,7 @@ Proof. { apply eqmod_mod_eq in C. unfold Int.modu. change (Int.unsigned (Int.repr 4096)) with (two_p 12). rewrite C. reflexivity. - apply two_p_gt_ZERO; omega. } + apply two_p_gt_ZERO; lia. } rewrite <- (Int.divu_pow2 m (Int.repr 4096) (Int.repr 12)) by auto. rewrite Int.shl_mul_two_p. change (two_p (Int.unsigned (Int.repr 12))) with 4096. @@ -684,18 +684,18 @@ Proof. unfold Val.cmp; destruct (rs#r1); simpl; auto. rewrite B1. unfold Int.lt. rewrite zlt_false. auto. change (Int.signed (Int.repr Int.max_signed)) with Int.max_signed. - generalize (Int.signed_range i); omega. + generalize (Int.signed_range i); lia. * exploit (opimm32_correct Psltw Psltiw (Val.cmp Clt)); eauto. intros (rs1 & A1 & B1 & C1). exists rs1; split. eexact A1. split; auto. rewrite B1. unfold Val.cmp; simpl; destruct (rs#r1); simpl; auto. unfold Int.lt. replace (Int.signed (Int.add n Int.one)) with (Int.signed n + 1). destruct (zlt (Int.signed n) (Int.signed i)). - rewrite zlt_false by omega. auto. - rewrite zlt_true by omega. auto. + rewrite zlt_false by lia. auto. + rewrite zlt_true by lia. auto. rewrite Int.add_signed. symmetry; apply Int.signed_repr. assert (Int.signed n <> Int.max_signed). { red; intros E. elim H1. rewrite <- (Int.repr_signed n). rewrite E. auto. } - generalize (Int.signed_range n); omega. + generalize (Int.signed_range n); lia. + apply DFL. + apply DFL. Qed. @@ -782,18 +782,18 @@ Proof. unfold Val.cmpl; destruct (rs#r1); simpl; auto. rewrite B1. unfold Int64.lt. rewrite zlt_false. auto. change (Int64.signed (Int64.repr Int64.max_signed)) with Int64.max_signed. - generalize (Int64.signed_range i); omega. + generalize (Int64.signed_range i); lia. * exploit (opimm64_correct Psltl Psltil (fun v1 v2 => Val.maketotal (Val.cmpl Clt v1 v2))); eauto. intros (rs1 & A1 & B1 & C1). exists rs1; split. eexact A1. split; auto. rewrite B1. unfold Val.cmpl; simpl; destruct (rs#r1); simpl; auto. unfold Int64.lt. replace (Int64.signed (Int64.add n Int64.one)) with (Int64.signed n + 1). destruct (zlt (Int64.signed n) (Int64.signed i)). - rewrite zlt_false by omega. auto. - rewrite zlt_true by omega. auto. + rewrite zlt_false by lia. auto. + rewrite zlt_true by lia. auto. rewrite Int64.add_signed. symmetry; apply Int64.signed_repr. assert (Int64.signed n <> Int64.max_signed). { red; intros E. elim H1. rewrite <- (Int64.repr_signed n). rewrite E. auto. } - generalize (Int64.signed_range n); omega. + generalize (Int64.signed_range n); lia. + apply DFL. + apply DFL. Qed. diff --git a/riscV/ConstpropOpproof.v b/riscV/ConstpropOpproof.v index 765aa035..8445e55f 100644 --- a/riscV/ConstpropOpproof.v +++ b/riscV/ConstpropOpproof.v @@ -333,7 +333,7 @@ Proof. Int.bit_solve. destruct (zlt i0 n0). replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)). rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto. - rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto. + rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto. rewrite Int.bits_not by auto. apply negb_involutive. rewrite H6 by auto. auto. econstructor; split; eauto. auto. diff --git a/riscV/Conventions1.v b/riscV/Conventions1.v index 0fc8295e..5dd50a4c 100644 --- a/riscV/Conventions1.v +++ b/riscV/Conventions1.v @@ -320,14 +320,14 @@ Proof. assert (AL: forall ofs ty, ofs >= 0 -> align ofs (typesize ty) >= 0). { intros. assert (ofs <= align ofs (typesize ty)) by (apply align_le; apply typesize_pos). - omega. } + lia. } assert (ALD: forall ofs ty, ofs >= 0 -> (typealign ty | align ofs (typesize ty))). { intros. eapply Z.divide_trans. apply typealign_typesize. apply align_divides. apply typesize_pos. } assert (SK: (if Archi.ptr64 then 2 else 1) > 0). - { destruct Archi.ptr64; omega. } + { destruct Archi.ptr64; lia. } assert (SKK: forall ty, (if Archi.ptr64 then 2 else