diff options
Diffstat (limited to 'src/versions/standard/Int63')
| -rw-r--r-- | src/versions/standard/Int63/Int63Axioms_standard.v | 313 | ||||
| -rw-r--r-- | src/versions/standard/Int63/Int63Native_standard.v | 143 | ||||
| -rw-r--r-- | src/versions/standard/Int63/Int63Op_standard.v | 334 | ||||
| -rw-r--r-- | src/versions/standard/Int63/Int63Properties_standard.v | 2768 | ||||
| -rw-r--r-- | src/versions/standard/Int63/Int63_standard.v | 23 |
5 files changed, 0 insertions, 3581 deletions
diff --git a/src/versions/standard/Int63/Int63Axioms_standard.v b/src/versions/standard/Int63/Int63Axioms_standard.v deleted file mode 100644 index 9625bce..0000000 --- a/src/versions/standard/Int63/Int63Axioms_standard.v +++ /dev/null @@ -1,313 +0,0 @@ -(**************************************************************************) -(* *) -(* SMTCoq *) -(* Copyright (C) 2011 - 2021 *) -(* *) -(* See file "AUTHORS" for the list of authors *) -(* *) -(* This file is distributed under the terms of the CeCILL-C licence *) -(* *) -(**************************************************************************) - - -Require Import Bvector. -(* Require Export BigNumPrelude. *) -Require Import Int31 Cyclic31. -Require Export Int63Native. -Require Export Int63Op. -Require Import Psatz. - -Local Open Scope Z_scope. - - -(* Taken from BigNumPrelude *) - - Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p. - Proof. - intros p x Hle;destruct (Z_le_gt_dec 0 p). - apply Zdiv_le_lower_bound;auto with zarith. - replace (2^p) with 0. - destruct x;compute;intro;discriminate. - destruct p;trivial;discriminate. - Qed. - - Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y. - Proof. - intros p x y H;destruct (Z_le_gt_dec 0 p). - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with y;auto with zarith. - rewrite <- (Z.mul_1_r y);apply Z.mul_le_mono_nonneg;auto with zarith. - assert (0 < 2^p);auto with zarith. - replace (2^p) with 0. - destruct x;change (0<y);auto with zarith. - destruct p;trivial;discriminate. - Qed. - - -(* Int63Axioms *) - -Definition wB := (2^(Z_of_nat size)). - -Notation "[| x |]" := (to_Z x) (at level 0, x at level 99) : int63_scope. - -Notation "[+| c |]" := - (interp_carry 1 wB to_Z c) (at level 0, c at level 99) : int63_scope. - -Notation "[-| c |]" := - (interp_carry (-1) wB to_Z c) (at level 0, c at level 99) : int63_scope. - -Notation "[|| x ||]" := - (zn2z_to_Z wB to_Z x) (at level 0, x at level 99) : int63_scope. - -Local Open Scope int63_scope. -Local Open Scope Z_scope. - -(* Bijection : int63 <-> Bvector size *) - -Theorem to_Z_inj : forall x y, [|x|] = [|y|] -> x = y. -Proof Ring31.Int31_canonic. - -Theorem of_to_Z : forall x, of_Z ([|x|]) = x. -Proof. exact phi_inv_phi. Qed. - -(* Comparisons *) -Theorem eqb_refl x : (x == x)%int = true. -Proof. now rewrite Ring31.eqb31_eq. Qed. - -Theorem ltb_spec x y : (x < y)%int = true <-> [|x|] < [|y|]. -Proof. - unfold ltb. rewrite spec_compare, <- Z.compare_lt_iff. - change (phi x ?= phi y) with ([|x|] ?= [|y|]). - case ([|x|] ?= [|y|]); intuition; discriminate. -Qed. - -Theorem leb_spec x y : (x <= y)%int = true <-> [|x|] <= [|y|]. -Proof. - unfold leb. rewrite spec_compare, <- Z.compare_le_iff. - change (phi x ?= phi y) with ([|x|] ?= [|y|]). - case ([|x|] ?= [|y|]); intuition; discriminate. -Qed. - - -(** Specification of logical operations *) -Lemma lsl_spec_alt p : - forall x, [| addmuldiv31_alt p x 0 |] = ([|x|] * 2^(Z.of_nat p)) mod wB. -Proof. - induction p as [ |p IHp]; simpl; intro x. - - rewrite Z.mul_1_r. symmetry. apply Zmod_small. apply phi_bounded. - - rewrite IHp, phi_twice, Zmult_mod_idemp_l, Z.double_spec, - Z.mul_comm, Z.mul_assoc, Z.mul_comm, - Z.pow_pos_fold, Zpos_P_of_succ_nat, <- Z.add_1_r, Z.pow_add_r. - * reflexivity. - * apply Zle_0_nat. - * exact Z.le_0_1. -Qed. - -Theorem lsl_spec x p : [| x << p |] = ([|x|] * 2^[|p|]) mod wB. -Proof. - unfold lsl. - rewrite addmuldiv31_equiv, lsl_spec_alt, Nat2Z.inj_abs_nat, Z.abs_eq. - - reflexivity. - - now destruct (phi_bounded p). -Qed. - - -Lemma div_greater (p x:int) (H:Z.of_nat Int31.size <= [|p|]) : [|x|] / 2 ^ [|p|] = 0. -Proof. - apply Z.div_small. destruct (phi_bounded x) as [H1 H2]. split; auto. - apply (Z.lt_le_trans _ _ _ H2). apply Z.pow_le_mono_r; auto. - exact Z.lt_0_2. -Qed. - -Theorem lsr_spec x p : [|x >> p|] = [|x|] / 2 ^ [|p|]. -Proof. - unfold lsr. case_eq (p < 31%int31)%int; intro Heq. - - assert (H : [|31%int31 - p|] = 31 - [|p|]). - * rewrite spec_sub. rewrite Zmod_small; auto. - split. - + rewrite ltb_spec in Heq. assert (forall x y, x < y -> 0 <= y - x) by (intros;lia); auto. - + assert (H:forall x y z, 0 <= y /\ x < z -> x - y < z) by (intros;lia). - apply H. destruct (phi_bounded p). destruct (phi_bounded (31%int31)). split; auto. - * rewrite spec_add_mul_div. - + rewrite Z.add_0_l. change (phi (31%int31 - p)) with [|31%int31 - p|]. rewrite H. replace (31 - (31 - [|p|])) with [|p|] by ring. apply Zmod_small. split. - ++ apply div_le_0. now destruct (phi_bounded x). - ++ apply div_lt. apply phi_bounded. - + change (phi (31%int31 - p)) with [|31%int31 - p|]. rewrite H. assert (forall x y, 0 <= y -> x - y <= x) by (intros;lia). apply H0. now destruct (phi_bounded p). - - rewrite div_greater; auto. - destruct (Z.le_gt_cases [|31%int31|] [|p|]); auto. - rewrite <- ltb_spec in H. rewrite H in Heq. discriminate. -Qed. - - -Lemma bit_testbit x i : bit x i = Z.testbit [|x|] [|i|]. -Admitted. -(* Proof. *) -(* case_eq [|i|]. *) -(* - simpl. change 0 with [|0|]. intro Heq. apply Ring31.Int31_canonic in Heq. subst i. *) -(* unfold bit. *) - - -Lemma Z_pos_xO_pow i x (Hi:0 <= i) : Z.pos x < 2 ^ i <-> Z.pos x~0 < 2 ^ (i+1). -Proof. rewrite Pos2Z.inj_xO, Z.pow_add_r; auto using Z.le_0_1; lia. Qed. - -Lemma Z_pos_xI_pow i x (Hi:0 <= i) : Z.pos x < 2 ^ i <-> Z.pos x~1 < 2 ^ (i+1). -Proof. rewrite Pos2Z.inj_xI, Z.pow_add_r; auto using Z.le_0_1; lia. Qed. - -Lemma pow_nonneg i (Hi : 1 <= 2 ^ i) : 0 <= i. -Proof. - destruct (Z.le_gt_cases 0 i); auto. - rewrite (Z.pow_neg_r _ _ H) in Hi. lia. -Qed. - -Lemma pow_pos i (Hi : 1 < 2 ^ i) : 0 < i. -Proof. - destruct (Z.lt_trichotomy 0 i) as [H|[H|H]]; auto. - - subst i. lia. - - rewrite (Z.pow_neg_r _ _ H) in Hi. lia. -Qed. - -Lemma pos_land_bounded : forall x y i, - Z.pos x < 2 ^ i -> Z.pos y < 2 ^ i -> Z.of_N (Pos.land x y) < 2 ^ i. -Proof. - induction x as [x IHx|x IHx| ]; intros [y|y| ] i; auto. - - simpl. intro H. - assert (H4:0 <= i - 1) by (assert (H4:0 < i); try lia; apply pow_pos; apply (Z.le_lt_trans _ (Z.pos x~1)); auto; lia). - generalize H. replace i with ((i-1)+1) at 1 2 by ring. rewrite <- !Z_pos_xI_pow; auto. intros H1 H2. - assert (H3:=IHx _ _ H1 H2). - unfold Pos.Nsucc_double. case_eq (Pos.land x y). - * intros _. eapply Z.le_lt_trans; [ |exact H]. clear. lia. - * intros p Hp. revert H3. rewrite Hp, N2Z.inj_pos, Z_pos_xI_pow; auto. - replace (i - 1 + 1) with i by ring. clear. lia. - - simpl. intro H. - assert (H4:0 <= i - 1) by (assert (H4:0 < i); try lia; apply pow_pos; apply (Z.le_lt_trans _ (Z.pos x~1)); auto; lia). - generalize H. replace i with ((i-1)+1) at 1 2 by ring. rewrite <- Z_pos_xI_pow, <- Z_pos_xO_pow; auto. intros H1 H2. - assert (H3:=IHx _ _ H1 H2). - unfold Pos.Ndouble. case_eq (Pos.land x y). - * intros _. eapply Z.le_lt_trans; [ |exact H]. clear. lia. - * intros p Hp. revert H3. rewrite Hp, N2Z.inj_pos, Z_pos_xO_pow; auto. - replace (i - 1 + 1) with i by ring. clear. lia. - - simpl. intro H. - assert (H4:0 <= i - 1) by (assert (H4:0 < i); try lia; apply pow_pos; apply (Z.le_lt_trans _ (Z.pos x~0)); auto; lia). - generalize H. replace i with ((i-1)+1) at 1 2 by ring. rewrite <- Z_pos_xI_pow, <- Z_pos_xO_pow; auto. intros H1 H2. - assert (H3:=IHx _ _ H1 H2). - unfold Pos.Ndouble. case_eq (Pos.land x y). - * intros _. eapply Z.le_lt_trans; [ |exact H]. clear. lia. - * intros p Hp. revert H3. rewrite Hp, N2Z.inj_pos, Z_pos_xO_pow; auto. - replace (i - 1 + 1) with i by ring. clear. lia. - - simpl. intro H. - assert (H4:0 <= i - 1) by (assert (H4:0 < i); try lia; apply pow_pos; apply (Z.le_lt_trans _ (Z.pos x~0)); auto; lia). - generalize H. replace i with ((i-1)+1) at 1 2 by ring. rewrite <- !Z_pos_xO_pow; auto. intros H1 H2. - assert (H3:=IHx _ _ H1 H2). - unfold Pos.Ndouble. case_eq (Pos.land x y). - * intros _. eapply Z.le_lt_trans; [ |exact H]. clear. lia. - * intros p Hp. revert H3. rewrite Hp, N2Z.inj_pos, Z_pos_xO_pow; auto. - replace (i - 1 + 1) with i by ring. clear. lia. - - simpl. intros H _. apply (Z.le_lt_trans _ (Z.pos x~0)); lia. - - simpl. intros H _. apply (Z.le_lt_trans _ 1); lia. -Qed. - - -Lemma Z_land_bounded i : forall x y, - 0 <= x < 2 ^ i -> 0 <= y < 2 ^ i -> 0 <= Z.land x y < 2 ^ i. -Proof. - intros [ |p|p] [ |q|q]; auto. - - intros [_ H1] [_ H2]. simpl. split. - * apply N2Z.is_nonneg. - * now apply pos_land_bounded. -Admitted. - -Theorem land_spec x y i : bit (x land y) i = bit x i && bit y i. -Proof. - rewrite !bit_testbit. change (x land y) with (land31 x y). unfold land31. - rewrite phi_phi_inv. rewrite Zmod_small. - - apply Z.land_spec. - - split. - * rewrite Z.land_nonneg. left. now destruct (phi_bounded x). - * now destruct (Z_land_bounded _ _ _ (phi_bounded x) (phi_bounded y)). -Qed. - - -Axiom lor_spec: forall x y i, bit (x lor y) i = bit x i || bit y i. - -Axiom lxor_spec: forall x y i, bit (x lxor y) i = xorb (bit x i) (bit y i). - -(** Specification of basic opetations *) - -(* Arithmetic modulo operations *) - -(* Remarque : les axiomes seraient plus simple si on utilise of_Z a la place : - exemple : add_spec : forall x y, of_Z (x + y) = of_Z x + of_Z y. *) - -Axiom add_spec : forall x y, [|x + y|] = ([|x|] + [|y|]) mod wB. - -Axiom sub_spec : forall x y, [|x - y|] = ([|x|] - [|y|]) mod wB. - -Axiom mul_spec : forall x y, [| x * y |] = [|x|] * [|y|] mod wB. - -Axiom mulc_spec : forall x y, [|x|] * [|y|] = [|fst (mulc x y)|] * wB + [|snd (mulc x y)|]. - -Axiom div_spec : forall x y, [|x / y|] = [|x|] / [|y|]. - -Axiom mod_spec : forall x y, [|x \% y|] = [|x|] mod [|y|]. - -(** Iterators *) - -Axiom foldi_cont_gt : forall A B f from to cont, - (to < from)%int = true -> foldi_cont (A:=A) (B:=B) f from to cont = cont. - -Axiom foldi_cont_eq : forall A B f from to cont, - from = to -> foldi_cont (A:=A) (B:=B) f from to cont = f from cont. - -Axiom foldi_cont_lt : forall A B f from to cont, - (from < to)%int = true-> - foldi_cont (A:=A) (B:=B) f from to cont = - f from (fun a' => foldi_cont f (from + 1%int) to cont a'). - -Axiom foldi_down_cont_lt : forall A B f from downto cont, - (from < downto)%int = true -> foldi_down_cont (A:=A) (B:=B) f from downto cont = cont. - -Axiom foldi_down_cont_eq : forall A B f from downto cont, - from = downto -> foldi_down_cont (A:=A) (B:=B) f from downto cont = f from cont. - -Axiom foldi_down_cont_gt : forall A B f from downto cont, - (downto < from)%int = true-> - foldi_down_cont (A:=A) (B:=B) f from downto cont = - f from (fun a' => foldi_down_cont f (from-1) downto cont a'). - -(** Print *) - -Axiom print_int_spec : forall x, x = print_int x. - -(** Axioms on operations which are just short cut *) - -Axiom compare_def_spec : forall x y, compare x y = compare_def x y. - -Axiom head0_spec : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ ([|head0 x|]) * [|x|] < wB. - -Axiom tail0_spec : forall x, 0 < [|x|] -> - (exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|tail0 x|]))%Z. - -Axiom addc_def_spec : forall x y, (x +c y)%int = addc_def x y. - -Axiom addcarryc_def_spec : forall x y, addcarryc x y = addcarryc_def x y. - -Axiom subc_def_spec : forall x y, (x -c y)%int = subc_def x y. - -Axiom subcarryc_def_spec : forall x y, subcarryc x y = subcarryc_def x y. - -Axiom diveucl_def_spec : forall x y, diveucl x y = diveucl_def x y. - -Axiom diveucl_21_spec : forall a1 a2 b, - let (q,r) := diveucl_21 a1 a2 b in - ([|q|],[|r|]) = Z.div_eucl ([|a1|] * wB + [|a2|]) [|b|]. - -Axiom addmuldiv_def_spec : forall p x y, - addmuldiv p x y = addmuldiv_def p x y. - - -(* - Local Variables: - coq-load-path: ((rec "../../.." "SMTCoq")) - End: -*) diff --git a/src/versions/standard/Int63/Int63Native_standard.v b/src/versions/standard/Int63/Int63Native_standard.v deleted file mode 100644 index 6600a27..0000000 --- a/src/versions/standard/Int63/Int63Native_standard.v +++ /dev/null @@ -1,143 +0,0 @@ -(**************************************************************************) -(* *) -(* SMTCoq *) -(* Copyright (C) 2011 - 2021 *) -(* *) -(* See file "AUTHORS" for the list of authors *) -(* *) -(* This file is distributed under the terms of the CeCILL-C licence *) -(* *) -(**************************************************************************) - - -Require Export DoubleType. -Require Import Int31 Cyclic31 Ring31. -Require Import ZArith. -Require Import Bool. - - -Definition size := size. - -Notation int := int31. - -Delimit Scope int63_scope with int. -Bind Scope int63_scope with int. - -(* Some constants *) -Notation "0" := 0%int31 : int63_scope. -Notation "1" := 1%int31 : int63_scope. -Notation "2" := 2%int31 : int63_scope. -Notation "3" := 3%int31 : int63_scope. - -(* Comparisons *) -Definition eqb := eqb31. -Notation "m '==' n" := (eqb m n) (at level 70, no associativity) : int63_scope. - -Definition ltb : int -> int -> bool := - fun i j => match compare31 i j with | Lt => true | _ => false end. -Notation "m < n" := (ltb m n) : int63_scope. - -Definition leb : int -> int -> bool := - fun i j => match compare31 i j with | Gt => false | _ => true end. -Notation "m <= n" := (leb m n) : int63_scope. - - -Lemma eqb_correct : forall i j, (i==j)%int = true -> i = j. -Proof. exact eqb31_correct. Qed. - -(* Logical operations *) -Definition lsl : int -> int -> int := - fun i j => addmuldiv31 j i 0. -Infix "<<" := lsl (at level 30, no associativity) : int63_scope. - -Definition lsr : int -> int -> int := - fun i j => if (j < 31%int31)%int then addmuldiv31 (31-j)%int31 0 i else 0%int31. -Infix ">>" := lsr (at level 30, no associativity) : int63_scope. - -Definition land : int -> int -> int := land31. -Global Arguments land i j : simpl never. -Global Opaque land. -Infix "land" := land (at level 40, left associativity) : int63_scope. - -Definition lor : int -> int -> int := lor31. -Global Arguments