typesize ty) > 0). - { intros. destruct Archi.ptr64. omega. apply typesize_pos. } + { intros. destruct Archi.ptr64. lia. apply typesize_pos. } assert (A: forall ri rf ofs ty f, OKF f -> ofs >= 0 -> OK (int_arg ri rf ofs ty f)). { intros until f; intros OF OO; red; unfold int_arg; intros. @@ -336,7 +336,7 @@ Proof. - eapply OF; eauto. - subst p; simpl. auto using align_divides, typealign_pos. - eapply OF; [idtac|eauto]. - generalize (AL ofs ty OO) (SKK ty); omega. + generalize (AL ofs ty OO) (SKK ty); lia. } assert (B: forall va ri rf ofs ty f, OKF f -> ofs >= 0 -> OK (float_arg va ri rf ofs ty f)). @@ -347,12 +347,12 @@ Proof. + subst p; simpl. apply CSF. eapply list_nth_z_in; eauto. + eapply OF; eauto. + subst p; repeat split; auto. - + eapply OF; [idtac|eauto]. generalize (AL ofs ty OO) (SKK ty); omega. + + eapply OF; [idtac|eauto]. generalize (AL ofs ty OO) (SKK ty); lia. + subst p; simpl. apply CSF. eapply list_nth_z_in; eauto. + eapply OF; eauto. - destruct H. + subst p; repeat split; auto. - + eapply OF; [idtac|eauto]. generalize (AL ofs ty OO) (SKK ty); omega. + + eapply OF; [idtac|eauto]. generalize (AL ofs ty OO) (SKK ty); lia. } assert (C: forall va ri rf ofs f, OKF f -> ofs >= 0 -> OK (split_long_arg va ri rf ofs f)). @@ -364,13 +364,13 @@ Proof. [destruct (list_nth_z int_param_regs (ri'+1)) as [r2|] eqn:NTH2 | idtac]. - red; simpl; intros; destruct H. + subst p; split; apply CSI; eauto using list_nth_z_in. - + eapply OF; [idtac|eauto]. omega. + + eapply OF; [idtac|eauto]. lia. - red; simpl; intros; destruct H. + subst p; split. split; auto using Z.divide_1_l. apply CSI; eauto using list_nth_z_in. - + eapply OF; [idtac|eauto]. omega. + + eapply OF; [idtac|eauto]. lia. - red; simpl; intros; destruct H. - + subst p; repeat split; auto using Z.divide_1_l. omega. - + eapply OF; [idtac|eauto]. omega. + + subst p; repeat split; auto using Z.divide_1_l. lia. + + eapply OF; [idtac|eauto]. lia. } cut (forall tyl fixed ri rf ofs, ofs >= 0 -> OK (loc_arguments_rec tyl fixed ri rf ofs)). unfold OK. eauto. @@ -392,7 +392,7 @@ Lemma loc_arguments_acceptable: forall (s: signature) (p: rpair loc), In p (loc_arguments s) -> forall_rpair loc_argument_acceptable p. Proof. - unfold loc_arguments; intros. eapply loc_arguments_rec_charact; eauto. omega. + unfold loc_arguments; intros. eapply loc_arguments_rec_charact; eauto. lia. Qed. Lemma loc_arguments_main: diff --git a/riscV/NeedOp.v b/riscV/NeedOp.v index 117bbcb4..4070431a 100644 --- a/riscV/NeedOp.v +++ b/riscV/NeedOp.v @@ -164,8 +164,8 @@ Lemma operation_is_redundant_sound: vagree v arg1' nv. Proof. intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst. -- apply sign_ext_redundant_sound; auto. omega. -- apply sign_ext_redundant_sound; auto. omega. +- apply sign_ext_redundant_sound; auto. lia. +- apply sign_ext_redundant_sound; auto. lia. - apply andimm_redundant_sound; auto. - apply orimm_redundant_sound; auto. Qed. diff --git a/riscV/SelectLongproof.v b/riscV/SelectLongproof.v index 78a1935d..3794e050 100644 --- a/riscV/SelectLongproof.v +++ b/riscV/SelectLongproof.v @@ -502,8 +502,8 @@ Proof. assert (LTU2: Int.ltu (Int.sub Int64.iwordsize' n) Int64.iwordsize' = true). { unfold Int.ltu; apply zlt_true. unfold Int.sub. change (Int.unsigned Int64.iwordsize') with 64. - rewrite Int.unsigned_repr. omega. - assert (64 < Int.max_unsigned) by reflexivity. omega. } + rewrite Int.unsigned_repr. lia. + assert (64 < Int.max_unsigned) by reflexivity. lia. } assert (X: eval_expr ge sp e m le (Eop (Oshrlimm (Int.repr (Int64.zwordsize - 1))) (a ::: Enil)) (Vlong (Int64.shr' i (Int.repr (Int64.zwordsize - 1))))). @@ -514,7 +514,7 @@ Proof. TrivialExists. constructor. EvalOp. simpl; eauto. constructor. simpl. unfold Int.ltu; rewrite zlt_true. rewrite Int64.shrx'_shr_2 by auto. reflexivity. - change (Int.unsigned Int64.iwordsize') with 64; omega. + change (Int.unsigned Int64.iwordsize') with 64; lia. *) Qed. diff --git a/riscV/SelectOpproof.v b/riscV/SelectOpproof.v index 593be1ed..b0b4b794 100644 --- a/riscV/SelectOpproof.v +++ b/riscV/SelectOpproof.v @@ -365,20 +365,20 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.signed i * Int.signed i0) 32)). - unfold Int.mulhs; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhs; f_equal. rewrite Zshiftr_div_two_p by lia. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. - assert (N1: 0 <= n < 64) by omega. + assert (N1: 0 <= n < 64) by lia. rewrite Int64.bits_loword by auto. rewrite Int64.bits_shr' by auto. change (Int.unsigned (Int.repr 32)) with 32. change Int64.zwordsize with 64. - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite Int.testbit_repr by auto. - unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; omega). + unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; lia). transitivity (Z.testbit (Int.signed i * Int.signed i0) (n + 32)). - rewrite Z.shiftr_spec by omega. auto. + rewrite Z.shiftr_spec by lia. auto. apply Int64.same_bits_eqm. apply Int64.eqm_mult; apply Int64.eqm_unsigned_repr. - change Int64.zwordsize with 64; omega. + change Int64.zwordsize with 64; lia. - TrivialExists. Qed. @@ -393,20 +393,20 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.unsigned i * Int.unsigned i0) 32)). - unfold Int.mulhu; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhu; f_equal. rewrite Zshiftr_div_two_p by lia. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. - assert (N1: 0 <= n < 64) by omega. + assert (N1: 0 <= n < 64) by lia. rewrite Int64.bits_loword by auto. rewrite Int64.bits_shru' by auto. change (Int.unsigned (Int.repr 32)) with 32. change Int64.zwordsize with 64. - rewrite zlt_true by omega. + rewrite zlt_true by lia. rewrite Int.testbit_repr by auto. - unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; omega). + unfold Int64.mul. rewrite Int64.testbit_repr by (change Int64.zwordsize with 64; lia). transitivity (Z.testbit (Int.unsigned i * Int.unsigned i0) (n + 32)). - rewrite Z.shiftr_spec by omega. auto. + rewrite Z.shiftr_spec by lia. auto. apply Int64.same_bits_eqm. apply Int64.eqm_mult; apply Int64.eqm_unsigned_repr. - change Int64.zwordsize with 64; omega. + change Int64.zwordsize with 64; lia. - TrivialExists. Qed. @@ -563,8 +563,8 @@ Proof. assert (LTU2: Int.ltu (Int.sub Int.iwordsize n) Int.iwordsize = true). { unfold Int.ltu; apply zlt_true. unfold Int.sub. change (Int.unsigned Int.iwordsize) with 32. - rewrite Int.unsigned_repr. omega. - assert (32 < Int.max_unsigned) by reflexivity. omega. } + rewrite Int.unsigned_repr. lia. + assert (32 < Int.max_unsigned) by reflexivity. lia. } assert (X: eval_expr ge sp e m le (Eop (Oshrimm (Int.repr (Int.zwordsize - 1))) (a ::: Enil)) (Vint (Int.shr i (Int.repr (Int.zwordsize - 1))))). @@ -575,7 +575,7 @@ Proof. TrivialExists. constructor. EvalOp. simpl; eauto. constructor. simpl. unfold Int.ltu; rewrite zlt_true. rewrite Int.shrx_shr_2 by auto. reflexivity. - change (Int.unsigned Int.iwordsize) with 32; omega. + change (Int.unsigned Int.iwordsize) with 32; lia. *) Qed. @@ -763,7 +763,7 @@ Qed. Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8). Proof. red; intros until x. unfold cast8unsigned. - rewrite Val.zero_ext_and. apply eval_andimm. omega. + rewrite Val.zero_ext_and. apply eval_andimm. lia. Qed. Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16). @@ -776,7 +776,7 @@ Qed. Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16). Proof. red; intros until x. unfold cast8unsigned. - rewrite Val.zero_ext_and. apply eval_andimm. omega. + rewrite Val.zero_ext_and. apply eval_andimm. lia. Qed. Theorem eval_intoffloat: diff --git a/riscV/Stacklayout.v b/riscV/Stacklayout.v index d0c6a526..25f02aab 100644 --- a/riscV/Stacklayout.v +++ b/riscV/Stacklayout.v @@ -68,15 +68,15 @@ Local Opaque Z.add Z.mul sepconj range. set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). replace (size_chunk Mptr) with w by (rewrite size_chunk_Mptr; auto). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + w <= oretaddr) by (unfold oretaddr; omega). - assert (oretaddr + w <= ocs) by (unfold ocs; omega). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + w <= oretaddr) by (unfold oretaddr; lia). + assert (oretaddr + w <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). (* Reorder as: outgoing back link @@ -89,11 +89,11 @@ Local Opaque Z.add Z.mul sepconj range. rewrite sep_swap45. (* Apply range_split and range_split2 repeatedly *) unfold fe_ofs_arg. - apply range_split_2. fold olink; omega. omega. - apply range_split. omega. - apply range_split. omega. - apply range_split_2. fold ol. omega. omega. - apply range_drop_right with ostkdata. omega. + apply range_split_2. fold olink; lia. lia. + apply range_split. lia. + apply range_split. lia. + apply range_split_2. fold ol. lia. lia. + apply range_drop_right with ostkdata. lia. eapply sep_drop2. eexact H. Qed. @@ -109,16 +109,16 @@ Proof. set (ocs := oretaddr + w). set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + w <= oretaddr) by (unfold oretaddr; omega). - assert (oretaddr + w <= ocs) by (unfold ocs; omega). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + w <= oretaddr) by (unfold oretaddr; lia). + assert (oretaddr + w <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). - split. omega. apply align_le. omega. + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). + split. lia. apply align_le. lia. Qed. Lemma frame_env_aligned: @@ -137,11 +137,11 @@ Proof. set (ocs := oretaddr + w). set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). replace (align_chunk Mptr) with w by (rewrite align_chunk_Mptr; auto). split. apply Z.divide_0_r. - split. apply align_divides; omega. - split. apply align_divides; omega. - split. apply align_divides; omega. - apply Z.divide_add_r. apply align_divides; omega. apply Z.divide_refl. + split. apply align_divides; lia. + split. apply align_divides; lia. + split. apply align_divides; lia. + apply Z.divide_add_r. apply align_divides; lia. apply Z.divide_refl. Qed. @@ -1193,7 +1193,7 @@ Ltac Equalities := split. auto. intros. destruct B; auto. subst. auto. - (* trace length *) red; intros; inv H; simpl. - omega. + lia. eapply external_call_trace_length; eauto. eapply external_call_trace_length; eauto. - (* initial states *) diff --git a/x86/Asmgenproof.v b/x86/Asmgenproof.v index f1fd41e3..67c42b2b 100644 --- a/x86/Asmgenproof.v +++ b/x86/Asmgenproof.v @@ -67,7 +67,7 @@ Lemma transf_function_no_overflow: transf_function f = OK tf -> list_length_z (fn_code tf) <= Ptrofs.max_unsigned. Proof. intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (list_length_z (fn_code x))); monadInv EQ0. - omega. + lia. Qed. Lemma exec_straight_exec: @@ -332,8 +332,8 @@ Proof. split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Ptrofs.unsigned_repr. replace (pos' - 0) with pos' in Q. - auto. omega. - generalize (transf_function_no_overflow _ _ H0). omega. + auto. lia. + generalize (transf_function_no_overflow _ _ H0). lia. intros. apply Pregmap.gso; auto. Qed. @@ -852,7 +852,7 @@ Transparent destroyed_by_jumptable. econstructor; eauto. unfold nextinstr. rewrite Pregmap.gss. repeat rewrite Pregmap.gso; auto with asmgen. rewrite ATPC. simpl. constructor; eauto. - unfold fn_code. eapply code_tail_next_int. simpl in g. omega. + unfold fn_code. eapply code_tail_next_int. simpl in g. lia. constructor. apply agree_nextinstr. eapply agree_change_sp; eauto. Transparent destroyed_at_function_entry. @@ -877,7 +877,7 @@ Transparent destroyed_at_function_entry. - (* return *) inv STACKS. simpl in *. - right. split. omega. split. auto. + right. split. lia. split. auto. econstructor; eauto. rewrite ATPC; eauto. congruence. Qed. diff --git a/x86/ConstpropOpproof.v b/x86/ConstpropOpproof.v index 82179fa4..09c6e91b 100644 --- a/x86/ConstpropOpproof.v +++ b/x86/ConstpropOpproof.v @@ -532,7 +532,7 @@ Proof. Int.bit_solve. destruct (zlt i0 n0). replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)). rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto. - rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto. + rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto. rewrite Int.bits_not by auto. apply negb_involutive. rewrite H6 by auto. auto. econstructor; split; eauto. auto. diff --git a/x86/Conventions1.v b/x86/Conventions1.v index 645aae90..803d162a 100644 --- a/x86/Conventions1.v +++ b/x86/Conventions1.v @@ -302,14 +302,14 @@ Remark loc_arguments_32_charact: In p (loc_arguments_32 tyl ofs) -> forall_rpair (loc_argument_32_charact ofs) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_32_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_32_charact ofs1 l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition lia. } induction tyl as [ | ty tyl]; simpl loc_arguments_32; intros. - contradiction. - destruct H. -+ destruct ty; subst p; simpl; omega. ++ destruct ty; subst p; simpl; lia. + apply IHtyl in H. generalize (typesize_pos ty); intros. destruct p; simpl in *. -* eapply X; eauto; omega. -* destruct H; split; eapply X; eauto; omega. +* eapply X; eauto; lia. +* destruct H; split; eapply X; eauto; lia. Qed. Remark loc_arguments_elf64_charact: @@ -317,7 +317,7 @@ Remark loc_arguments_elf64_charact: In p (loc_arguments_elf64 tyl ir fr ofs) -> (2 | ofs) -> forall_rpair (loc_argument_elf64_charact ofs) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_elf64_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_elf64_charact ofs1 l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition lia. } assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_elf64_charact ofs2) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_elf64_charact ofs1) p). { destruct p; simpl; intuition eauto. } assert (Z: forall ofs, (2 | ofs) -> (2 | ofs + 2)). @@ -334,8 +334,8 @@ Opaque list_nth_z. { intros. destruct (list_nth_z int_param_regs_elf64 ir) as [r|] eqn:E; destruct H1. subst. left. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. assumption. - eapply Y; eauto. omega. } + subst. split. lia. assumption. + eapply Y; eauto. lia. } assert (B: forall ty, In p match list_nth_z float_param_regs_elf64 fr with | Some ireg => One (R ireg) :: loc_arguments_elf64 tyl ir (fr + 1) ofs @@ -345,8 +345,8 @@ Opaque list_nth_z. { intros. destruct (list_nth_z float_param_regs_elf64 fr) as [r|] eqn:E; destruct H1. subst. right. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. assumption. - eapply Y; eauto. omega. } + subst. split. lia. assumption. + eapply Y; eauto. lia. } destruct a; eauto. Qed. @@ -355,7 +355,7 @@ Remark loc_arguments_win64_charact: In p (loc_arguments_win64 tyl r ofs) -> (2 | ofs) -> forall_rpair (loc_argument_win64_charact ofs) p. Proof. assert (X: forall ofs1 ofs2 l, loc_argument_win64_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_win64_charact ofs1 l). - { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. } + { destruct l; simpl; intros; auto. destruct sl; auto. intuition lia. } assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_win64_charact ofs2) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_win64_charact ofs1) p). { destruct p; simpl; intuition eauto. } assert (Z: forall ofs, (2 | ofs) -> (2 | ofs + 2)). @@ -372,8 +372,8 @@ Opaque list_nth_z. { intros. destruct (list_nth_z int_param_regs_win64 r) as [r'|] eqn:E; destruct H1. subst. left. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. assumption. - eapply Y; eauto. omega. } + subst. split. lia. assumption. + eapply Y; eauto. lia. } assert (B: forall ty, In p match list_nth_z float_param_regs_win64 r with | Some ireg => One (R ireg) :: loc_arguments_win64 tyl (r + 1) ofs @@ -383,8 +383,8 @@ Opaque list_nth_z. { intros. destruct (list_nth_z float_param_regs_win64 r) as [r'|] eqn:E; destruct H1. subst. right. eapply list_nth_z_in; eauto. eapply IHtyl; eauto. - subst. split. omega. assumption. - eapply Y; eauto. omega. } + subst. split. lia. assumption. + eapply Y; eauto. lia. } destruct a; eauto. Qed. diff --git a/x86/NeedOp.v b/x86/NeedOp.v index d9a58fbb..775a23db 100644 --- a/x86/NeedOp.v +++ b/x86/NeedOp.v @@ -206,9 +206,9 @@ Proof. unfold needs_of_operation; intros; destruct op; try (eapply default_needs_of_operation_sound; eauto; fail); simpl in *; FuncInv; InvAgree; TrivialExists. - apply sign_ext_sound; auto. compute; auto. -- apply zero_ext_sound; auto. omega. +- apply zero_ext_sound; auto. lia. - apply sign_ext_sound; auto. compute; auto. -- apply zero_ext_sound; auto. omega. +- apply zero_ext_sound; auto. lia. - apply neg_sound; auto. - apply mul_sound; auto. - apply mul_sound; auto with na. @@ -246,10 +246,10 @@ Lemma operation_is_redundant_sound: vagree v arg1' nv. Proof. intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst. -- apply sign_ext_redundant_sound; auto. omega. -- apply zero_ext_redundant_sound; auto. omega. -- apply sign_ext_redundant_sound; auto. omega. -- apply zero_ext_redundant_sound; auto. omega. +- apply sign_ext_redundant_sound; auto. lia. +- apply zero_ext_redundant_sound; auto. lia. +- apply sign_ext_redundant_sound; auto. lia. +- apply zero_ext_redundant_sound; auto. lia. - apply andimm_redundant_sound; auto. - apply orimm_redundant_sound; auto. Qed. diff --git a/x86/SelectOpproof.v b/x86/SelectOpproof.v index 961f602c..d8ab32a4 100644 --- a/x86/SelectOpproof.v +++ b/x86/SelectOpproof.v @@ -381,9 +381,9 @@ Proof. - TrivialExists. simpl. rewrite Int.and_commut; auto. - TrivialExists. simpl. rewrite Val.and_assoc. rewrite Int.and_commut. auto. - rewrite Val.zero_ext_and. TrivialExists. rewrite Val.and_assoc. - rewrite Int.and_commut. auto. omega. + rewrite Int.and_commut. auto. lia. - rewrite Val.zero_ext_and. TrivialExists. rewrite Val.and_assoc. - rewrite Int.and_commut. auto. omega. + rewrite Int.and_commut. auto. lia. - TrivialExists. Qed. @@ -743,7 +743,7 @@ Proof. red; intros until x. unfold cast8unsigned. destruct (cast8unsigned_match a); intros; InvEval. TrivialExists. subst. rewrite Val.zero_ext_and. rewrite Val.and_assoc. - rewrite Int.and_commut. apply eval_andimm; auto. omega. + rewrite Int.and_commut. apply eval_andimm; auto. lia. TrivialExists. Qed. @@ -759,7 +759,7 @@ Proof. red; intros until x. unfold cast16unsigned. destruct (cast16unsigned_match a); intros; InvEval. TrivialExists. subst. rewrite Val.zero_ext_and. rewrite Val.and_assoc. - rewrite Int.and_commut. apply eval_andimm; auto. omega. + rewrite Int.and_commut. apply eval_andimm; auto. lia. TrivialExists. Qed. @@ -860,7 +860,7 @@ Proof. simpl. rewrite Heqo; reflexivity. simpl. unfold Int64.loword. rewrite Int64.unsigned_repr, Int.repr_unsigned; auto. assert (Int.modulus < Int64.max_unsigned) by reflexivity. - generalize (Int.unsigned_range n); omega. + generalize (Int.unsigned_range n); lia. Qed. Theorem eval_floatofintu: diff --git a/x86/Stacklayout.v b/x86/Stacklayout.v index 4f68cf26..002b86bf 100644 --- a/x86/Stacklayout.v +++ b/x86/Stacklayout.v @@ -69,16 +69,16 @@ Local Opaque Z.add Z.mul sepconj range. set (ostkdata := align (ol + 4 * b.