lor i j : simpl never. -Global Opaque lor. -Infix "lor" := lor (at level 40, left associativity) : int63_scope. - -Definition lxor : int -> int -> int := lxor31. -Global Arguments lxor i j : simpl never. -Global Opaque lxor. -Infix "lxor" := lxor (at level 40, left associativity) : int63_scope. - -(* Arithmetic modulo operations *) -Notation "n + m" := (add31 n m) : int63_scope. -Notation "n - m" := (sub31 n m) : int63_scope. -Notation "n * m" := (mul31 n m) : int63_scope. - -Definition mulc : int -> int -> int * int := - fun i j => match mul31c i j with - | W0 => (0%int, 0%int) - | WW h l => (h, l) - end. - -Definition div : int -> int -> int := - fun i j => let (q,_) := div31 i j in q. -Notation "n / m" := (div n m) : int63_scope. - -Definition modulo : int -> int -> int := - fun i j => let (_,r) := div31 i j in r. -Notation "n '\%' m" := (modulo n m) (at level 40, left associativity) : int63_scope. - - -(* Iterators *) - -Definition firstr i := if ((i land 1) == 0)%int then D0 else D1. -Fixpoint recr_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) - (i:int31) : A := - match n with - | O => case0 - | S next => - if (i == 0)%int then - case0 - else - let si := (i >> 1)%int in - caserec (firstr i) si (recr_aux next A case0 caserec si) - end. -Definition recr := recr_aux size. -Definition iter_int31 i A f := - recr (A->A) (fun x => x) - (fun b si rec => match b with - | D0 => fun x => rec (rec x) - | D1 => fun x => f (rec (rec x)) - end) - i. - -Definition foldi_cont - {A B : Type} - (f : int -> (A -> B) -> A -> B) - (from to : int) - (cont : A -> B) : A -> B := - if ltb to from then - cont - else - let (_,r) := iter_int31 (to - from) _ (fun (jy: (int * (A -> B))%type) => - let (j,y) := jy in ((j-1)%int, f j y) - ) (to, cont) in - f from r. - -Definition foldi_down_cont - {A B : Type} - (f : int -> (A -> B) -> A -> B) - (from downto : int) - (cont : A -> B) : A -> B := - if ltb from downto then - cont - else - let (_,r) := iter_int31 (from - downto) _ (fun (jy: (int * (A -> B))%type) => - let (j,y) := jy in ((j+1)%int, f j y) - ) (downto, cont) in - f from r. - -(* Fake print *) - -Definition print_int : int -> int := fun i => i. diff --git a/src/versions/standard/Int63/Int63Op_standard.v b/src/versions/standard/Int63/Int63Op_standard.v deleted file mode 100644 index bb7d9a1..0000000 --- a/src/versions/standard/Int63/Int63Op_standard.v +++ /dev/null @@ -1,334 +0,0 @@ -(**************************************************************************) -(* *) -(* SMTCoq *) -(* Copyright (C) 2011 - 2021 *) -(* *) -(* See file "AUTHORS" for the list of authors *) -(* *) -(* This file is distributed under the terms of the CeCILL-C licence *) -(* *) -(**************************************************************************) - - -Require Import Int31 Cyclic31. -Require Export Int63Native. -(* Require Import BigNumPrelude. *) -Require Import Bvector. - - -Local Open Scope int63_scope. - -(** The number of digits as a int *) -Definition digits := 31%int31. - -(** The bigger int *) -Definition max_int := Eval vm_compute in 0 - 1. - -(** Access to the nth digits *) -Definition get_digit x p := (0 < (x land (1 << p))). - -Definition set_digit x p (b:bool) := - if (0 <= p) && (p < digits) then - if b then x lor (1 << p) - else x land (max_int lxor (1 << p)) - else x. - -(** Equality to 0 *) -Definition is_zero (i:int) := i == 0. - -(** Parity *) -Definition is_even (i:int) := is_zero (i land 1). - -(** Bit *) - -Definition bit i n := negb (is_zero ((i >> n) << (digits - 1))). -(* Register bit as PrimInline. *) - -(** Extra modulo operations *) -Definition opp (i:int) := 0 - i. -Notation "- x" := (opp x) : int63_scope. - -Definition oppcarry i := max_int - i. - -Definition succ i := i + 1. - -Definition pred i := i - 1. - -Definition addcarry i j := i + j + 1. - -Definition subcarry i j := i - j - 1. - -(** Exact arithmetic operations *) - -Definition addc_def x y := - let r := x + y in - if r < x then C1 r else C0 r. -(* the same but direct implementation for efficiancy *) -Definition addc : int -> int -> carry int := add31c. -Notation "n '+c' m" := (addc n m) (at level 50, no associativity) : int63_scope. - -Definition addcarryc_def x y := - let r := addcarry x y in - if r <= x then C1 r else C0 r. -(* the same but direct implementation for efficiancy *) -Definition addcarryc : int -> int -> carry int := add31carryc. - -Definition subc_def x y := - if y <= x then C0 (x - y) else C1 (x - y). -(* the same but direct implementation for efficiancy *) -Definition subc : int -> int -> carry int := sub31c. -Notation "n '-c' m" := (subc n m) (at level 50, no associativity) : int63_scope. - -Definition subcarryc_def x y := - if y < x then C0 (x - y - 1) else C1 (x - y - 1). -(* the same but direct implementation for efficiancy *) -Definition subcarryc : int -> int -> carry int := sub31carryc. - -Definition diveucl_def x y := (x/y, x\%y). -(* the same but direct implementation for efficiancy *) -Definition diveucl : int -> int -> int * int := div31. - -Definition diveucl_21 : int -> int -> int -> int * int := div3121. - -Definition addmuldiv_def p x y := - (x << p) lor (y >> (digits - p)). -(* the same but direct implementation for efficiancy *) -Definition addmuldiv : int -> int -> int -> int := addmuldiv31. - -Definition oppc (i:int) := 0 -c i. - -Definition succc i := i +c 1. - -Definition predc i := i -c 1. - -(** Comparison *) -Definition compare_def x y := - if x < y then Lt - else if (x == y) then Eq else Gt. - -Definition compare : int -> int -> comparison := compare31. -Notation "n ?= m" := (compare n m) (at level 70, no associativity) : int63_scope. - -(** Exotic operations *) - -(** I should add the definition (like for compare) *) -Definition head0 : int -> int := head031. -Definition tail0 : int -> int := tail031. - -(** Iterators *) - -Definition foldi {A} (f:int -> A -> A) from to := - foldi_cont (fun i cont a => cont (f i a)) from to (fun a => a). - -Definition fold {A} (f: A -> A) from to := - foldi_cont (fun i cont a => cont (f a)) from to (fun a => a). - -Definition foldi_down {A} (f:int -> A -> A) from downto := - foldi_down_cont (fun i cont a => cont (f i a)) from downto (fun a => a). - -Definition fold_down {A} (f:A -> A) from downto := - foldi_down_cont (fun i cont a => cont (f a)) from downto (fun a => a). - -Definition forallb (f:int -> bool) from to := - foldi_cont (fun i cont _ => if f i then cont tt else false) from to (fun _ => true) tt. - -Definition existsb (f:int -> bool) from to := - foldi_cont (fun i cont _ => if f i then true else cont tt) from to (fun _ => false) tt. - -(** Translation to Z *) - -(* Fixpoint to_Z_rec (n:nat) (i:int) := *) -(* match n with *) -(* | O => 0%Z *) -(* | S n => *) -(* (if is_even i then Zdouble else Zdouble_plus_one) (to_Z_rec n (i >> 1)) *) -(* end. *) - -(* Definition to_Z := to_Z_rec size. *) - -Definition to_Z := phi. -Definition of_Z := phi_inv. - -(* Fixpoint of_pos_rec (n:nat) (p:positive) := *) -(* match n, p with *) -(* | O, _ => 0 *) -(* | S n, xH => 1 *) -(* | S n, xO p => (of_pos_rec n p) << 1 *) -(* | S n, xI p => (of_pos_rec n p) << 1 lor 1 *) -(* end. *) - -(* Definition of_pos := of_pos_rec size. *) - -(* Definition of_Z z := *) -(* match z with *) -(* | Zpos p => of_pos p *) -(* | Z0 => 0 *) -(* | Zneg p => - (of_pos p) *) -(* end. *) - -(** Gcd **) -Fixpoint gcd_rec (guard:nat) (i j:int) {struct guard} := - match guard with - | O => 1 - | S p => if j == 0 then i else gcd_rec p j (i \% j) - end. - -Definition gcd := gcd_rec (2*size). - -(** Square root functions using newton iteration **) - -Definition sqrt_step (rec: int -> int -> int) (i j: int) := - let quo := i/j in - if quo < j then rec i ((j + (i/j)%int) >> 1) - else j. - -Definition iter_sqrt := - Eval lazy beta delta [sqrt_step] in - fix iter_sqrt (n: nat) (rec: int -> int -> int) - (i j: int) {struct n} : int := - sqrt_step - (fun i j => match n with - O => rec i j - | S n => (iter_sqrt n (iter_sqrt n rec)) i j - end) i j. - -Definition sqrt i := - match compare 1 i with - Gt => 0 - | Eq => 1 - | Lt => iter_sqrt size (fun i j => j) i (i >> 1) - end. - -Definition high_bit := 1 << (digits - 1). - -Definition sqrt2_step (rec: int -> int -> int -> int) - (ih il j: int) := - if ih < j then - let (quo,_) := diveucl_21 ih il j in - if quo < j then - match j +c quo with - | C0 m1 => rec ih il (m1 >> 1) - | C1 m1 => rec ih il ((m1 >> 1) + high_bit) - end - else j - else j. - -Definition iter2_sqrt := - Eval lazy beta delta [sqrt2_step] in - fix iter2_sqrt (n: nat) - (rec: int -> int -> int -> int) - (ih il j: int) {struct n} : int := - sqrt2_step - (fun ih il j => - match n with - | O => rec ih il j - | S n => (iter2_sqrt n (iter2_sqrt n rec)) ih il j - end) ih il j. - -Definition sqrt2 ih il := - let s := iter2_sqrt size (fun ih il j => j) ih il max_int in - let (ih1, il1) := mulc s s in - match il -c il1 with - | C0 il2 => - if ih1 < ih then (s, C1 il2) else (s, C0 il2) - | C1 il2 => - if ih1 < (ih - 1) then (s, C1 il2) else (s, C0 il2) - end. - -(* Extra function on equality *) - -Definition cast_digit d1 d2 : - option (forall P : Int31.digits -> Type, P d1 -> P d2) := - match d1, d2 with - | D0, D0 | D1, D1 => Some (fun P h => h) - | _, _ => None - end. - -(* May need to improve this definition, but no reported efficiency problem for the moment *) -Definition cast i j : - option (forall P : int -> Type, P i -> P j) := - match i, j return option (forall P : int -> Type, P i -> P j) with - | I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d28 d29 d30, I31 d'0 d'1 d'2 d'3 d'4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30 => - match - cast_digit d0 d'0, - cast_digit d1 d'1, - cast_digit d2 d'2, - cast_digit d3 d'3, - cast_digit d4 d'4, - cast_digit d5 d'5, - cast_digit d6 d'6, - cast_digit d7 d'7, - cast_digit d8 d'8, - cast_digit d9 d'9, - cast_digit d10 d'10, - cast_digit d11 d'11, - cast_digit d12 d'12, - cast_digit d13 d'13, - cast_digit d14 d'14, - cast_digit d15 d'15, - cast_digit d16 d'16, - cast_digit d17 d'17, - cast_digit d18 d'18, - cast_digit d19 d'19, - cast_digit d20 d'20, - cast_digit d21 d'21, - cast_digit d22 d'22, - cast_digit d23 d'23, - cast_digit d24 d'24, - cast_digit d25 d'25, - cast_digit d26 d'26, - cast_digit d27 d'27, - cast_digit d28 d'28, - cast_digit d29 d'29, - cast_digit d30 d'30 - with - | Some k0, - Some k1, - Some k2, - Some k3, - Some k4, - Some k5, - Some k6, - Some k7, - Some k8, - Some k9, - Some k10, - Some k11, - Some k12, - Some k13, - Some k14, - Some k15, - Some k16, - Some k17, - Some k18, - Some k19, - Some k20, - Some k21, - Some k22, - Some k23, - Some k24, - Some k25, - Some k26, - Some k27, - Some k28, - Some k29, - Some k30 => - Some (fun P h => - k0 (fun d0 => P (I31 d0 d'1 d'2 d'3 d'4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k1 (fun d1 => P (I31 d0 d1 d'2 d'3 d'4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k2 (fun d2 => P (I31 d0 d1 d2 d'3 d'4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k3 (fun d3 => P (I31 d0 d1 d2 d3 d'4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k4 (fun d4 => P (I31 d0 d1 d2 d3 d4 d'5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k5 (fun d5 => P (I31 d0 d1 d2 d3 d4 d5 d'6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k6 (fun d6 => P (I31 d0 d1 d2 d3 d4 d5 d6 d'7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k7 (fun d7 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d'8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k8 (fun d8 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d'9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k9 (fun d9 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d'10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k10 (fun d10 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d'11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k11 (fun d11 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d'12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k12 (fun d12 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d'13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k13 (fun d13 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d'14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k14 (fun d14 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d'15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k15 (fun d15 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d'16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k16 (fun d16 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d'17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k17 (fun d17 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d'18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k18 (fun d18 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d'19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k19 (fun d19 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d'20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k20 (fun d20 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d'21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k21 (fun d21 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d'22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k22 (fun d22 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d'23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k23 (fun d23 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d'24 d'25 d'26 d'27 d'28 d'29 d'30)) (k24 (fun d24 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d'25 d'26 d'27 d'28 d'29 d'30)) (k25 (fun d25 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d'26 d'27 d'28 d'29 d'30)) (k26 (fun d26 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d'27 d'28 d'29 d'30)) (k27 (fun d27 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d'28 d'29 d'30)) (k28 (fun d28 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d28 d'29 d'30)) (k29 (fun d29 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d28 d29 d'30)) (k30 (fun d30 => P (I31 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d28 d29 d30)) h))))))))))))))))))))))))))))))) - | _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _ => None - end - end. - - -Definition eqo i j : option (i = j) := - match cast i j with - | Some k => Some (k (fun j => i = j) (refl_equal i)) - | None => None - end. - - -(* - Local Variables: - coq-load-path: ((rec "../../.." "SMTCoq")) - End: -*) diff --git a/src/versions/standard/Int63/Int63Properties_standard.v b/src/versions/standard/Int63/Int63Properties_standard.v deleted file mode 100644 index feb19b8..0000000 --- a/src/versions/standard/Int63/Int63Properties_standard.v +++ /dev/null @@ -1,2768 +0,0 @@ -(**************************************************************************) -(* *) -(* SMTCoq *) -(* Copyright (C) 2011 - 2021 *) -(* *) -(* See file "AUTHORS" for the list of authors *) -(* *) -(* This file is distributed under the terms of the CeCILL-C licence *) -(* *) -(**************************************************************************) - - -Require Import Zgcd_alt. -Require Import Bvector. -Require Import Int31 Cyclic31. -Require Export Int63Axioms. -Require Import Eqdep_dec. -Require Import Psatz. -Require Import Znumtheory Zpow_facts. - -Local Open Scope int63_scope. -Local Open Scope Z_scope. - - -Notation Zpower_2 := Z.pow_2_r. -Notation Zpower_Zsucc := Z.pow_succ_r. - - -(* Taken from BigNumPrelude *) - -Lemma Zlt0_not_eq : forall n, 0<n -> n<>0. -Proof. - auto with zarith. -Qed. - -Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H). - -Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. - Proof. - intros a b H H1;case (Z_mod_lt a b);auto with zarith;intros H2 H3;split;auto. - case (Z.le_gt_cases b a); intros H4; auto with zarith. - rewrite Zmod_small; auto with zarith. - Qed. - - -(** Trivial lemmas without axiom *) - -Lemma wB_diff_0 : wB <> 0. -Proof. compute;discriminate. Qed. - -Lemma wB_pos : 0 < wB. -Proof. reflexivity. Qed. - -Lemma to_Z_0 : [|0|] = 0. -Proof. reflexivity. Qed. - -Lemma to_Z_1 : [|1|] = 1. -Proof. reflexivity. Qed. - -(** equality *) -Lemma eqb_complete : forall x y, x = y -> (x == y) = true. -Proof. - intros x y H;rewrite H, eqb_refl;trivial. -Qed. - -Lemma eqb_spec : forall x y, (x == y) = true <-> x = y. -Proof. - split;auto using eqb_correct, eqb_complete. -Qed. - -Lemma eqb_false_spec : forall x y, (x == y) = false <-> x <> y. -Proof. - intros;rewrite <- not_true_iff_false, eqb_spec;split;trivial. -Qed. - -Lemma eqb_false_complete : forall x y, x <> y -> (x == y) = false. -Proof. - intros x y;rewrite eqb_false_spec;trivial. -Qed. - -Lemma eqb_false_correct : forall x y, (x == y) = false -> x <> y. -Proof. - intros x y;rewrite eqb_false_spec;trivial. -Qed. - -Definition eqs (i j : int) : {i = j} + { i <> j } := - (if i == j as b return ((b = true -> i = j) -> (b = false -> i <> j) -> {i=j} + {i <> j} ) - then fun (Heq : true = true -> i = j) _ => left _ (Heq (eq_refl true)) - else fun _ (Hdiff : false = false -> i <> j) => right _ (Hdiff (eq_refl false))) - (eqb_correct i j) - (eqb_false_correct i j). - -Lemma eq_dec : forall i j:int, i = j \/ i <> j. -Proof. - intros i j;destruct (eqs i j);auto. -Qed. - -(* TODO: fill these proofs *) -Lemma cast_refl : forall i, cast i i = Some (fun P H => H). -Admitted. -(* Proof. *) -(* unfold cast;intros. *) -(* generalize (eqb_correct i i). *) -(* rewrite eqb_refl;intros. *) -(* rewrite (eq_proofs_unicity eq_dec (e (eq_refl true)) (eq_refl i));trivial. *) -(* Qed. *) - -Lemma cast_diff : forall i j, i == j = false -> cast i j = None. -Admitted. -(* Proof. *) -(* intros;unfold cast;intros; generalize (eqb_correct i j). *) -(* rewrite H;trivial. *) -(* Qed. *) - -Lemma eqo_refl : forall i, eqo i i = Some (eq_refl i). -Admitted. -(* Proof. *) -(* unfold eqo;intros. *) -(* generalize (eqb_correct i i). *) -(* rewrite eqb_refl;intros. *) -(* rewrite (eq_proofs_unicity eq_dec (e (eq_refl true)) (eq_refl i));trivial. *) -(* Qed. *) - -Lemma eqo_diff : forall i j, i == j = false -> eqo i j = None. -Admitted. -(* Proof. *) -(* unfold eqo;intros; generalize (eqb_correct i j). *) -(* rewrite H;trivial. *) -(* Qed. *) - -(** translation with Z *) -Require Import Ndigits. - -Lemma Z_of_N_double : forall n, Z_of_N (N.double n) = Z.double (Z_of_N n). -Proof. - destruct n;simpl;trivial. -Qed. - -Lemma Z_of_N_double_plus_one : forall n, Z_of_N (Ndouble_plus_one n) = Zdouble_plus_one (Z_of_N n). -Proof. - destruct n;simpl;trivial. -Qed. - -Lemma to_Z_bounded : forall x, 0 <= [|x|] < wB. -Proof. apply phi_bounded. Qed. -(* unfold to_Z, wB;induction size;intros. *) -(* simpl;auto with zarith. *) -(* rewrite inj_S;simpl;assert (W:= IHn (x >> 1)%int). *) -(* rewrite Zpower_Zsucc;auto with zarith. *) -(* destruct (is_even x). *) -(* rewrite Z.double_mult;auto with zarith. *) -(* rewrite Zdouble_plus_one_mult;auto with zarith. *) -(* Qed. *) - -(* TODO: move_this *) -(* Lemma orb_true_iff : forall b1 b2, b1 || b2 = true <-> b1 = true \/ b2 = true. *) -(* Proof. *) -(* split;intros;[apply orb_prop | apply orb_true_intro];trivial. *) -(* Qed. *) - -Lemma to_Z_eq : forall x y, [|x|] = [|y|] <-> x = y. -Proof. - split;intros;subst;trivial. - apply to_Z_inj;trivial. -Qed. - -Lemma leb_ltb_eqb : forall x y, ((x <= y) = (x < y) || (x == y))%int. -Proof. - intros. - apply eq_true_iff_eq. - rewrite leb_spec, orb_true_iff, ltb_spec, eqb_spec, <- to_Z_eq;omega. -Qed. - - -(** Comparison *) - -Lemma compare_spec : - forall x y, compare x y = ([|x|] ?= [|y|]). -Proof. - intros;rewrite compare_def_spec;unfold compare_def. - case_eq (x < y)%int;intros Heq. - rewrite ltb_spec in Heq. - red in Heq;rewrite Heq;trivial. - rewrite <- not_true_iff_false, ltb_spec in Heq. - case_eq (x == y)%int;intros Heq1. - rewrite eqb_spec in Heq1;rewrite Heq1, Z.compare_refl;trivial. - rewrite <- not_true_iff_false, eqb_spec in Heq1. - symmetry;change ([|x|] > [|y|]);rewrite <- to_Z_eq in Heq1;omega. -Qed. - -Lemma is_zero_spec : forall x : int, is_zero x = true <-> x = 0%int. -Proof. - unfold is_zero;intros;apply eqb_spec. -Qed. - - -(** Addition *) - -Lemma addc_spec : forall x y, [+|x +c y|] = [|x|] + [|y|]. -Proof. - intros;rewrite addc_def_spec;unfold addc_def. - assert (W1 := to_Z_bounded x); assert (W2 := to_Z_bounded y). - case_eq ((x + y < x)%int). - rewrite ltb_spec;intros. - change (wB + [|x+y|] = [|x|] + [|y|]). - rewrite add_spec in H |- *. - assert ([|x|] + [|y|] >= wB). - destruct (Z_lt_ge_dec ([|x|] + [|y|]) wB);auto with zarith. - elimtype False;rewrite Zmod_small in H;auto with zarith. - assert (([|x|] + [|y|]) mod wB = [|x|] + [|y|] - wB). - symmetry;apply Zmod_unique with 1;auto with zarith. - rewrite H1;ring. - rewrite <- not_true_iff_false, ltb_spec;intros. - change ([|x+y|] = [|x|] + [|y|]). - rewrite add_spec in *. - assert ([|x|] + [|y|] < wB). - destruct (Z_lt_ge_dec ([|x|] + [|y|]) wB);auto with zarith. - assert (([|x|] + [|y|]) mod wB = [|x|] + [|y|] - wB). - symmetry;apply Zmod_unique with 1;auto with zarith. - elim H;omega. - rewrite Zmod_small;auto with zarith. -Qed. - - -Lemma succc_spec : forall x, [+|succc x|] = [|x|] + 1. -Proof. intros; unfold succc; apply addc_spec. Qed. - -Lemma addcarry_spec : forall x y, [|addcarry x y|] = ([|x|] + [|y|] + 1) mod wB. -Proof. - unfold addcarry;intros. - rewrite add_spec,add_spec,Zplus_mod_idemp_l;trivial. -Qed. - -Lemma addcarryc_spec : forall x y, [+|addcarryc x y|] = [|x|] + [|y|] + 1. -Proof. - intros;rewrite addcarryc_def_spec;unfold addcarryc_def. - assert (W1 := to_Z_bounded x); assert (W2 := to_Z_bounded y). - case_eq ((addcarry x y <= x)%int). - rewrite leb_spec;intros. - change (wB + [|(addcarry x y)|] = [|x|] + [|y|] + 1). - rewrite addcarry_spec in H |- *. - assert ([|x|] + [|y|] + 1 >= wB). - destruct (Z_lt_ge_dec ([|x|] + [|y|] + 1) wB);auto with zarith. - elimtype False;rewrite Zmod_small in H;auto with zarith. - assert (([|x|] + [|y|] + 1) mod wB = [|x|] + [|y|] + 1 - wB). - symmetry;apply Zmod_unique with 1;auto with zarith. - rewrite H1;ring. - rewrite <- not_true_iff_false, leb_spec;intros. - change ([|addcarry x y|] = [|x|] + [|y|] + 1). - rewrite addcarry_spec in *. - assert ([|x|] + [|y|] + 1 < wB). - destruct (Z_lt_ge_dec ([|x|] + [|y|] + 1) wB);auto with zarith. - assert (([|x|] + [|y|] + 1) mod wB = [|x|] + [|y|] + 1 - wB). - symmetry;apply Zmod_unique with 1;auto with zarith. - elim H;omega. - rewrite Zmod_small;auto with zarith. -Qed. - -Lemma succ_spec : forall x, [|succ x|] = ([|x|] + 1) mod wB. -Proof. intros; apply add_spec. Qed. - -(** Subtraction *) -Lemma subc_spec : forall x y, [-|x -c y|] = [|x|] - [|y|]. -Proof. - intros;rewrite subc_def_spec;unfold subc_def. - assert (W1 := to_Z_bounded x); assert (W2 := to_Z_bounded y). - case_eq (y <= x)%int. - rewrite leb_spec;intros. - change ([|x - y|] = [|x|] - [|y|]). - rewrite sub_spec. - rewrite Zmod_small;auto with zarith. - rewrite <- not_true_iff_false, leb_spec;intros. - change (-wB + [|x - y|] = [|x|] - [|y|]). - rewrite sub_spec. - assert (([|x|] - [|y|]) mod wB = [|x|] - [|y|] + wB). - symmetry;apply Zmod_unique with (-1);auto with zarith. - rewrite H0;ring. -Qed. - -Lemma subcarry_spec : - forall x y, [|subcarry x y|] = ([|x|] - [|y|] - 1) mod wB. -Proof. - unfold subcarry; intros. - rewrite sub_spec,sub_spec,Zminus_mod_idemp_l;trivial. -Qed. - -Lemma subcarryc_spec : forall x y, [-|subcarryc x y|] = [|x|] - [|y|] - 1. - intros;rewrite subcarryc_def_spec;unfold subcarryc_def. - assert (W1 := to_Z_bounded x); assert (W2 := to_Z_bounded y). - (* fold (subcarry x y). *) - replace ((x - y - 1)%int) with (subcarry x y) by reflexivity. - case_eq (y < x)%int. - rewrite ltb_spec;intros. - change ([|subcarry x y|] = [|x|] - [|y|] - 1). - rewrite subcarry_spec. - rewrite Zmod_small;auto with zarith. - rewrite <- not_true_iff_false, ltb_spec;intros. - change (-wB + [|subcarry x y|] = [|x|] - [|y|] - 1). - rewrite subcarry_spec. - assert (([|x|] - [|y|] - 1) mod wB = [|x|] - [|y|] - 1 + wB). - symmetry;apply Zmod_unique with (-1);auto with zarith. - rewrite H0;ring. -Qed. - -Lemma oppc_spec : forall x : int, [-|oppc x|] = - [|x|]. -Proof. - unfold oppc;intros;rewrite subc_spec, to_Z_0;trivial. -Qed. - -Lemma opp_spec : forall x : int, [|- x|] = - [|x|] mod wB. -Proof. - unfold opp;intros. rewrite sub_spec, to_Z_0;trivial. -Qed. - -Lemma oppcarry_spec : forall x, [|oppcarry x|] = wB - [|x|] - 1. -Proof. - unfold oppcarry;intros. - rewrite sub_spec. - change [|max_int|] with (wB - 1). - rewrite <- Zminus_plus_distr, Zplus_comm, Zminus_plus_distr. - apply Zmod_small. - generalize (to_Z_bounded x);auto with zarith. -Qed. - -Lemma predc_spec : forall x, [-|predc x|] = [|x|] - 1. -Proof. intros; unfold predc; apply subc_spec. Qed. - -Lemma pred_spec : forall x, [|pred x|] = ([|x|] - 1) mod wB. -Proof. intros; unfold pred; apply sub_spec. Qed. - -Lemma diveucl_spec : - forall x y, - let (q,r) := diveucl x y in - ([|q|],[|r|]) = Z.div_eucl [|x|] [|y|]. -Proof. - intros;rewrite diveucl_def_spec. - unfold diveucl_def;rewrite div_spec, mod_spec. - unfold Z.div, Zmod;destruct (Z.div_eucl [|x|] [|y|]);trivial. -Qed. - -(* Sqrt *) - - (* Direct transcription of an old proof - of a fortran program in boyer-moore *) - -Lemma quotient_by_2 a: a - 1 <= (a/2) + (a/2). -Proof. - case (Z_mod_lt a 2); auto with zarith. - intros H1; rewrite Zmod_eq_full; auto with zarith. -Qed. - -Lemma sqrt_main_trick j k: 0 <= j -> 0 <= k -> - (j * k) + j <= ((j + k)/2 + 1) ^ 2. -Proof. - intros Hj; generalize Hj k; pattern j; apply natlike_ind; - auto; clear k j Hj. - intros _ k Hk; repeat rewrite Zplus_0_l. - apply Zmult_le_0_compat; generalize (Z_div_pos k 2); auto with zarith. - intros j Hj Hrec _ k Hk; pattern k; apply natlike_ind; auto; clear k Hk. - rewrite Zmult_0_r, Zplus_0_r, Zplus_0_l. - generalize (sqr_pos (Z.succ j / 2)) (quotient_by_2 (Z.succ j)); - unfold Z.succ. - rewrite Zpower_2, Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. - auto with zarith. - intros k Hk _. - replace ((Z.succ j + Z.succ k) / 2) with ((j + k)/2 + 1). - generalize (Hrec Hj k Hk) (quotient_by_2 (j + k)). - unfold Z.succ; repeat rewrite Zpower_2; - repeat rewrite Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. - repeat rewrite Zmult_1_l; repeat rewrite Zmult_1_r. - auto with zarith. - rewrite Zplus_comm, <- Z_div_plus_full_l; auto with zarith. - apply f_equal2 with (f := Z.div); auto with zarith. -Qed. - -Lemma sqrt_main i j: 0 <= i -> 0 < j -> i < ((j + (i/j))/2 + 1) ^ 2. -Proof. - intros Hi Hj. - assert (Hij: 0 <= i/j) by (apply Z_div_pos; auto with zarith). - apply Z.lt_le_trans with (2 := sqrt_main_trick _ _ (Zlt_le_weak _ _ Hj) Hij). - pattern i at 1; rewrite (Z_div_mod_eq i j); case (Z_mod_lt i j); auto with zarith. -Qed. - -Lemma sqrt_init i: 1 < i -> i < (i/2 + 1) ^ 2. -Proof. - intros Hi. - assert (H1: 0 <= i - 2) by auto with zarith. - assert (H2: 1 <= (i / 2) ^ 2); auto with zarith. - replace i with (1* 2 + (i - 2)); auto with zarith. - rewrite Zpower_2, Z_div_plus_full_l; auto with zarith. - generalize (sqr_pos ((i - 2)/ 2)) (Z_div_pos (i - 2) 2). - rewrite Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. - auto with zarith. - generalize (quotient_by_2 i). - rewrite Zpower_2 in H2 |- *; - repeat (rewrite Zmult_plus_distr_l || - rewrite Zmult_plus_distr_r || - rewrite Zmult_1_l || rewrite Zmult_1_r). - auto with zarith. -Qed. - -Lemma sqrt_test_true i j: 0 <= i -> 0 < j -> i/j >= j -> j ^ 2 <= i. -Proof. - intros Hi Hj Hd; rewrite Zpower_2. - apply Z.le_trans with (j * (i/j)); auto with zarith. - apply Z_mult_div_ge; auto with zarith. -Qed. - -Lemma sqrt_test_false i j: 0 <= i -> 0 < j -> i/j < j -> (j + (i/j))/2 < j. -Proof. - intros Hi Hj H; case (Zle_or_lt j ((j + (i/j))/2)); auto. - intros H1; contradict H; apply Zle_not_lt. - assert (2 * j <= j + (i/j)); auto with zarith. - apply Z.le_trans with (2 * ((j + (i/j))/2)); auto with zarith. - apply Z_mult_div_ge; auto with zarith. -Qed. - - -Lemma sqrt_step_correct rec i j: - 0 < [|i|] -> 0 < [|j|] -> [|i|] < ([|j|] + 1) ^ 2 -> - 2 * [|j|] < wB -> - (forall j1 : int, - 0 < [|j1|] < [|j|] -> [|i|] < ([|j1|] + 1) ^ 2 -> - [|rec i j1|] ^ 2 <= [|i|] < ([|rec i j1|] + 1) ^ 2) -> - [|sqrt_step rec i j|] ^ 2 <= [|i|] < ([|sqrt_step rec i j|] + 1) ^ 2. -Proof. - assert (Hp2: 0 < [|2|]) by exact (refl_equal Lt). - intros Hi Hj Hij H31 Hrec. - unfold sqrt_step. - case_eq ((i / j < j)%int);[ | rewrite <- Bool.not_true_iff_false]; - rewrite ltb_spec, div_spec;intros. - assert ([| j + (i / j)%int|] = [|j|] + [|i|]/[|j|]). - { - rewrite add_spec, Zmod_small;rewrite div_spec; auto with zarith. - split. - - apply Z.add_nonneg_nonneg. - + apply Z.lt_le_incl; apply Z.le_lt_trans with (2 := H). apply Z_div_pos. - * apply Z.lt_gt. abstract omega. - * abstract omega. - + apply Z_div_pos. - * apply Z.lt_gt. assumption. - * abstract omega. - - abstract omega. - } - apply Hrec;rewrite lsr_spec, H0, to_Z_1;change (2^1) with 2. - split; [ | apply sqrt_test_false;auto with zarith]. - replace ([|j|] + [|i|]/[|j|]) with - (1 * 2 + (([|j|] - 2) + [|i|] / [|j|]));[ | ring]. - rewrite Z_div_plus_full_l; auto with zarith. - assert (0 <= [|i|]/ [|j|]) by (apply Z_div_pos; auto with zarith). - assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / 2) ; auto with zarith. - case (Zle_lt_or_eq 1 [|j|]); auto with zarith. - { - intro. apply Z_div_pos. - - apply Zgt_pos_0. - - apply Z.add_nonneg_nonneg. - + abstract omega. - + assumption. - } - intros Hj1. - rewrite <- Hj1, Zdiv_1_r. - assert (0 <= ([|i|] - 1) /2)%Z;[ |apply Z_div_pos]; auto with zarith. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply sqrt_main;auto with zarith. - split;[apply sqrt_test_true | ];auto with zarith. -Qed. - -Lemma iter_sqrt_correct n rec i j: 0 < [|i|] -> 0 < [|j|] -> - [|i|] < ([|j|] + 1) ^ 2 -> 2 * [|j|] < wB -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> - [|i|] < ([|j1|] + 1) ^ 2 -> 2 * [|j1|] < wB -> - [|rec i j1|] ^ 2 <= [|i|] < ([|rec i j1|] + 1) ^ 2) -> - [|iter_sqrt n rec i j|] ^ 2 <= [|i|] < ([|iter_sqrt n rec i j|] + 1) ^ 2. -Proof. - revert rec i j; elim n; unfold iter_sqrt; fold iter_sqrt; clear n. - intros rec i j Hi Hj Hij H31 Hrec. replace (and (Z.le (Z.pow (to_Z match ltb (div i j) j return int with | true => rec i (lsr (add31 j (div i j)) In) | false => j end) (Zpos (xO xH))) (to_Z i)) (Z.lt (to_Z i) (Z.pow (Z.add (to_Z match ltb (div i j) j return int with | true => rec i (lsr (add31 j (div i j)) In) | false => j end) (Zpos xH)) (Zpos (xO xH))))) with ([|sqrt_step rec i j|] ^ 2 <= [|i|] < ([|sqrt_step rec i j|] + 1) ^ 2) by reflexivity. apply sqrt_step_correct; auto with zarith. - intros; apply Hrec; auto with zarith. - rewrite Zpower_0_r; auto with zarith. - intros n Hrec rec i j Hi Hj Hij H31 HHrec. - replace (and (Z.le (Z.pow (to_Z match ltb (div i j) j return int with | true => iter_sqrt n (iter_sqrt n rec) i (lsr (add31 j (div i j)) In) | false => j end) (Zpos (xO xH))) (to_Z i)) (Z.lt (to_Z i) (Z.pow (Z.add (to_Z match ltb (div i j) j return int with | true => iter_sqrt n (iter_sqrt n rec) i (lsr (add31 j (div i j)) In) | false => j end) (Zpos xH)) (Zpos (xO xH))))) with ([|sqrt_step (iter_sqrt n (iter_sqrt n rec)) i j|] ^ 2 <= [|i|] < ([|sqrt_step (iter_sqrt n (iter_sqrt n rec)) i j|] + 1) ^ 2) by reflexivity. - apply sqrt_step_correct; auto. - intros j1 Hj1 Hjp1; apply Hrec; auto with zarith. - intros j2 Hj2 H2j2 Hjp2 Hj31; apply Hrec; auto with zarith. - intros j3 Hj3 Hpj3. - apply HHrec; auto. - rewrite inj_S, Zpower_Zsucc. - apply Z.le_trans with (2 ^Z_of_nat n + [|j2|]); auto with zarith. - apply Zle_0_nat. -Qed. - -Lemma sqrt_spec : forall x, - [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2. -Proof. - intros i; unfold sqrt. - rewrite compare_spec. case Z.compare_spec; rewrite to_Z_1; - intros Hi; auto with zarith. - repeat rewrite Zpower_2; auto with zarith. - apply iter_sqrt_correct; auto with zarith; - rewrite lsr_spec, to_Z_1; change (2^1) with 2; auto with zarith. - replace ([|i|]) with (1 * 2 + ([|i|] - 2))%Z; try ring. - assert (0 <= ([|i|] - 2)/2)%Z by (apply Z_div_pos; auto with zarith). - rewrite Z_div_plus_full_l; auto with zarith. - apply sqrt_init; auto. - assert (W:= Z_mult_div_ge [|i|] 2);assert (W':= to_Z_bounded i);auto with zarith. - intros j2 H1 H2; contradict H2; apply Zlt_not_le. - fold wB;assert (W:=to_Z_bounded i). - apply Z.le_lt_trans with ([|i|]); auto with zarith. - assert (0 <= [|i|]/2)%Z by (apply Z_div_pos; auto with zarith). - apply Z.le_trans with (2 * ([|i|]/2)); auto with zarith. - apply Z_mult_div_ge; auto with zarith. - case (to_Z_bounded i); repeat rewrite Zpower_2; auto with zarith. -Qed. - -Lemma sqrt2_step_def rec ih il j: - sqrt2_step rec ih il j = - if (ih < j)%int then - let quo := fst (diveucl_21 ih il j) in - if (quo < j)%int then - let m := - match j +c quo with - | C0 m1 => m1 >> 1 - | C1 m1 => (m1 >> 1 + 1 << (digits -1))%int - end in - rec ih il m - else j - else j. -Proof. - unfold sqrt2_step; case diveucl_21; intros;simpl. - case (j +c i);trivial. -Qed. - -Lemma sqrt2_lower_bound ih il j: - [|| WW ih il||] < ([|j|] + 1) ^ 2 -> [|ih|] <= [|j|]. -Proof. - intros H1. - case (to_Z_bounded j); intros Hbj _. - case (to_Z_bounded il); intros Hbil _. - case (to_Z_bounded ih); intros Hbih Hbih1. - assert (([|ih|] < [|j|] + 1)%Z); auto with zarith. - apply Zlt_square_simpl; auto with zarith. - simpl zn2z_to_Z in H1. - repeat rewrite <-Zpower_2; apply Z.le_lt_trans with (2 := H1). - apply Z.le_trans with ([|ih|] * wB)%Z;try rewrite Zpower_2; auto with zarith. -Qed. - - -Lemma div2_phi ih il j: - [|fst (diveucl_21 ih il j)|] = [|| WW ih il||] /[|j|]. -Proof. - generalize (diveucl_21_spec ih il j). - case diveucl_21; intros q r Heq. - simpl zn2z_to_Z;unfold Z.div;rewrite <- Heq;trivial. -Qed. - -Lemma zn2z_to_Z_pos ih il : 0 <= [||WW ih il||]. -Proof. - simpl zn2z_to_Z;destruct (to_Z_bounded ih);destruct (to_Z_bounded il);auto with zarith. -Qed. - - -Lemma sqrt2_step_correct rec ih il j: - 2 ^ (Z_of_nat (size - 2)) <= [|ih|] -> - 0 < [|j|] -> [|| WW ih il||] < ([|j|] + 1) ^ 2 -> - (forall j1, 0 < [|j1|] < [|j|] -> [|| WW ih il||] < ([|j1|] + 1) ^ 2 -> - [|rec ih il j1|] ^ 2 <= [||WW ih il||] < ([|rec ih il j1|] + 1) ^ 2) -> - [|sqrt2_step rec ih il j|] ^ 2 <= [||WW ih il ||] - < ([|sqrt2_step rec ih il j|] + 1) ^ 2. -Proof. - assert (Hp2: (0 < [|2|])%Z) by exact (refl_equal Lt). - intros Hih Hj Hij Hrec; rewrite sqrt2_step_def. - assert (H1: ([|ih|] <= [|j|])%Z) by (apply sqrt2_lower_bound with il; auto). - case (to_Z_bounded ih); intros Hih1 _. - case (to_Z_bounded il); intros Hil1 _. - case (to_Z_bounded j); intros _ Hj1. - assert (Hp3: (0 < [||WW ih il||])). - simpl zn2z_to_Z;apply Z.lt_le_trans with ([|ih|] * wB)%Z; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. - apply Z.lt_le_trans with (2:= Hih); auto with zarith. - cbv zeta. - case_eq (ih < j)%int;intros Heq. - rewrite ltb_spec in Heq. - 2: rewrite <-not_true_iff_false, ltb_spec in Heq. - 2: split; auto. - 2: apply sqrt_test_true; auto with zarith. - 2: unfold zn2z_to_Z; replace [|ih|] with [|j|]; auto with zarith. - 2: assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith). - 2: rewrite Zmult_comm, Z_div_plus_full_l; unfold base; auto with zarith. - case (Zle_or_lt (2^(Z_of_nat size -1)) [|j|]); intros Hjj. - case_eq (fst (diveucl_21 ih il j) < j)%int;intros Heq0. - 2: rewrite <-not_true_iff_false, ltb_spec, div2_phi in Heq0. - 2: split; auto; apply sqrt_test_true; auto with zarith. - rewrite ltb_spec, div2_phi in Heq0. - match goal with |- context[rec _ _ ?X] => - set (u := X) - end. - assert (H: [|u|] = ([|j|] + ([||WW ih il||])/([|j|]))/2). - unfold u; generalize (addc_spec j (fst (diveucl_21 ih il j))); - case addc;unfold interp_carry;rewrite div2_phi;simpl zn2z_to_Z. - intros i H;rewrite lsr_spec, H;trivial. - intros i H;rewrite <- H. - case (to_Z_bounded i); intros H1i H2i. - rewrite add_spec, Zmod_small, lsr_spec. - change (1 * wB) with ([|(1 << (digits -1))|] * 2)%Z. - rewrite Z_div_plus_full_l; auto with zarith. - change wB with (2 * (wB/2))%Z; auto. - replace [|(1 << (digits - 1))|] with (wB/2); auto. - rewrite lsr_spec; auto. - replace (2^[|1|]) with 2%Z; auto. - split. - { - apply Z.add_nonneg_nonneg. - - apply Z_div_pos. - + apply Zgt_pos_0. - + assumption. - - apply Z_div_pos. - + apply Zgt_pos_0. - + abstract omega. - } - assert ([|i|]/2 < wB/2); auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. - apply Hrec; rewrite H; clear u H. - assert (Hf1: 0 <= [||WW ih il||]/ [|j|]) by (apply Z_div_pos; auto with zarith). - case (Zle_lt_or_eq 1 ([|j|])); auto with zarith; intros Hf2. - 2: contradict Heq0; apply Zle_not_lt; rewrite <- Hf2, Zdiv_1_r; assert (H10: forall (x:Z), 0 < x -> 1 <= x) by (intros; omega); auto. - split. - replace ([|j|] + [||WW ih il||]/ [|j|])%Z with - (1 * 2 + (([|j|] - 2) + [||WW ih il||] / [|j|])); try ring. - rewrite Z_div_plus_full_l; auto with zarith. - assert (0 <= ([|j|] - 2 + [||WW ih il||] / [|j|]) / 2) ; auto with zarith. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - apply Z.add_nonneg_nonneg. - + abstract omega. - + assumption. - } - apply sqrt_test_false; auto with zarith. - apply sqrt_main; auto with zarith. - contradict Hij; apply Zle_not_lt. - assert ((1 + [|j|]) <= 2 ^ (Z_of_nat size - 1)); auto with zarith. - apply Z.le_trans with ((2 ^ (Z_of_nat size - 1)) ^2); auto with zarith. - assert (0 <= 1 + [|j|]); auto with zarith. - apply Zmult_le_compat; auto with zarith. - change ((2 ^ (Z_of_nat size - 1))^2) with (2 ^ (Z_of_nat size - 2) * wB). - apply Z.le_trans with ([|ih|] * wB); auto with zarith. - unfold zn2z_to_Z, wB; auto with zarith. -Qed. - - - -Lemma iter2_sqrt_correct n rec ih il j: - 2^(Z_of_nat (size - 2)) <= [|ih|] -> 0 < [|j|] -> [||WW ih il||] < ([|j|] + 1) ^ 2 -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> - [||WW ih il||] < ([|j1|] + 1) ^ 2 -> - [|rec ih il j1|] ^ 2 <= [||WW ih il||] < ([|rec ih il j1|] + 1) ^ 2) -> - [|iter2_sqrt n rec ih il j|] ^ 2 <= [||WW ih il||] - < ([|iter2_sqrt n rec ih il j|] + 1) ^ 2. -Proof. - revert rec ih il j; elim n; unfold iter2_sqrt; fold iter2_sqrt; clear n. - intros rec ih il j Hi Hj Hij Hrec; apply sqrt2_step_correct; auto with zarith. - intros; apply Hrec; auto with zarith. - rewrite Zpower_0_r; auto with zarith. - intros n Hrec rec ih il j Hi Hj Hij HHrec. - apply sqrt2_step_correct; auto. - intros j1 Hj1 Hjp1; apply Hrec; auto with zarith. - intros j2 Hj2 H2j2 Hjp2; apply Hrec; auto with zarith. - intros j3 Hj3 Hpj3. - apply HHrec; auto. - rewrite inj_S, Zpower_Zsucc. - apply Z.le_trans with (2 ^Z_of_nat n + [|j2|])%Z; auto with zarith. - apply Zle_0_nat. -Qed. - - -Lemma sqrt2_spec : forall x y, - wB/ 4 <= [|x|] -> - let (s,r) := sqrt2 x y in - [||WW x y||] = [|s|] ^ 2 + [+|r|] /\ - [+|r|] <= 2 * [|s|]. - Proof. - intros ih il Hih; unfold sqrt2. - change [||WW ih il||] with ([||WW ih il||]). - assert (Hbin: forall s, s * s + 2* s + 1 = (s + 1) ^ 2) by - (intros s; ring). - assert (Hb: 0 <= wB) by (red; intros HH; discriminate). - assert (Hi2: [||WW ih il ||] < ([|max_int|] + 1) ^ 2). - apply Z.le_lt_trans with ((wB - 1) * wB + (wB - 1)); auto with zarith. - 2: apply refl_equal. - case (to_Z_bounded ih); case (to_Z_bounded il); intros H1 H2 H3 H4. - unfold zn2z_to_Z; auto with zarith. - case (iter2_sqrt_correct size (fun _ _ j => j) ih il max_int); auto with zarith. - apply refl_equal. - intros j1 _ HH; contradict HH. - apply Zlt_not_le. - case (to_Z_bounded j1); auto with zarith. - change (2 ^ Z_of_nat size) with ([|max_int|]+1)%Z; auto with zarith. - set (s := iter2_sqrt size (fun _ _ j : int=> j) ih il max_int). - intros Hs1 Hs2. - generalize (mulc_spec s s); case mulc. - simpl fst; simpl snd; intros ih1 il1 Hihl1. - generalize (subc_spec il il1). - case subc; intros il2 Hil2. - simpl interp_carry in Hil2. - case_eq (ih1 < ih)%int; [idtac | rewrite <- not_true_iff_false]; - rewrite ltb_spec; intros Heq. - unfold interp_carry; rewrite Zmult_1_l. - rewrite Zpower_2, Hihl1, Hil2. - case (Zle_lt_or_eq ([|ih1|] + 1) ([|ih|])); auto with zarith. - intros H2; contradict Hs2; apply Zle_not_lt. - replace (([|s|] + 1) ^ 2) with ([||WW ih1 il1||] + 2 * [|s|] + 1). - unfold zn2z_to_Z. - case (to_Z_bounded il); intros Hpil _. - assert (Hl1l: [|il1|] <= [|il|]). - case (to_Z_bounded il2); rewrite Hil2; auto with zarith. - assert ([|ih1|] * wB + 2 * [|s|] + 1 <= [|ih|] * wB); auto with zarith. - case (to_Z_bounded s); intros _ Hps. - case (to_Z_bounded ih1); intros Hpih1 _; auto with zarith. - apply Z.le_trans with (([|ih1|] + 2) * wB); auto with zarith. - rewrite Zmult_plus_distr_l. - assert (2 * [|s|] + 1 <= 2 * wB); auto with zarith. - unfold zn2z_to_Z; rewrite <-Hihl1, Hbin; auto. - intros H2; split. - unfold zn2z_to_Z; rewrite <- H2; ring. - replace (wB + ([|il|] - [|il1|])) with ([||WW ih il||] - ([|s|] * [|s|])). - rewrite <-Hbin in Hs2; assert (([||WW ih il||] < [|s|] * [|s|] + 2 * [|s|] + 1) -> ([||WW ih il||] - [|s|] * [|s|] <= 2 * [|s|])) by omega; auto. - rewrite Hihl1; unfold zn2z_to_Z; rewrite <- H2; ring. - unfold interp_carry. - case (Zle_lt_or_eq [|ih|] [|ih1|]); auto with zarith; intros H. - contradict Hs1. - apply Zlt_not_le; rewrite Zpower_2, Hihl1. - unfold zn2z_to_Z. - case (to_Z_bounded il); intros _ H2. - apply Z.lt_le_trans with (([|ih|] + 1) * wB + 0). - rewrite Zmult_plus_distr_l, Zplus_0_r; auto with zarith. - case (to_Z_bounded il1); intros H3 _. - apply Zplus_le_compat; auto with zarith. - split. - rewrite Zpower_2, Hihl1. - unfold zn2z_to_Z; ring[Hil2 H]. - replace [|il2|] with ([||WW ih il||] - [||WW ih1 il1||]). - unfold zn2z_to_Z at 2; rewrite <-Hihl1. - rewrite <-Hbin in Hs2; assert (([||WW ih il||] < [|s|] * [|s|] + 2 * [|s|] + 1) -> ([||WW ih il||] - [|s|] * [|s|] <= 2 * [|s|])) by omega; auto. - unfold zn2z_to_Z; rewrite H, Hil2; ring. - unfold interp_carry in Hil2 |- *. - assert (Hsih: [|ih - 1|] = [|ih|] - 1). - rewrite sub_spec, Zmod_small; auto; replace [|1|] with 1; auto. - case (to_Z_bounded ih); intros H1 H2. - split; auto with zarith. - apply Z.le_trans with (wB/4 - 1); auto with zarith. - case_eq (ih1 < ih - 1)%int; [idtac | rewrite <- not_true_iff_false]; - rewrite ltb_spec, Hsih; intros Heq. - rewrite Zpower_2, Hihl1. - case (Zle_lt_or_eq ([|ih1|] + 2) [|ih|]); auto with zarith. - intros H2; contradict Hs2; apply Zle_not_lt. - replace (([|s|] + 1) ^ 2) with ([||WW ih1 il1||] + 2 * [|s|] + 1). - unfold zn2z_to_Z. - assert ([|ih1|] * wB + 2 * [|s|] + 1 <= [|ih|] * wB + ([|il|] - [|il1|])); - auto with zarith. - rewrite <-Hil2. - case (to_Z_bounded il2); intros Hpil2 _. - apply Z.le_trans with ([|ih|] * wB + - wB); auto with zarith. - case (to_Z_bounded s); intros _ Hps. - assert (2 * [|s|] + 1 <= 2 * wB); auto with zarith. - apply Z.le_trans with ([|ih1|] * wB + 2 * wB); auto with zarith. - assert (Hi: ([|ih1|] + 3) * wB <= [|ih|] * wB); auto with zarith. - rewrite Zmult_plus_distr_l in Hi; auto with zarith. - unfold zn2z_to_Z; rewrite <-Hihl1, Hbin; auto. - intros H2; unfold zn2z_to_Z; rewrite <-H2. - split. - replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring. - rewrite <-Hil2; ring. - replace (1 * wB + [|il2|]) with ([||WW ih il||] - [||WW ih1 il1||]). - unfold zn2z_to_Z at 2; rewrite <-Hihl1. - rewrite <-Hbin in Hs2; assert (([||WW ih il||] < [|s|] * [|s|] + 2 * [|s|] + 1) -> ([||WW ih il||] - [|s|] * [|s|] <= 2 * [|s|])) by omega; auto. - unfold zn2z_to_Z; rewrite <-H2. - replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring. - rewrite <-Hil2; ring. - case (Zle_lt_or_eq ([|ih|] - 1) ([|ih1|])); auto with zarith; intros H1. - assert (He: [|ih|] = [|ih1|]). - apply Zle_antisym; auto with zarith. - case (Zle_or_lt [|ih1|] [|ih|]); auto; intros H2. - contradict Hs1; apply Zlt_not_le; rewrite Zpower_2, Hihl1. - unfold zn2z_to_Z. - case (to_Z_bounded il); intros _ Hpil1. - apply Z.lt_le_trans with (([|ih|] + 1) * wB). - rewrite Zmult_plus_distr_l, Zmult_1_l; auto with zarith. - case (to_Z_bounded il1); intros Hpil2 _. - apply Z.le_trans with (([|ih1|]) * wB); auto with zarith. - contradict Hs1; apply Zlt_not_le; rewrite Zpower_2, Hihl1. - unfold zn2z_to_Z; rewrite He. - assert ([|il|] - [|il1|] < 0); auto with zarith. - rewrite <-Hil2. - case (to_Z_bounded il2); auto with zarith. - split. - rewrite Zpower_2, Hihl1. - unfold zn2z_to_Z; rewrite <-H1. - apply trans_equal with ([|ih|] * wB + [|il1|] + ([|il|] - [|il1|])). - ring. - rewrite <-Hil2; ring. - replace [|il2|] with ([||WW ih il||] - [||WW ih1 il1||]). - unfold zn2z_to_Z at 2; rewrite <- Hihl1. - rewrite <-Hbin in Hs2; assert (([||WW ih il||] < [|s|] * [|s|] + 2 * [|s|] + 1) -> ([||WW ih il||] - [|s|] * [|s|] <= 2 * [|s|])) by omega; auto. - unfold zn2z_to_Z. - rewrite <-H1. - ring_simplify. - apply trans_equal with (wB + ([|il|] - [|il1|])). - ring. - rewrite <-Hil2; ring. -Qed. - -Lemma to_Z_gcd : forall i j, - [|gcd i j|] = Zgcdn (2*size) [|j|] [|i|]. -Proof. - unfold gcd. - induction (2*size)%nat; intros. - reflexivity. - simpl. - generalize (to_Z_bounded j)(to_Z_bounded i); intros. - case_eq (j == 0)%int. - rewrite eqb_spec;intros H1;rewrite H1. - replace [|0|] with 0;trivial;rewrite Z.abs_eq;auto with zarith. - rewrite <- not_true_iff_false, eqb_spec;intros. - case_eq [|j|]; intros. - elim H1;apply to_Z_inj;assumption. - rewrite IHn, <- H2, mod_spec;trivial. - rewrite H2 in H;destruct H as (H, _);elim H;trivial. -Qed. - -Lemma gcd_spec : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|]. -Proof. - intros. - rewrite to_Z_gcd. - apply Zis_gcd_sym. - apply Zgcdn_is_gcd. - unfold Zgcd_bound. - generalize (to_Z_bounded b). - destruct [|b|]. - unfold size; intros _; change Int31.size with 31%nat; omega. - intros (_,H). - cut (Psize p <= size)%nat; [ omega | rewrite <- Zpower2_Psize; auto]. - intros (H,_); compute in H; elim H; auto. -Qed. - -Lemma head00_spec: forall x, [|x|] = 0 -> [|head0 x|] = [|digits|]. -Proof. - change 0 with [|0|];intros x Heq. - apply to_Z_inj in Heq;rewrite Heq;trivial. -Qed. - -Lemma tail00_spec: forall x, [|x|] = 0 -> [|tail0 x|] = [|digits|]. -Proof. - change 0 with [|0|];intros x Heq. - apply to_Z_inj in Heq;rewrite Heq;trivial. -Qed. - -(* lsr lsl *) -Lemma lsl_0_l i: 0 << i = 0%int. -Proof. - apply to_Z_inj. - generalize (lsl_spec 0 i). - rewrite to_Z_0, Zmult_0_l, Zmod_0_l; auto. -Qed. - -Lemma lsl_0_r i: i << 0 = i. -Proof. - apply to_Z_inj. - rewrite lsl_spec, to_Z_0, Zmult_1_r. - apply Zmod_small; apply (to_Z_bounded i). -Qed. - -Lemma lsl_M_r x i (H: (digits <= i = true)%int) : x << i = 0%int. -Proof. - apply to_Z_inj. - rewrite lsl_spec, to_Z_0. - rewrite leb_spec in H. - unfold wB; change (Z_of_nat size) with [|digits|]. - replace ([|i|]) with (([|i|] - [|digits|]) + [|digits|])%Z; try ring. - rewrite Zpower_exp, Zmult_assoc, Z_mod_mult; auto with arith. - apply Z.le_ge; auto with zarith. - case (to_Z_bounded digits); auto with zarith. -Qed. - -Lemma lsr_0_l i: 0 >> i = 0%int. -Proof. - apply to_Z_inj. - generalize (lsr_spec 0 i). - rewrite to_Z_0, Zdiv_0_l; auto. -Qed. - -Lemma lsr_0_r i: i >> 0 = i. -Proof. - apply to_Z_inj. - rewrite lsr_spec, to_Z_0, Zdiv_1_r; auto. -Qed. - -Lemma lsr_M_r x i (H: (digits <= i = true)%int) : x >> i = 0%int. -Proof. - apply to_Z_inj. - rewrite lsr_spec, to_Z_0. - case (to_Z_bounded x); intros H1x H2x. - case (to_Z_bounded digits); intros H1d H2d. - rewrite leb_spec in H. - apply Zdiv_small; split; auto. - apply Z.lt_le_trans with (1 := H2x). - unfold wB; change (Z_of_nat size) with [|digits|]. - apply Zpower_le_monotone; auto with zarith. -Qed. - -Lemma add_le_r m n: - if (n <= m + n)%int then ([|m|] + [|n|] < wB)%Z else (wB <= [|m|] + [|n|])%Z. -Proof. - case (to_Z_bounded m); intros H1m H2m. - case (to_Z_bounded n); intros H1n H2n. - case (Zle_or_lt wB ([|m|] + [|n|])); intros H. - assert (H1: ([| m + n |] = [|m|] + [|n|] - wB)%Z). - rewrite add_spec. - replace (([|m|] + [|n|]) mod wB)%Z with (((([|m|] + [|n|]) - wB) + wB) mod wB)%Z. - rewrite Zplus_mod, Z_mod_same_full, Zplus_0_r, !Zmod_small; auto with zarith. - rewrite !Zmod_small; auto with zarith. - apply f_equal2 with (f := Zmod); auto with zarith. - case_eq (n <= m + n)%int; auto. - rewrite leb_spec, H1; auto with zarith. - assert (H1: ([| m + n |] = [|m|] + [|n|])%Z). - rewrite add_spec, Zmod_small; auto with zarith. - replace (n <= m + n)%int with true; auto. - apply sym_equal; rewrite leb_spec, H1; auto with zarith. -Qed. - -Lemma lsr_add i m n: ((i >> m) >> n = if n <= m + n then i >> (m + n) else 0)%int. -Proof. - case (to_Z_bounded m); intros H1m H2m. - case (to_Z_bounded n); intros H1n H2n. - case (to_Z_bounded i); intros H1i H2i. - generalize (add_le_r m n); case (n <= m + n)%int; intros H. - apply to_Z_inj; rewrite !lsr_spec, Zdiv_Zdiv, <- Zpower_exp; auto with zarith. - rewrite add_spec, Zmod_small; auto with zarith. - apply to_Z_inj; rewrite !lsr_spec, Zdiv_Zdiv, <- Zpower_exp; auto with zarith. - apply Zdiv_small; split; auto with zarith. - apply Z.lt_le_trans with (1 := H2i). - apply Z.le_trans with (1 := H). - apply Zpower2_le_lin; auto with zarith. -Qed. - -Lemma lsl_add i m n: ((i << m) << n = if n <= m + n then i << (m + n) else 0)%int. -Proof. - case (to_Z_bounded m); intros H1m H2m. - case (to_Z_bounded n); intros H1n H2n. - case (to_Z_bounded i); intros H1i H2i. - generalize (add_le_r m n); case (n <= m + n)%int; intros H. - apply to_Z_inj; rewrite !lsl_spec, Zmult_mod, Zmod_mod, <- Zmult_mod. - rewrite <-Zmult_assoc, <- Zpower_exp; auto with zarith. - apply f_equal2 with (f := Zmod); auto. - rewrite add_spec, Zmod_small; auto with zarith. - apply to_Z_inj; rewrite !lsl_spec, Zmult_mod, Zmod_mod, <- Zmult_mod. - rewrite <-Zmult_assoc, <- Zpower_exp; auto with zarith. - unfold wB. - replace ([|m|] + [|n|])%Z with - ((([|m|] + [|n|]) - Z_of_nat size) + Z_of_nat size)%Z. - 2: ring. - rewrite Zpower_exp, Zmult_assoc, Z_mod_mult; auto with zarith. - assert (Z_of_nat size < wB)%Z; auto with zarith. - apply Zpower2_lt_lin; auto with zarith. -Qed. - - -Coercion b2i (b: bool) : int := if b then 1%int else 0%int. - -Lemma bit_0 n : bit 0 n = false. -Proof. unfold bit; rewrite lsr_0_l; auto. Qed. - -Lemma lsr_1 n : 1 >> n = (n == 0). -Proof. - case_eq (n == 0). - rewrite eqb_spec; intros H; rewrite H, lsr_0_r. - apply refl_equal. - intros Hn. - assert (H1n : (1 >> n = 0)%int); auto. - apply to_Z_inj; rewrite lsr_spec. - apply Zdiv_small; rewrite to_Z_1; split; auto with zarith. - change 1%Z with (2^0)%Z. - apply Zpower_lt_monotone; split; auto with zarith. - case (Zle_lt_or_eq 0 [|n|]); auto. - case (to_Z_bounded n); auto. - intros H1. - assert ((n == 0) = true). - rewrite eqb_spec; apply to_Z_inj; rewrite <-H1, to_Z_0; auto. - generalize H; rewrite Hn; discriminate. -Qed. - -Lemma bit_1 n : bit 1 n = (n == 0). -Proof. - unfold bit; rewrite lsr_1. - case (n == 0). - apply refl_equal. - rewrite lsl_0_l; apply refl_equal. -Qed. - -Lemma bit_M i n (H: (digits <= n = true)%int): bit i n = false. -Proof. unfold bit; rewrite lsr_M_r; auto. Qed. - -Lemma bit_half i n (H: (n < digits = true)%int) : bit (i>>1) n = bit i (n+1). -Proof. - unfold bit. - rewrite lsr_add. - case_eq (n <= (1 + n))%int. - replace (1+n)%int with (n+1)%int; [auto|idtac]. - apply to_Z_inj; rewrite !add_spec, Zplus_comm; auto. - intros H1; assert (H2: n = max_int). - 2: generalize H; rewrite H2; discriminate. - case (to_Z_bounded n); intros H1n H2n. - case (Zle_lt_or_eq [|n|] (wB - 1)); auto with zarith; - intros H2; apply to_Z_inj; auto. - generalize (add_le_r 1 n); rewrite H1. - change [|max_int|] with (wB - 1)%Z. - replace [|1|] with 1%Z; auto with zarith. -Qed. - -Lemma bit_0_spec i: [|bit i 0|] = [|i|] mod 2. -Proof. - unfold bit, is_zero; rewrite lsr_0_r. - assert (Hbi: ([|i|] mod 2 < 2)%Z). - apply Z_mod_lt; auto with zarith. - case (to_Z_bounded i); intros H1i H2i. - case (Zmod_le_first [|i|] 2); auto with zarith; intros H3i H4i. - assert (H2b: (0 < 2 ^ [|digits - 1|])%Z). - apply Zpower_gt_0; auto with zarith. - case (to_Z_bounded (digits -1)); auto with zarith. - assert (H: [|i << (digits -1)|] = ([|i|] mod 2 * 2^ [|digits -1|])%Z). - rewrite lsl_spec. - rewrite (Z_div_mod_eq [|i|] 2) at 1; auto with zarith. - rewrite Zmult_plus_distr_l, <-Zplus_mod_idemp_l. - rewrite (Zmult_comm 2), <-Zmult_assoc. - replace (2 * 2 ^ [|digits - 1|])%Z with wB; auto. - rewrite Z_mod_mult, Zplus_0_l; apply Zmod_small. - split; auto with zarith. - replace wB with (2 * 2 ^ [|digits -1|])%Z; auto. - apply Zmult_lt_compat_r; auto with zarith. - case (Zle_lt_or_eq 0 ([|i|] mod 2)); auto with zarith; intros Hi. - 2: generalize H; rewrite <-Hi, Zmult_0_l. - 2: replace 0%Z with [|0|]; auto. - 2: rewrite to_Z_eq, <-eqb_spec; intros H1; rewrite H1; auto. - generalize H; replace ([|i|] mod 2) with 1%Z; auto with zarith. - rewrite Zmult_1_l. - intros H1. - assert (H2: [|i << (digits - 1)|] <> [|0|]). - replace [|0|] with 0%Z; auto with zarith. - generalize (eqb_spec (i << (digits - 1)) 0). - case (i << (digits - 1) == 0); auto. - intros (H3,_); case H2. - rewrite to_Z_eq; auto. -Qed. - -Lemma bit_split i : (i = (i>>1)<<1 + bit i 0)%int. -Proof. - apply to_Z_inj. - rewrite add_spec, lsl_spec, lsr_spec, bit_0_spec, Zplus_mod_idemp_l. - replace (2 ^ [|1|]) with 2%Z; auto with zarith. - rewrite Zmult_comm, <-Z_div_mod_eq; auto with zarith. - rewrite Zmod_small; auto; case (to_Z_bounded i); auto. -Qed. - - -Lemma bit_eq i1 i2: - i1 = i2 <-> forall i, bit i1 i = bit i2 i. -Admitted. (* Too slow *) -(* Proof. *) -(* split; try (intros; subst; auto; fail). *) -(* case (to_Z_bounded i2); case (to_Z_bounded i1). *) -(* unfold wB; generalize i1 i2; elim size; clear i1 i2. *) -(* replace (2^Z_of_nat 0) with 1%Z; auto with zarith. *) -(* intros; apply to_Z_inj; auto with zarith. *) -(* intros n IH i1 i2 H1i1 H2i1 H1i2 H2i2 H. *) -(* rewrite (bit_split i1), (bit_split i2). *) -(* rewrite H. *) -(* apply f_equal2 with (f := add31); auto. *) -(* apply f_equal2 with (f := lsl); auto. *) -(* apply IH; try rewrite lsr_spec; *) -(* replace (2^[|1|]) with 2%Z; auto with zarith. *) -(* apply Zdiv_lt_upper_bound; auto with zarith. *) -(* generalize H2i1; rewrite inj_S. *) -(* unfold Z.succ; rewrite Zpower_exp; auto with zarith. *) -(* apply Zdiv_lt_upper_bound; auto with zarith. *) -(* generalize H2i2; rewrite inj_S. *) -(* unfold Z.succ; rewrite Zpower_exp; auto with zarith. *) -(* intros i. *) -(* case (Zle_or_lt [|digits|] [|i|]); intros Hi. *) -(* rewrite !bit_M; auto; rewrite leb_spec; auto. *) -(* rewrite !bit_half; auto; rewrite ltb_spec; auto with zarith. *) -(* Qed. *) - -Lemma bit_lsr x i j : - (bit (x >> i) j = if j <= i + j then bit x (i + j) else false)%int. -Proof. - unfold bit; rewrite lsr_add; case leb; auto. -Qed. - -Lemma bit_lsl x i j : bit (x << i) j = -(if (j < i) || (digits <= j) then false else bit x (j - i))%int. -Proof. - assert (F1: 1 >= 0) by discriminate. - case_eq (digits <= j)%int; intros H. - rewrite orb_true_r, bit_M; auto. - set (d := [|digits|]). - case (Zle_or_lt d [|j|]); intros H1. - case (leb_spec digits j); rewrite H; auto with zarith. - intros _ HH; generalize (HH H1); discriminate. - clear H. - generalize (ltb_spec j i); case ltb; intros H2; unfold bit; [change (if true || false then false else negb (is_zero ((x >> (j - i)) << (digits - 1)))) with false | change (if false || false then false else negb (is_zero ((x >> (j - i)) << (digits - 1)))) with (negb (is_zero ((x >> (j - i)) << (digits - 1))))]. - assert (F2: ([|j|] < [|i|])%Z) by (case H2; auto); clear H2. - replace (is_zero (((x << i) >> j) << (digits - 1))) with true; auto. - case (to_Z_bounded j); intros H1j H2j. - apply sym_equal; rewrite is_zero_spec; apply to_Z_inj. - rewrite lsl_spec, lsr_spec, lsl_spec. - replace wB with (2^d); auto. - pattern d at 1; replace d with ((d - ([|j|] + 1)) + ([|j|] + 1))%Z. - 2: ring. - rewrite Zpower_exp; auto with zarith. - replace [|i|] with (([|i|] - ([|j|] + 1)) + ([|j|] + 1))%Z. - 2: ring. - rewrite Zpower_exp, Zmult_assoc; auto with zarith. - rewrite Zmult_mod_distr_r. - rewrite Zplus_comm, Zpower_exp, !Zmult_assoc; auto with zarith. - rewrite Z_div_mult_full; auto with zarith. - 2: assert (0 < 2 ^ [|j|])%Z; auto with zarith. - rewrite <-Zmult_assoc, <-Zpower_exp; auto with zarith. - replace (1 + [|digits - 1|])%Z with d; auto with zarith. - rewrite Z_mod_mult; auto. - case H2; intros _ H3; case (Zle_or_lt [|i|] [|j|]); intros F2. - 2: generalize (H3 F2); discriminate. - clear H2 H3. - apply f_equal with (f := negb). - apply f_equal with (f := is_zero). - apply to_Z_inj. - rewrite !lsl_spec, !lsr_spec, !lsl_spec. - pattern wB at 2 3; replace wB with (2^(1+ [|digits - 1|])); auto. - rewrite Zpower_exp, Zpower_1_r; auto with zarith. - rewrite !Zmult_mod_distr_r. - apply f_equal2 with (f := Zmult); auto. - replace wB with (2^ d); auto with zarith. - replace d with ((d - [|i|]) + [|i|])%Z. - 2: ring. - case (to_Z_bounded i); intros H1i H2i. - rewrite Zpower_exp; [ |apply Z.le_ge; lia|apply Z.le_ge; assumption]. - rewrite Zmult_mod_distr_r. - case (to_Z_bounded j); intros