(bound_local)) 8). set (oretaddr := align (ostkdata + b.(bound_stack_data)) w). replace (size_chunk Mptr) with w by (rewrite size_chunk_Mptr; auto). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= fe_ofs_arg) by (unfold fe_ofs_arg; destruct Archi.win64; omega). - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (fe_ofs_arg + 4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + w <= ocs) by (unfold ocs; omega). + assert (0 <= fe_ofs_arg) by (unfold fe_ofs_arg; destruct Archi.win64; lia). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (fe_ofs_arg + 4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + w <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). - assert (ostkdata + bound_stack_data b <= oretaddr) by (apply align_le; omega). + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). + assert (ostkdata + bound_stack_data b <= oretaddr) by (apply align_le; lia). (* Reorder as: outgoing back link @@ -90,13 +90,13 @@ Local Opaque Z.add Z.mul sepconj range. rewrite sep_swap45. rewrite sep_swap34. (* Apply range_split and range_split2 repeatedly *) - apply range_drop_left with 0. omega. - apply range_split_2. fold olink. omega. omega. - apply range_split. omega. - apply range_split_2. fold ol. omega. omega. - apply range_drop_right with ostkdata. omega. + apply range_drop_left with 0. lia. + apply range_split_2. fold olink. lia. lia. + apply range_split. lia. + apply range_split_2. fold ol. lia. lia. + apply range_drop_right with ostkdata. lia. rewrite sep_swap. - apply range_drop_left with (ostkdata + bound_stack_data b). omega. + apply range_drop_left with (ostkdata + bound_stack_data b). lia. rewrite sep_swap. exact H. Qed. @@ -113,17 +113,17 @@ Proof. set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). set (oretaddr := align (ostkdata + b.(bound_stack_data)) w). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros. - assert (0 <= fe_ofs_arg) by (unfold fe_ofs_arg; destruct Archi.win64; omega). - assert (0 <= 4 * b.(bound_outgoing)) by omega. - assert (fe_ofs_arg + 4 * b.(bound_outgoing) <= olink) by (apply align_le; omega). - assert (olink + w <= ocs) by (unfold ocs; omega). + assert (0 <= fe_ofs_arg) by (unfold fe_ofs_arg; destruct Archi.win64; lia). + assert (0 <= 4 * b.(bound_outgoing)) by lia. + assert (fe_ofs_arg + 4 * b.(bound_outgoing) <= olink) by (apply align_le; lia). + assert (olink + w <= ocs) by (unfold ocs; lia). assert (ocs <= size_callee_save_area b ocs) by (apply size_callee_save_area_incr). - assert (size_callee_save_area b ocs <= ol) by (apply align_le; omega). - assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; omega). - assert (ostkdata + bound_stack_data b <= oretaddr) by (apply align_le; omega). - split. omega. omega. + assert (size_callee_save_area b ocs <= ol) by (apply align_le; lia). + assert (ol + 4 * b.(bound_local) <= ostkdata) by (apply align_le; lia). + assert (ostkdata + bound_stack_data b <= oretaddr) by (apply align_le; lia). + split. lia. lia. Qed. Lemma frame_env_aligned: @@ -142,11 +142,11 @@ Proof. set (ol := align (size_callee_save_area b ocs) 8). set (ostkdata := align (ol + 4 * b.(bound_local)) 8). set (oretaddr := align (ostkdata + b.(bound_stack_data)) w). - assert (0 < w) by (unfold w; destruct Archi.ptr64; omega). + assert (0 < w) by (unfold w; destruct Archi.ptr64; lia). replace (align_chunk Mptr) with w by (rewrite align_chunk_Mptr; auto). split. exists (fe_ofs_arg / 8). unfold fe_ofs_arg; destruct Archi.win64; reflexivity. - split. apply align_divides; omega. - split. apply align_divides; omega. - split. apply align_divides; omega. - apply align_divides; omega. + split. apply align_divides; lia. + split. apply align_divides; lia. + split. apply align_divides; lia. + apply align_divides; lia. Qed. |