H1j H2j. - replace [|j - i|] with ([|j|] - [|i|])%Z. - 2: rewrite sub_spec, Zmod_small; auto with zarith. - set (d1 := (d - [|i|])%Z). - set (d2 := ([|j|] - [|i|])%Z). - pattern [|j|] at 1; - replace [|j|] with (d2 + [|i|])%Z. - 2: unfold d2; ring. - rewrite Zpower_exp; auto with zarith. - rewrite Zdiv_mult_cancel_r. - 2: (apply Zlt0_not_eq; apply Z.pow_pos_nonneg; [apply Pos2Z.is_pos|assumption]). - rewrite (Z_div_mod_eq [|x|] (2^d1)) at 2; auto with zarith. - 2: apply Z.lt_gt; apply Zpower_gt_0; unfold d1; lia. - pattern d1 at 2; - replace d1 with (d2 + (1+ (d - [|j|] - 1)))%Z. - 2: unfold d1, d2; ring. - rewrite Zpower_exp; auto with zarith. - rewrite <-Zmult_assoc, Zmult_comm. - rewrite Z_div_plus_l; auto with zarith. - rewrite Zpower_exp, Zpower_1_r; auto with zarith. - rewrite <-Zplus_mod_idemp_l. - rewrite <-!Zmult_assoc, Zmult_comm, Z_mod_mult, Zplus_0_l; auto. -Qed. - - -Lemma bit_b2i (b: bool) i : bit b i = (i == 0) && b. -Proof. - case b; unfold bit; simpl b2i. - 2: rewrite lsr_0_l, lsl_0_l, andb_false_r; auto. - rewrite lsr_1; case (i == 0); auto. -Qed. - -Lemma bit_or_split i : (i = (i>>1)<<1 lor bit i 0)%int. -Proof. - rewrite bit_eq. - intros n; rewrite lor_spec. - rewrite bit_lsl, bit_lsr, bit_b2i. - case (to_Z_bounded n); intros Hi _. - case (Zle_lt_or_eq _ _ Hi). - 2: replace 0%Z with [|0|]; auto; rewrite to_Z_eq. - 2: intros H; rewrite <-H. - 2: replace (0 < 1)%int with true; auto. - intros H; clear Hi. - case_eq (n == 0). - rewrite eqb_spec; intros H1; generalize H; rewrite H1; discriminate. - intros _; rewrite orb_false_r. - case_eq (n < 1)%int. - rewrite ltb_spec, to_Z_1; intros HH; contradict HH; auto with zarith. - intros _. - generalize (@bit_M i n); case leb. - intros H1; rewrite H1; auto. - intros _. - case (to_Z_bounded n); intros H1n H2n. - assert (F1: [|n - 1|] = ([|n|] - 1)%Z). - rewrite sub_spec, Zmod_small; rewrite to_Z_1; auto with zarith. - generalize (add_le_r 1 (n - 1)); case leb; rewrite F1, to_Z_1; intros HH. - replace (1 + (n -1))%int with n. change (bit i n = bit i n). reflexivity. - apply to_Z_inj; rewrite add_spec, F1, Zmod_small; rewrite to_Z_1; - auto with zarith. - rewrite bit_M; auto; rewrite leb_spec. - replace [|n|] with wB; try discriminate; auto with zarith. -Qed. - -(* is_zero *) -Lemma is_zero_0: is_zero 0 = true. -Proof. apply refl_equal. Qed. - -(* is_even *) -Lemma is_even_bit i : is_even i = negb (bit i 0). -Proof. - unfold is_even. - replace (i land 1) with (b2i (bit i 0)). - case bit; auto. - apply bit_eq; intros n. - rewrite bit_b2i, land_spec, bit_1. - generalize (eqb_spec n 0). - case (n == 0); auto. - intros(H,_); rewrite andb_true_r, H; auto. - rewrite andb_false_r; auto. -Qed. - -Lemma is_even_0: is_even 0 = true. -Proof. apply refl_equal. Qed. - -Lemma is_even_lsl_1 i: is_even (i << 1) = true. -Proof. - rewrite is_even_bit, bit_lsl; auto. -Qed. - -Lemma is_even_spec : forall x, - if is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1. -Proof. -intros x; rewrite is_even_bit. -generalize (bit_0_spec x); case bit; simpl; auto. -Qed. - -(* More land *) - -Lemma land_0_l i: 0 land i = 0%int. -Proof. - apply bit_eq; intros n. - rewrite land_spec, bit_0; auto. -Qed. - -Lemma land_0_r i: i land 0 = 0%int. -Proof. - apply bit_eq; intros n. - rewrite land_spec, bit_0, andb_false_r; auto. -Qed. - -Lemma land_assoc i1 i2 i3 : - i1 land (i2 land i3) = i1 land i2 land i3. -Proof. - apply bit_eq; intros n. - rewrite !land_spec, andb_assoc; auto. -Qed. - - -Lemma land_comm i j : i land j = j land i. -Proof. - apply bit_eq; intros n. - rewrite !land_spec, andb_comm; auto. -Qed. - -Lemma lor_comm i1 i2 : i1 lor i2 = i2 lor i1. -Proof. - apply bit_eq; intros n. - rewrite !lor_spec, orb_comm; auto. -Qed. - -Lemma lor_assoc i1 i2 i3 : - i1 lor (i2 lor i3) = i1 lor i2 lor i3. -Proof. - apply bit_eq; intros n. - rewrite !lor_spec, orb_assoc; auto. -Qed. - -Lemma land_lor_distrib_r i1 i2 i3 : - i1 land (i2 lor i3) = (i1 land i2) lor (i1 land i3). -Proof. - apply bit_eq; intros n. - rewrite !land_spec, !lor_spec, !land_spec, andb_orb_distrib_r; auto. -Qed. - -Lemma land_lor_distrib_l i1 i2 i3 : - (i1 lor i2) land i3 = (i1 land i3) lor (i2 land i3). -Proof. - apply bit_eq; intros n. - rewrite !land_spec, !lor_spec, !land_spec, andb_orb_distrib_l; auto. -Qed. - -Lemma lor_land_distrib_r i1 i2 i3: - i1 lor (i2 land i3) = (i1 lor i2) land (i1 lor i3). -Proof. - apply bit_eq; intros n. - rewrite !land_spec, !lor_spec, !land_spec, orb_andb_distrib_r; auto. -Qed. - -Lemma lor_land_distrib_l i1 i2 i3: - (i1 land i2) lor i3 = (i1 lor i3) land (i2 lor i3). -Proof. - apply bit_eq; intros n. - rewrite !land_spec, !lor_spec, !land_spec, orb_andb_distrib_l; auto. -Qed. - -Lemma absoption_land i1 i2 : i1 land (i1 lor i2) = i1. -Proof. - apply bit_eq; intros n. - rewrite land_spec, lor_spec, absoption_andb; auto. -Qed. - -Lemma absoption_lor i1 i2: i1 lor (i1 land i2) = i1. -Proof. - apply bit_eq; intros n. - rewrite lor_spec, land_spec, absoption_orb; auto. -Qed. - -Lemma land_lsl i1 i2 i: (i1 land i2) << i = (i1 << i) land (i2 << i). -Proof. - apply bit_eq; intros n. - rewrite land_spec, !bit_lsl, land_spec. - case (_ || _); auto. -Qed. - -Lemma lor_lsl i1 i2 i: (i1 lor i2) << i = (i1 << i) lor (i2 << i). -Proof. - apply bit_eq; intros n. - rewrite lor_spec, !bit_lsl, lor_spec. - case (_ || _); auto. -Qed. - -Lemma lxor_lsl i1 i2 i: (i1 lxor i2) << i = (i1 << i) lxor (i2 << i). -Proof. - apply bit_eq; intros n. - rewrite lxor_spec, !bit_lsl, lxor_spec. - case (_ || _); auto. -Qed. - -Lemma land_lsr i1 i2 i: (i1 land i2) >> i = (i1 >> i) land (i2 >> i). -Proof. - apply bit_eq; intros n. - rewrite land_spec, !bit_lsr, land_spec. - case (_ <= _)%int; auto. -Qed. - -Lemma lor_lsr i1 i2 i: (i1 lor i2) >> i = (i1 >> i) lor (i2 >> i). -Proof. - apply bit_eq; intros n. - rewrite lor_spec, !bit_lsr, lor_spec. - case (_ <= _)%int; auto. -Qed. - -Lemma lxor_lsr i1 i2 i: (i1 lxor i2) >> i = (i1 >> i) lxor (i2 >> i). -Proof. - apply bit_eq; intros n. - rewrite lxor_spec, !bit_lsr, lxor_spec. - case (_ <= _)%int; auto. -Qed. - -Lemma is_even_and i j : is_even (i land j) = is_even i || is_even j. -Proof. - rewrite !is_even_bit, land_spec; case bit; auto. -Qed. - -Lemma is_even_or i j : is_even (i lor j) = is_even i && is_even j. -Proof. - rewrite !is_even_bit, lor_spec; case bit; auto. -Qed. - -Lemma is_even_xor i j : is_even (i lxor j) = negb (xorb (is_even i) (is_even j)). -Proof. - rewrite !is_even_bit, lxor_spec; do 2 case bit; auto. -Qed. - -Lemma lsl_add_distr x y n: (x + y) << n = ((x << n) + (y << n))%int. -Proof. - apply to_Z_inj; rewrite !lsl_spec, !add_spec, Zmult_mod_idemp_l. - rewrite !lsl_spec, <-Zplus_mod. - apply f_equal2 with (f := Zmod); auto with zarith. -Qed. - -Lemma add_assoc x y z: (x + (y + z) = (x + y) + z)%int. -Proof. - apply to_Z_inj; rewrite !add_spec. - rewrite Zplus_mod_idemp_l, Zplus_mod_idemp_r, Zplus_assoc; auto. -Qed. - -Lemma add_comm x y: (x + y = y + x)%int. -Proof. - apply to_Z_inj; rewrite !add_spec, Zplus_comm; auto. -Qed. - -Lemma lsr_add_distr x y n: (x + y) << n = ((x << n) + (y << n))%int. -Proof. - apply to_Z_inj. - rewrite add_spec, !lsl_spec, add_spec. - rewrite Zmult_mod_idemp_l, <-Zplus_mod. - apply f_equal2 with (f := Zmod); auto with zarith. -Qed. - -Lemma is_even_add x y : - is_even (x + y) = negb (xorb (negb (is_even x)) (negb (is_even y))). -Proof. - assert (F : [|x + y|] mod 2 = ([|x|] mod 2 + [|y|] mod 2) mod 2). - assert (F1: (2 | wB)) by (apply Zpower_divide; apply refl_equal). - assert (F2: 0 < wB) by (apply refl_equal). - case (to_Z_bounded x); intros H1x H2x. - case (to_Z_bounded y); intros H1y H2y. - rewrite add_spec, <-Zmod_div_mod; auto with zarith. - rewrite (Z_div_mod_eq [|x|] 2) at 1; auto with zarith. - rewrite (Z_div_mod_eq [|y|] 2) at 1; auto with zarith. - rewrite Zplus_mod. - rewrite Zmult_comm, (fun x => Zplus_comm (x * 2)), Z_mod_plus; auto with zarith. - rewrite Zmult_comm, (fun x => Zplus_comm (x * 2)), Z_mod_plus; auto with zarith. - rewrite !Zmod_mod, <-Zplus_mod; auto. - generalize (is_even_spec (x + y)) (is_even_spec x) (is_even_spec y). - do 3 case is_even; auto; rewrite F; intros H1 H2 H3; - generalize H1; rewrite H2, H3; try discriminate. -Qed. - -Lemma bit_add_0 x y: bit (x + y) 0 = xorb (bit x 0) (bit y 0). -Proof. - rewrite <-(fun x => (negb_involutive (bit x 0))). - rewrite <-is_even_bit, is_even_add, !is_even_bit. - do 2 case bit; auto. -Qed. - -Lemma add_cancel_l x y z : (x + y = x + z)%int -> y = z. -Proof. - intros H; case (to_Z_bounded x); case (to_Z_bounded y); case (to_Z_bounded z); - intros H1z H2z H1y H2y H1x H2x. - generalize (add_le_r y x) (add_le_r z x); rewrite (add_comm y x), H, (add_comm z x). - case_eq (x <= x + z)%int; intros H1 H2 H3. - apply to_Z_inj; generalize H; rewrite <-to_Z_eq, !add_spec, !Zmod_small; auto with zarith. - apply to_Z_inj; assert ([|x|] + [|y|] = [|x|] + [|z|]); auto with zarith. - assert (F1: wB > 0) by apply refl_equal. - rewrite (Z_div_mod_eq ([|x|] + [|y|]) wB), (Z_div_mod_eq ([|x|] + [|z|]) wB); auto. - rewrite <-to_Z_eq, !add_spec in H; rewrite H. - replace (([|x|] + [|y|])/wB) with 1. - replace (([|x|] + [|z|])/wB) with 1; auto with zarith. - apply Zle_antisym. - apply Zdiv_le_lower_bound; auto with zarith. - assert (F2: [|x|] + [|z|] < 2 * wB); auto with zarith. - generalize (Zdiv_lt_upper_bound _ _ _ (Z.gt_lt _ _ F1) F2); auto with zarith. - apply Zle_antisym. - apply Zdiv_le_lower_bound; auto with zarith. - assert (F2: [|x|] + [|y|] < 2 * wB); auto with zarith. - generalize (Zdiv_lt_upper_bound _ _ _ (Z.gt_lt _ _ F1) F2); auto with zarith. -Qed. - -Lemma add_cancel_r x y z : (y + x = z + x)%int -> y = z. -Proof. - rewrite !(fun t => add_comm t x); intros Hl; apply (add_cancel_l x); auto. -Qed. - -Lemma to_Z_split x : [|x|] = [|(x >> 1)|] * 2 + [|bit x 0|]. -Proof. - case (to_Z_bounded x); intros H1x H2x. - case (to_Z_bounded (bit x 0)); intros H1b H2b. - assert (F1: 0 <= [|x >> 1|] < wB/2). - rewrite lsr_spec, to_Z_1, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - assumption. - } - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite (bit_split x) at 1. - rewrite add_spec, Zmod_small, lsl_spec, to_Z_1, Zpower_1_r, Zmod_small; - split; auto with zarith. - change wB with ((wB/2)*2); auto with zarith. - rewrite lsl_spec, to_Z_1, Zpower_1_r, Zmod_small; auto with zarith. - change wB with ((wB/2)*2); auto with zarith. - rewrite lsl_spec, to_Z_1, Zpower_1_r, Zmod_small; auto with zarith. - 2: change wB with ((wB/2)*2); auto with zarith. - change wB with (((wB/2 - 1) * 2 + 1) + 1). - assert ([|bit x 0|] <= 1); auto with zarith. - case bit; discriminate. -Qed. - -Lemma lor_le x y : (y <= x lor y)%int = true. -Proof. - generalize x y (to_Z_bounded x) (to_Z_bounded y); clear x y. - unfold wB; elim size. - replace (2^Z_of_nat 0) with 1%Z; auto with zarith. - intros x y Hx Hy; replace x with 0%int. - replace y with 0%int; auto. - apply to_Z_inj; rewrite to_Z_0; auto with zarith. - apply to_Z_inj; rewrite to_Z_0; auto with zarith. - intros n IH x y; rewrite inj_S. - unfold Z.succ; rewrite Zpower_exp, Zpower_1_r; auto with zarith. - intros Hx Hy. - rewrite leb_spec. - rewrite (to_Z_split y) at 1; rewrite (to_Z_split (x lor y)). - assert ([|y>>1|] <= [|(x lor y) >> 1|]). - rewrite lor_lsr, <-leb_spec; apply IH. - rewrite lsr_spec, to_Z_1, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite lsr_spec, to_Z_1, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - assert ([|bit y 0|] <= [|bit (x lor y) 0|]); auto with zarith. - rewrite lor_spec; do 2 case bit; try discriminate. -Qed. - - -Lemma bit_add_or x y: - (forall n, bit x n = true -> bit y n = true -> False) <-> (x + y)%int= x lor y. -Proof. - generalize x y (to_Z_bounded x) (to_Z_bounded y); clear x y. - unfold wB; elim size. - replace (2^Z_of_nat 0) with 1%Z; auto with zarith. - intros x y Hx Hy; replace x with 0%int. - replace y with 0%int. - split; auto; intros _ n; rewrite !bit_0; discriminate. - apply to_Z_inj; rewrite to_Z_0; auto with zarith. - apply to_Z_inj; rewrite to_Z_0; auto with zarith. - intros n IH x y; rewrite inj_S. - unfold Z.succ; rewrite Zpower_exp, Zpower_1_r; auto with zarith. - intros Hx Hy. - split. - intros Hn. - assert (F1: ((x >> 1) + (y >> 1))%int = (x >> 1) lor (y >> 1)). - apply IH. - rewrite lsr_spec, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite lsr_spec, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - intros m H1 H2. - case_eq (digits <= m)%int; [idtac | rewrite <- not_true_iff_false]; - intros Heq. - rewrite bit_M in H1; auto; discriminate. - rewrite leb_spec in Heq. - apply (Hn (m + 1)%int); - rewrite <-bit_half; auto; rewrite ltb_spec; auto with zarith. - rewrite (bit_split (x lor y)), lor_lsr, <- F1, lor_spec. - replace (b2i (bit x 0 || bit y 0)) with (bit x 0 + bit y 0)%int. - 2: generalize (Hn 0%int); do 2 case bit; auto; intros [ ]; auto. - rewrite lsl_add_distr. - rewrite (bit_split x) at 1; rewrite (bit_split y) at 1. - rewrite <-!add_assoc; apply f_equal2 with (f := add31); auto. - rewrite add_comm, <-!add_assoc; apply f_equal2 with (f := add31); auto. - rewrite add_comm; auto. - intros Heq. - generalize (add_le_r x y); rewrite Heq, lor_le; intro Hb. - generalize Heq; rewrite (bit_split x) at 1; rewrite (bit_split y )at 1; clear Heq. - rewrite (fun y => add_comm y (bit x 0)), <-!add_assoc, add_comm, - <-!add_assoc, (add_comm (bit y 0)), add_assoc, <-lsr_add_distr. - rewrite (bit_split (x lor y)), lor_spec. - intros Heq. - assert (F: (bit x 0 + bit y 0)%int = (bit x 0 || bit y 0)). - assert (F1: (2 | wB)) by (apply Zpower_divide; apply refl_equal). - assert (F2: 0 < wB) by (apply refl_equal). - assert (F3: [|bit x 0 + bit y 0|] mod 2 = [|bit x 0 || bit y 0|] mod 2). - apply trans_equal with (([|(x>>1 + y>>1) << 1|] + [|bit x 0 + bit y 0|]) mod 2). - rewrite lsl_spec, Zplus_mod, <-Zmod_div_mod; auto with zarith. - rewrite Zpower_1_r, Z_mod_mult, Zplus_0_l, Zmod_mod; auto with zarith. - rewrite (Zmod_div_mod 2 wB), <-add_spec, Heq; auto with zarith. - rewrite add_spec, <-Zmod_div_mod; auto with zarith. - rewrite lsl_spec, Zplus_mod, <-Zmod_div_mod; auto with zarith. - rewrite Zpower_1_r, Z_mod_mult, Zplus_0_l, Zmod_mod; auto with zarith. - generalize F3; do 2 case bit; try discriminate; auto. - case (IH (x >> 1) (y >> 1)). - rewrite lsr_spec, to_Z_1, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite lsr_spec, to_Z_1, Zpower_1_r; split. - { - apply Z_div_pos. - - apply Zgt_pos_0. - - abstract omega. - } - apply Zdiv_lt_upper_bound; auto with zarith. - intros _ HH m; case (to_Z_bounded m); intros H1m H2m. - case_eq (digits <= m)%int. - intros Hlm; rewrite bit_M; auto; discriminate. - rewrite <- not_true_iff_false, leb_spec; intros Hlm. - case (Zle_lt_or_eq 0 [|m|]); auto; intros Hm. - replace m with ((m -1) + 1)%int. - rewrite <-(bit_half x), <-(bit_half y); auto with zarith. - apply HH. - rewrite <-lor_lsr. - assert (0 <= [|bit (x lor y) 0|] <= 1) by (case bit; split; discriminate). - rewrite F in Heq; generalize (add_cancel_r _ _ _ Heq). - intros Heq1; apply to_Z_inj. - generalize Heq1; rewrite <-to_Z_eq, lsl_spec, to_Z_1, Zpower_1_r, Zmod_small. - rewrite lsl_spec, to_Z_1, Zpower_1_r, Zmod_small; auto with zarith. - case (to_Z_bounded (x lor y)); intros H1xy H2xy. - rewrite lsr_spec, to_Z_1, Zpower_1_r; auto with zarith. - change wB with ((wB/2)*2); split. - { - apply Z.mul_nonneg_nonneg. - - apply Z_div_pos. - + apply Zgt_pos_0. - + assumption. - - apply Pos2Z.is_nonneg. - } - assert ([|x lor y|] / 2 < wB / 2); auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. - split. - case (to_Z_bounded (x >> 1 + y >> 1)); auto with zarith. - rewrite add_spec. - apply Z.le_lt_trans with (([|x >> 1|] + [|y >> 1|]) * 2); auto with zarith. - case (Zmod_le_first ([|x >> 1|] + [|y >> 1|]) wB); auto with zarith. - case (to_Z_bounded (x >> 1)); case (to_Z_bounded (y >> 1)); auto with zarith. - generalize Hb; rewrite (to_Z_split x) at 1; rewrite (to_Z_split y) at 1. - case (to_Z_bounded (bit x 0)); case (to_Z_bounded (bit y 0)); auto with zarith. - rewrite ltb_spec, sub_spec, to_Z_1, Zmod_small; auto with zarith. - rewrite ltb_spec, sub_spec, to_Z_1, Zmod_small; auto with zarith. - apply to_Z_inj. - rewrite add_spec, sub_spec, Zplus_mod_idemp_l, to_Z_1, Zmod_small; auto with zarith. - replace m with 0%int. - intros Hbx Hby; generalize F; rewrite <-to_Z_eq, Hbx, Hby; discriminate. - apply to_Z_inj; auto. -Qed. - -Lemma addmuldiv_spec : forall x y p, [|p|] <= [|digits|] -> - [| addmuldiv p x y |] = - ([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ([|digits|] - [|p|]))) mod wB. -Proof. - intros x y p H. - assert (Fp := to_Z_bounded p); assert (Fd := to_Z_bounded digits). - rewrite addmuldiv_def_spec; unfold addmuldiv_def. - case (bit_add_or (x << p) (y >> (digits - p))); intros HH _. - rewrite <-HH, add_spec, lsl_spec, lsr_spec, Zplus_mod_idemp_l, sub_spec. - rewrite (fun x y => Zmod_small (x - y)); auto with zarith. - intros n; rewrite bit_lsl, bit_lsr. - generalize (add_le_r (digits - p) n). - case leb; try discriminate. - rewrite sub_spec, Zmod_small; auto with zarith; intros H1. - case_eq (n < p)%int; try discriminate. - rewrite <- not_true_iff_false, ltb_spec; intros H2. - case leb; try discriminate. - intros _; rewrite bit_M; try discriminate. - rewrite leb_spec, add_spec, Zmod_small, sub_spec, Zmod_small; auto with zarith. - rewrite sub_spec, Zmod_small; auto with zarith. -Qed. - -Lemma lxor_comm: forall i1 i2 : int, i1 lxor i2 = i2 lxor i1. -Proof. - intros;apply bit_eq;intros. - rewrite !lxor_spec;apply xorb_comm. -Qed. - -Lemma lxor_assoc: forall i1 i2 i3 : int, i1 lxor (i2 lxor i3) = i1 lxor i2 lxor i3. -Proof. - intros;apply bit_eq;intros. - rewrite !lxor_spec, xorb_assoc;trivial. -Qed. - -Lemma lxor_0_l : forall i, 0 lxor i = i. -Proof. - intros;apply bit_eq;intros. - rewrite lxor_spec, bit_0, xorb_false_l;trivial. -Qed. - -Lemma lxor_0_r : forall i, i lxor 0 = i. -Proof. - intros;rewrite lxor_comm;apply lxor_0_l. -Qed. - -Lemma lxor_nilpotent: forall i, i lxor i = 0%int. -Proof. - intros;apply bit_eq;intros. - rewrite lxor_spec, xorb_nilpotent, bit_0;trivial. -Qed. - -Lemma lor_0_l : forall i, 0 lor i = i. -Proof. - intros;apply bit_eq;intros. - rewrite lor_spec, bit_0, orb_false_l;trivial. -Qed. - -Lemma lor_0_r : forall i, i lor 0 = i. -Proof. - intros;rewrite lor_comm;apply lor_0_l. -Qed. - -Lemma reflect_leb : forall i j, reflect ([|i|] <= [|j|])%Z (i <= j)%int. -Proof. - intros; apply iff_reflect. - symmetry;apply leb_spec. -Qed. - -Lemma reflect_eqb : forall i j, reflect (i = j)%Z (i == j). -Proof. - intros; apply iff_reflect. - symmetry;apply eqb_spec. -Qed. - -Lemma reflect_ltb : forall i j, reflect ([|i|] < [|j|])%Z (i < j)%int. -Proof. - intros; apply iff_reflect. - symmetry;apply ltb_spec. -Qed. - -Lemma lsr_is_even_eq : forall i j, - i >> 1 = j >> 1 -> - is_even i = is_even j -> - i = j. -Proof. - intros;apply bit_eq. - intros n;destruct (reflect_eqb n 0). - rewrite <- (negb_involutive (bit i n)), <- (negb_involutive (bit j n)). - rewrite e, <- !is_even_bit, H0;trivial. - assert (W1 : [|n|] <> 0) by (intros Heq;apply n0;apply to_Z_inj;trivial). - assert (W2 := to_Z_bounded n);clear n0. - assert (W3 : [|n-1|] = [|n|] - 1). - rewrite sub_spec, to_Z_1, Zmod_small;trivial;omega. - assert (H1 : n = ((n-1)+1)%int). - apply to_Z_inj;rewrite add_spec, W3. - rewrite Zmod_small;rewrite to_Z_1; omega. - destruct (reflect_ltb (n-1) digits). - rewrite <- ltb_spec in l. - rewrite H1, <- !bit_half, H;trivial. - assert ((digits <= n)%int = true). - rewrite leb_spec;omega. - rewrite !bit_M;trivial. -Qed. - -Lemma lsr1_bit : forall i k, (bit i k >> 1 = 0)%int. -Proof. - intros;destruct (bit i k);trivial. -Qed. - -Lemma bit_xor_split: forall i : int, i = (i >> 1) << 1 lxor bit i 0. -Proof. - intros. - rewrite bit_or_split at 1. - apply lsr_is_even_eq. - rewrite lxor_lsr, lor_lsr, lsr1_bit, lxor_0_r, lor_0_r;trivial. - rewrite is_even_or, is_even_xor. - rewrite is_even_lsl_1;trivial. - rewrite (xorb_true_l (is_even (bit i 0))), negb_involutive;trivial. -Qed. - -(** Order *) -Local Open Scope int63_scope. - -Lemma succ_max_int : forall x, - (x < max_int)%int = true -> (0 < x + 1)%int = true. -Proof. - intros x;rewrite ltb_spec, ltb_spec, add_spec. - intros; assert (W:= to_Z_bounded x); assert (W1:= to_Z_bounded max_int). - change [|0|] with 0%Z;change [|1|] with 1%Z. - rewrite Zmod_small;omega. -Qed. - -Lemma leb_max_int : forall x, (x <= max_int)%int = true. -Proof. - intros x;rewrite leb_spec;assert (W:= to_Z_bounded x). - change [|max_int|] with (wB - 1)%Z;omega. -Qed. - -Lemma leb_0 : forall x, 0 <= x = true. -Proof. - intros x;rewrite leb_spec;destruct (to_Z_bounded x);trivial. -Qed. - -Lemma ltb_0 : forall x, ~ (x < 0 = true). -Proof. - intros x;rewrite ltb_spec, to_Z_0;destruct (to_Z_bounded x);omega. -Qed. - -Lemma leb_trans : forall x y z, x <= y = true -> y <= z = true -> x <= z = true. -Proof. - intros x y z;rewrite !leb_spec;apply Z.le_trans. -Qed. - -Lemma ltb_trans : forall x y z, x < y = true -> y < z = true -> x < z = true. -Proof. - intros x y z;rewrite !ltb_spec;apply Z.lt_trans. -Qed. - -Lemma ltb_leb_trans : forall x y z, x < y = true -> y <= z = true -> x < z = true. -Proof. - intros x y z;rewrite leb_spec, !ltb_spec;apply Z.lt_le_trans. -Qed. - -Lemma leb_ltb_trans : forall x y z, x <= y = true -> y < z = true -> x < z = true. -Proof. - intros x y z;rewrite leb_spec, !ltb_spec;apply Z.le_lt_trans. -Qed. - -Lemma gtb_not_leb : forall n m, m < n = true -> ~(n <= m = true). -Proof. - intros n m; rewrite ltb_spec, leb_spec;omega. -Qed. - -Lemma leb_not_gtb : forall n m, m <= n = true -> ~(n < m = true). -Proof. - intros n m; rewrite ltb_spec, leb_spec;omega. -Qed. - -Lemma leb_refl : forall n, n <= n = true. -Proof. - intros n;rewrite leb_spec;apply Z.le_refl. -Qed. - -Lemma leb_negb_gtb : forall x y, x <= y = negb (y < x). -Proof. - intros x y;apply Bool.eq_true_iff_eq;split;intros. - apply Bool.eq_true_not_negb;apply leb_not_gtb;trivial. - rewrite Bool.negb_true_iff, <- Bool.not_true_iff_false in H. - rewrite leb_spec; rewrite ltb_spec in H;omega. -Qed. - -Lemma ltb_negb_geb : forall x y, x < y = negb (y <= x). -Proof. - intros;rewrite leb_negb_gtb, Bool.negb_involutive;trivial. -Qed. - -Lemma to_Z_sub_gt : forall x y, y <= x = true -> [|x - y|] = ([|x|] - [|y|])%Z. -Proof. - intros x y;assert (W:= to_Z_bounded x);assert (W0:= to_Z_bounded y); - rewrite leb_spec;intros;rewrite sub_spec, Zmod_small;omega. -Qed. - -Lemma not_0_ltb : forall x, x <> 0 <-> 0 < x = true. -Proof. - intros x;rewrite ltb_spec, to_Z_0;assert (W:=to_Z_bounded x);split. - intros Hd;assert ([|x|] <> 0)%Z;[ | omega]. - intros Heq;elim Hd;apply to_Z_inj;trivial. - intros Hlt Heq;elimtype False. - assert ([|x|] = 0)%Z;[ rewrite Heq, to_Z_0;trivial | omega]. -Qed. - -Lemma not_ltb_refl : forall i, ~(i < i = true). -Proof. - intros;rewrite ltb_spec;omega. -Qed. - -Lemma to_Z_sub_1 : forall x y, y < x = true -> ([| x - 1|] = [|x|] - 1)%Z. -Proof. - intros;apply to_Z_sub_gt. - generalize (leb_ltb_trans _ _ _ (leb_0 y) H). - rewrite ltb_spec, leb_spec, to_Z_0, to_Z_1;auto with zarith. -Qed. - -Lemma to_Z_sub_1_diff : forall x, x <> 0 -> ([| x - 1|] = [|x|] - 1)%Z. -Proof. - intros x;rewrite not_0_ltb;apply to_Z_sub_1. -Qed. - -Lemma to_Z_add_1 : forall x y, x < y = true -> [|x+1|] = ([|x|] + 1)%Z. -Proof. - intros x y;assert (W:= to_Z_bounded x);assert (W0:= to_Z_bounded y); - rewrite ltb_spec;intros;rewrite add_spec, to_Z_1, Zmod_small;omega. -Qed. - -Lemma ltb_leb_sub1 : forall x i, x <> 0 -> (i < x = true <-> i <= x - 1 = true). -Proof. - intros x i Hdiff. - rewrite ltb_spec, leb_spec, to_Z_sub_1_diff;trivial. - split;auto with zarith. -Qed. - -Lemma ltb_leb_add1 : forall x y i, i < y = true -> (i < x = true <-> i + 1 <= x = true). -Proof. - intros x y i Hlt. - rewrite ltb_spec, leb_spec. - rewrite (to_Z_add_1 i y);trivial. - split;auto with zarith. -Qed. - -(** Iterators *) - -Lemma foldi_gt : forall A f from to (a:A), - (to < from)%int = true -> foldi f from to a = a. -Proof. - intros;unfold foldi;rewrite foldi_cont_gt;trivial. -Qed. - -Lemma foldi_eq : forall A f from to (a:A), - from = to -> foldi f from to a = f from a. -Proof. - intros;unfold foldi;rewrite foldi_cont_eq;trivial. -Qed. - -Lemma foldi_lt : forall A f from to (a:A), - (from < to)%int = true -> foldi f from to a = foldi f (from + 1) to (f from a). -Proof. - intros;unfold foldi;rewrite foldi_cont_lt;trivial. -Qed. - -Lemma fold_gt : forall A f from to (a:A), - (to < from)%int = true -> fold f from to a = a. -Proof. - intros;apply foldi_gt;trivial. -Qed. - -Lemma fold_eq : forall A f from to (a:A), - from = to -> fold f from to a = f a. -Proof. - intros;apply foldi_eq;trivial. -Qed. - -Lemma fold_lt : forall A f from to (a:A), - (from < to)%int = true -> fold f from to a = fold f (from + 1) to (f a). -Proof. - intros;apply foldi_lt;trivial. -Qed. - -Lemma foldi_down_lt : forall A f from downto (a:A), - (from < downto)%int = true -> foldi_down f from downto a = a. -Proof. - intros;unfold foldi_down;rewrite foldi_down_cont_lt;trivial. -Qed. - -Lemma foldi_down_eq : forall A f from downto (a:A), - from = downto -> foldi_down f from downto a = f from a. -Proof. - intros;unfold foldi_down;rewrite foldi_down_cont_eq;trivial. -Qed. - -Lemma foldi_down_gt : forall A f from downto (a:A), - (downto < from)%int = true-> - foldi_down f from downto a = - foldi_down f (from-1) downto (f from a). -Proof. - intros;unfold foldi_down;rewrite foldi_down_cont_gt;trivial. -Qed. - -Lemma fold_down_lt : forall A f from downto (a:A), - (from < downto)%int = true -> fold_down f from downto a = a. -Proof. - intros;apply foldi_down_lt;trivial. -Qed. - -Lemma fold_down_eq : forall A f from downto (a:A), - from = downto -> fold_down f from downto a = f a. -Proof. - intros;apply foldi_down_eq;trivial. -Qed. - -Lemma fold_down_gt : forall A f from downto (a:A), - (downto < from)%int = true-> - fold_down f from downto a = - fold_down f (from-1) downto (f a). -Proof. - intros;apply foldi_down_gt;trivial. -Qed. - -Require Import Wf_Z. - -Lemma int_ind : forall (P:int -> Type), - P 0%int -> - (forall i, (i < max_int)%int = true -> P i -> P (i + 1)%int) -> - forall i, P i. -Proof. - intros P HP0 Hrec. - assert (forall z, (0 <= z)%Z -> forall i, z = [|i|] -> P i). - intros z H;pattern z;apply natlike_rec2;intros;trivial. - rewrite <- (of_to_Z i), <- H0;exact HP0. - assert (W:= to_Z_bounded i). - assert ([|i - 1|] = [|i|] - 1)%Z. - rewrite sub_spec, Zmod_small;rewrite to_Z_1;auto with zarith. - assert (i = i - 1 + 1)%int. - apply to_Z_inj. - rewrite add_spec, H2. - rewrite Zmod_small;rewrite to_Z_1;auto with zarith. - rewrite H3;apply Hrec. - rewrite ltb_spec, H2;change [|max_int|] with (wB - 1)%Z;auto with zarith. - apply X;auto with zarith. - intros;apply (X [|i|]);trivial. - destruct (to_Z_bounded i);trivial. -Qed. - -Lemma int_ind_bounded : forall (P:int-> Type) min max, - min <= max =true -> - P max -> - (forall i, min <= i + 1 = true-> i < max =true-> P (i + 1) -> P i) -> - P min. -Proof. - intros P min max Hle. - intros Hmax Hrec. - assert (W1:= to_Z_bounded max);assert (W2:= to_Z_bounded min). - assert (forall z, (0 <= z)%Z -> (z <= [|max|] - [|min|])%Z -> forall i, z = [|i|] -> P (max - i)%int). - intros z H1;pattern z;apply natlike_rec2;intros;trivial. - assert (max - i = max)%int. - apply to_Z_inj;rewrite sub_spec, <- H0, Zminus_0_r, Zmod_small;auto using to_Z_bounded. - rewrite H2;trivial. - assert (W3:= to_Z_bounded i);apply Hrec. - rewrite leb_spec,add_spec, sub_spec, to_Z_1, (Zmod_small ([|max|] - [|i|])), Zmod_small;auto with zarith. - rewrite ltb_spec, sub_spec, Zmod_small;auto with zarith. - assert (max - i + 1 = max - (i - 1))%int. - apply to_Z_inj;rewrite add_spec, !sub_spec, to_Z_1. - rewrite (Zmod_small ([|max|] - [|i|]));auto with zarith. - rewrite (Zmod_small ([|i|] - 1));auto with zarith. - apply f_equal2;auto with zarith. - rewrite H3;apply X;auto with zarith. - rewrite sub_spec, to_Z_1, <- H2, Zmod_small;auto with zarith. - rewrite leb_spec in Hle;assert (min = max - (max - min))%int. - apply to_Z_inj. - rewrite !sub_spec, !Zmod_small;auto with zarith. - rewrite Zmod_small;auto with zarith. - rewrite H;apply (X [| max - min |]);trivial;rewrite sub_spec, Zmod_small;auto with zarith. -Qed. - -Lemma foldi_cont_ZInd : forall A B (P: Z -> (A -> B) -> Prop) (f:int -> (A -> B) -> (A -> B)) min max cont, - (forall z, ([|max|] < z)%Z -> P z cont) -> - (forall i cont, min <= i = true -> i <= max = true -> P ([|i|] + 1)%Z cont -> P [|i|] (f i cont)) -> - P [|min|] (foldi_cont f min max cont). -Proof. - intros A B P f min max cont Ha Hf. - assert (Bmax:= to_Z_bounded max);assert (Bmin:= to_Z_bounded min). - case_eq (min <= max);intros Heq. - generalize (leb_refl min). - assert (P ([|max|] + 1)%Z cont) by (apply Ha;auto with zarith). - clear Ha;revert cont H. - pattern min at 2 3 4;apply int_ind_bounded with max;trivial. - intros;rewrite foldi_cont_eq;auto using leb_refl. - intros i Hle Hlt Hr cont Hcont Hle'. - rewrite foldi_cont_lt;[ | trivial]. - apply Hf;trivial. rewrite leb_spec;rewrite ltb_spec in Hlt;auto with zarith. - assert ([|i|] + 1 = [|i + 1|])%Z. - rewrite ltb_spec in Hlt;assert (W:= to_Z_bounded i);rewrite add_spec, to_Z_1, Zmod_small;omega. - rewrite H;apply Hr;trivial. - assert (max < min = true) by (rewrite ltb_negb_geb,Heq;trivial). - rewrite foldi_cont_gt;trivial;apply Ha;rewrite <- ltb_spec;trivial. -Qed. - - -(* Lemma of_pos_spec : forall p, [|of_pos p|] = Zpos p mod wB. *) -(* Proof. *) -(* unfold of_pos. *) -(* unfold wB. *) -(* assert (forall k, (k <= size)%nat -> *) -(* forall p : positive, [|of_pos_rec k p|] = Zpos p mod 2 ^ Z_of_nat k). *) -(* induction k. *) -(* simpl;intros;rewrite to_Z_0,Zmod_1_r;trivial. *) -(* Opaque Z_of_nat. *) -(* destruct p;simpl. *) -(* destruct (bit_add_or (of_pos_rec k p << 1) 1) as (H1, _). *) -(* rewrite <- H1;clear H1. *) -(* change (Zpos p~1) with (2*(Zpos p) + 1)%Z. *) -(* rewrite add_spec,lsl_spec, IHk, to_Z_1. *) -(* rewrite Zmult_comm, Zplus_mod_idemp_l, Zmod_small. *) -(* change 2%Z with (2^1)%Z. *) -(* rewrite Zmod_distr. *) -(* rewrite inj_S, Zpower_Zsucc;[ | apply Zle_0_nat]. *) -(* repeat change (2^1)%Z with 2%Z. *) -(* rewrite Zmult_mod_distr_l;trivial. *) -(* Transparent Z_of_nat. *) -(* rewrite inj_S;omega. *) -(* discriminate. *) -(* split;[discriminate | trivial]. *) -(* compute;trivial. *) -(* assert (W:0 <= Zpos p mod 2 ^ Z_of_nat k < 2 ^ Z_of_nat k). *) -(* apply Z.mod_pos_bound;auto with zarith. *) -(* change (2^1)%Z with 2%Z;split;try omega. *) -(* apply Z.lt_le_trans with (2 ^ Z_of_nat (S k)). *) -(* rewrite inj_S, Zpower_Zsucc;omega. *) -(* unfold wB;apply Zpower_le_monotone;auto with zarith. *) -(* split;auto using inj_le with zarith. *) -(* auto with zarith. *) -(* intros n H1 H2. *) -(* rewrite bit_1, eqb_spec in H2;subst. *) -(* rewrite bit_lsl in H1;discriminate H1. *) - -(* change (Zpos p~0) with (2*(Zpos p))%Z. *) -(* rewrite lsl_spec, IHk, to_Z_1. *) -(* rewrite Zmult_comm, Zmod_small. *) -(* rewrite inj_S, Zpower_Zsucc;[ | apply Zle_0_nat]. *) -(* rewrite Zmult_mod_distr_l;trivial. *) -(* assert (W:0 <= Zpos p mod 2 ^ Z_of_nat k < 2 ^ Z_of_nat k). *) -(* apply Z.mod_pos_bound;auto with zarith. *) -(* change (2^1)%Z with 2%Z;split;try omega. *) -(* apply Z.lt_le_trans with (2 ^ Z_of_nat (S k)). *) -(* rewrite inj_S, Zpower_Zsucc;omega. *) -(* unfold wB;apply Zpower_le_monotone;auto with zarith. *) -(* split;auto using inj_le with zarith. *) -(* auto with zarith. *) - -(* rewrite to_Z_1, Zmod_small;trivial. *) -(* split;auto with zarith. *) -(* apply Zpower_gt_1;auto with zarith. *) -(* rewrite inj_S;auto with zarith. *) - -(* apply H;auto with zarith. *) -(* Qed. *) - -Lemma of_Z_spec : forall z, [|of_Z z|] = z mod wB. -Admitted. (* no more of_pos *) -(* Proof. *) -(* unfold of_Z;destruct z. *) -(* assert (W:= to_Z_bounded 0);rewrite Zmod_small;trivial. *) -(* apply of_pos_spec. *) -(* rewrite opp_spec, of_pos_spec. *) -(* rewrite <- Zmod_opp_opp. *) -(* change (- Zpos p)%Z with (Zneg p). *) -(* destruct (Z_eq_dec (Zneg p mod wB) 0). *) -(* rewrite e, Z_mod_zero_opp_r;trivial. *) -(* rewrite Z_mod_nz_opp_r, Zminus_mod, Z_mod_same_full, Zmod_mod, Zminus_0_r, Zmod_mod;trivial. *) -(* Qed. *) - -Lemma foldi_cont_Ind : forall A B (P: int -> (A -> B) -> Prop) (f:int -> (A -> B) -> (A -> B)) min max cont, - max < max_int = true -> - (forall z, max < z = true -> P z cont) -> - (forall i cont, min <= i = true -> i <= max = true -> P (i + 1) cont -> P i (f i cont)) -> - P min (foldi_cont f min max cont). -Proof. - intros. - set (P' z cont := (0 <= z < wB)%Z -> P (of_Z z) cont). - assert (P' [|min|] (foldi_cont f min max cont)). - apply foldi_cont_ZInd;unfold P';intros. - assert ([|(of_Z z)|] = z). - rewrite of_Z_spec, Zmod_small;trivial. - apply H0;rewrite ltb_spec, H4;trivial. - rewrite of_to_Z;apply H1;trivial. - assert (i < max_int = true). - apply leb_ltb_trans with max;trivial. - rewrite <- (to_Z_add_1 _ _ H6), of_to_Z in H4;apply H4. - apply to_Z_bounded. - unfold P' in H2;rewrite of_to_Z in H2;apply H2;apply to_Z_bounded. -Qed. - -Lemma foldi_cont_ind : forall A B (P: (A -> B) -> Prop) (f:int -> (A -> B) -> (A -> B)) min max cont, - P cont -> - (forall i cont, min <= i = true -> i <= max = true -> P cont -> P (f i cont)) -> - P (foldi_cont f min max cont). -Proof. - intros A B P f min max cont Ha Hf. - set (P2 := fun (z:Z) b => P b);change (P2 [|min|] (foldi_cont f min max cont)). - apply foldi_cont_ZInd;trivial. -Qed. - -Lemma foldi_ZInd : forall A (P : Z -> A -> Prop) f min max a, - (max < min = true -> P ([|max|] + 1)%Z a) -> - P [|min|] a -> - (forall i a, min <= i = true -> i <= max = true -> - P [|i|] a -> P ([|i|] + 1)%Z (f i a)) -> - P ([|max|]+1)%Z (foldi f min max a). -Proof. - unfold foldi;intros A P f min max a Hlt;intros. - set (P' z cont := - if Zlt_bool [|max|] z then cont = (fun a0 : A => a0) - else forall a, P z a -> P ([|max|]+1)%Z (cont a)). - assert (P' [|min|] (foldi_cont (fun (i : int) (cont : A -> A) (a0 : A) => cont (f i a0)) min - max (fun a0 : A => a0))). - apply foldi_cont_ZInd;intros;red. - rewrite Zlt_is_lt_bool in H1;rewrite H1;trivial. - case_eq (Zlt_bool [|max|] [|i|]);intros. - rewrite <- Zlt_is_lt_bool in H4;rewrite leb_spec in H2;elimtype False;omega. - clear H4; revert H3;unfold P'. - case_eq (Zlt_bool [|max|] ([|i|] + 1));intros;auto. - rewrite <- Zlt_is_lt_bool in H3; assert ([|i|] = [|max|]) by (rewrite leb_spec in H2;omega). - rewrite H4, <- H6;apply H0;trivial. - revert H1;unfold P'. - case_eq (Zlt_bool [|max|] [|min|]);auto. - rewrite <- Zlt_is_lt_bool, <- ltb_spec;intros;rewrite foldi_cont_gt;auto. -Qed. - -Lemma foldi_Ind : forall A (P : int -> A -> Prop) f min max a, - (max < max_int = true) -> - (max < min = true -> P (max + 1) a) -> - P min a -> - (forall i a, min <= i = true -> i <= max = true -> - P i a -> P (i + 1) (f i a)) -> - P (max+1) (foldi f min max a). -Proof. - intros. - set (P' z a := (0 <= z < wB)%Z -> P (of_Z z) a). - assert (W:= to_Z_add_1 _ _ H). - assert (P' ([|max|]+1)%Z (foldi f min max a)). - apply foldi_ZInd;unfold P';intros. - rewrite <- W, of_to_Z;auto. - rewrite of_to_Z;trivial. - assert (i < max_int = true). - apply leb_ltb_trans with max;trivial. - rewrite <- (to_Z_add_1 _ _ H7), of_to_Z;apply H2;trivial. - rewrite of_to_Z in H5;apply H5;apply to_Z_bounded. - unfold P' in H3;rewrite <- W, of_to_Z in H3;apply H3;apply to_Z_bounded. -Qed. - -Lemma foldi_ind : forall A (P: A -> Prop) (f:int -> A -> A) min max a, - P a -> - (forall i a, min <= i = true -> i <= max = true -> P a -> P (f i a)) -> - P (foldi f min max a). -Proof. - unfold foldi;intros A P f min max a Ha Hr;revert a Ha. - apply foldi_cont_ind;auto. -Qed. - -Lemma fold_ind : forall A (P: A -> Prop) (f: A -> A) min max a, - P a -> (forall a, P a -> P (f a)) -> P (fold f min max a). -Proof. - unfold fold;intros A P f min max a Ha Hr;revert a Ha. - apply foldi_cont_ind;auto. -Qed. - -Lemma foldi_down_cont_ZInd : - forall A B (P: Z -> (A -> B) -> Prop) (f:int -> (A -> B) -> (A -> B)) max min cont, - (forall z, (z < [|min|])%Z -> P z cont) -> - (forall i cont, min <= i = true -> i <= max = true -> P ([|i|] - 1)%Z cont -> P [|i|] (f i cont)) -> - P [|max|] (foldi_down_cont f max min cont). -Proof. - intros A B P f max min cont Ha Hf. - assert (Bmax:= to_Z_bounded max);assert (Bmin:= to_Z_bounded min). - case_eq (min <= max);intros Heq. - generalize (leb_refl max). - assert (P ([|min|] -1)%Z cont) by (apply Ha;auto with zarith). - clear Ha;revert cont H Heq. - pattern max at 1 2 4 5;apply int_ind;trivial. - intros; assert (0 = min). - apply to_Z_inj;revert Heq;rewrite leb_spec, to_Z_0;omega. - rewrite foldi_down_cont_eq;subst;auto. - intros i Hmaxi Hr cont Hcont Hmin Hmax. - generalize Hmin;rewrite leb_ltb_eqb;case_eq (min < i+1);simpl;intros Hlt Hmin'. - rewrite foldi_down_cont_gt;[ | trivial]. - apply Hf;trivial. - assert ([|i|] + 1 = [|i + 1|])%Z. - assert (W:= to_Z_bounded i);rewrite ltb_spec in Hmaxi; - assert (W2 := to_Z_bounded max_int);rewrite add_spec, to_Z_1, Zmod_small;auto with zarith. - assert (i + 1 - 1 = i). - rewrite leb_spec in *;rewrite ltb_spec in *. - assert (W1:= to_Z_bounded i); apply to_Z_inj;rewrite sub_spec,to_Z_1, Zmod_small;try omega. - assert ([|i|] = [|i+1|]-1)%Z. - rewrite <- H;ring. - rewrite <- H1, H0;apply Hr;trivial. - rewrite ltb_spec in Hlt;rewrite leb_spec;omega. - rewrite leb_spec in Hmax |- *;omega. - rewrite eqb_spec in Hmin';subst;rewrite foldi_down_cont_eq;auto. - assert (max < min = true) by (rewrite ltb_negb_geb,Heq;trivial). - rewrite foldi_down_cont_lt;trivial. - apply Ha;rewrite <- ltb_spec;trivial. -Qed. - -Lemma foldi_down_cont_ind : forall A B (P: (A -> B) -> Prop) (f:int -> (A -> B) -> (A -> B)) max min cont, - P cont -> - (forall i cont, min <= i = true -> i <= max = true -> P cont -> P (f i cont)) -> - P (foldi_down_cont f max min cont). -Proof. - intros A B P f max min cont Ha Hf. - set (P2 := fun (z:Z) b => P b);change (P2 [|max|] (foldi_down_cont f max min cont)). - apply foldi_down_cont_ZInd;trivial. -Qed. - -Lemma foldi_down_ZInd : - forall A (P: Z -> A -> Prop) (f:int -> A -> A) max min a, - (max < min = true -> P ([|min|] - 1)%Z a) -> - (P [|max|] a) -> - (forall i a, min <= i = true -> i <= max = true -> P [|i|]%Z a -> P ([|i|]-1)%Z (f i a)) -> - P ([|min|] - 1)%Z (foldi_down f max min a). -Proof. - unfold foldi_down;intros A P f max min a Hlt;intros. - set (P' z cont := - if Zlt_bool z [|min|] then cont = (fun a0 : A => a0) - else forall a, P z a -> P ([|min|] - 1)%Z (cont a)). - assert (P' [|max|] (foldi_down_cont (fun (i : int) (cont : A -> A) (a0 : A) => cont (f i a0)) max - min (fun a0 : A => a0))). - apply foldi_down_cont_ZInd;intros;red. - rewrite Zlt_is_lt_bool in H1;rewrite H1;trivial. - case_eq (Zlt_bool [|i|] [|min|]);intros. - rewrite <- Zlt_is_lt_bool in H4;rewrite leb_spec in H1;elimtype False;omega. - clear H4;revert H3;unfold P'. - case_eq (Zlt_bool ([|i|] - 1) [|min|]);intros;auto. - rewrite <- Zlt_is_lt_bool in H3; assert ([|i|] = [|min|]) by (rewrite leb_spec in H1;omega). - rewrite H4, <- H6. apply H0;trivial. - revert H1;unfold P'. - case_eq (Zlt_bool [|max|] [|min|]);auto. - rewrite <- Zlt_is_lt_bool, <- ltb_spec;intros;rewrite foldi_down_cont_lt;auto. -Qed. - -Lemma foldi_down_ind : forall A (P: A -> Prop) (f:int -> A -> A) max min a, - P a -> - (forall i a, min <= i = true -> i <= max = true -> P a -> P (f i a)) -> - P (foldi_down f max min a). -Proof. - unfold foldi_down;intros A P f max min a Ha Hr;revert a Ha. - apply foldi_down_cont_ind;auto. -Qed. - -Lemma fold_down_ind : forall A (P: A -> Prop) (f: A -> A) max min a, - P a -> (forall a, P a -> P (f a)) -> P (fold_down f max min a). -Proof. - unfold fold_down;intros A P f max min a Ha Hr;revert a Ha. - apply foldi_down_cont_ind;auto. -Qed. - -Lemma foldi_down_Ind : - forall A (P: int -> A -> Prop) (f:int -> A -> A) max min a, - 0 < min = true -> - (max < min = true -> P (min - 1) a) -> - (P max a) -> - (forall i a, min <= i = true -> i <= max = true -> P i a -> P (i - 1) (f i a)) -> - P (min - 1) (foldi_down f max min a). -Proof. - intros. - set (P' z a := (0 <= z < wB)%Z -> P (of_Z z) a). - assert (W:= to_Z_sub_1 _ _ H). - assert (P' ([|min|]-1)%Z (foldi_down f max min a)). - apply foldi_down_ZInd;unfold P';intros. - rewrite <- W, of_to_Z;auto. - rewrite of_to_Z;trivial. - assert (0 < i = true). - apply ltb_leb_trans with min;trivial. - rewrite <- (to_Z_sub_1 _ _ H7), of_to_Z;apply H2;trivial. - rewrite of_to_Z in H5;apply H5;apply to_Z_bounded. - unfold P' in H3;rewrite <- W, of_to_Z in H3;apply H3;apply to_Z_bounded. -Qed. - -Lemma foldi_down_min : - forall A f min max (a:A), - min < max_int = true-> - (min <= max) = true -> - foldi_down f max min a = f min (foldi_down f max (min + 1) a). -Proof. - intros. - set (P:= fun i => i <= max - min = true -> - forall a, foldi_down f (min + i) min a = f min (foldi_down f (min + i) (min + 1) a)). - assert (min < min + 1 = true). - rewrite ltb_leb_add1 with (y:=max_int), leb_refl;trivial. - assert (P (max - min)). - apply int_ind;unfold P. - replace (min + 0) with min. - intros _ a'; rewrite foldi_down_eq, foldi_down_lt;trivial. - apply to_Z_inj;rewrite add_spec, to_Z_0, Zplus_0_r, Zmod_small;auto using to_Z_bounded. - intros i Hi Hrec Hi1 a'. - rewrite add_assoc. - assert (Wi:= to_Z_add_1 _ _ Hi). - assert (Wmin:= to_Z_add_1 _ _ H). - assert ((min + 1) <= (min + i + 1) = true). - assert (W1 := to_Z_bounded min); assert (W2:= to_Z_bounded max); assert (W3:= to_Z_bounded i). - replace (min + i + 1) with (min + 1 + i). - rewrite leb_spec, (add_spec (min+1)). - unfold is_true in Hi1;rewrite leb_spec in *; rewrite ltb_spec in *. - rewrite sub_spec in Hi1;rewrite Zmod_small in Hi1;[ | omega]. - rewrite Zmod_small;omega. - rewrite <- !add_assoc, (add_comm 1 i);trivial. - rewrite leb_ltb_eqb in H2;revert H2. - case_eq (min + 1 < min + i + 1). - intros Hlt _;rewrite foldi_down_gt. - rewrite foldi_down_gt with (from := min + i + 1);trivial. - replace (min + i + 1 - 1) with (min + i). - apply Hrec. - apply leb_trans with (i+1);[rewrite leb_spec;omega | trivial]. - apply to_Z_inj;rewrite sub_spec, (add_spec (min + i)), to_Z_1, Zminus_mod_idemp_l. - assert (H100: forall (x:Z), (x + 1 - 1)%Z = x) by (intros; ring). rewrite H100. - rewrite Zmod_small;auto using to_Z_bounded. - apply leb_ltb_trans with (2:= Hlt). - rewrite leb_spec;omega. - simpl;rewrite eqb_spec;intros _ Heq. - rewrite <- Heq. - rewrite foldi_down_gt. - replace (min + 1 - 1) with min. - rewrite !foldi_down_eq;trivial. - apply to_Z_inj;rewrite sub_spec, add_spec, to_Z_1, Zminus_mod_idemp_l. - replace ([|min|] + 1 - 1)%Z with [|min|] by ring. - rewrite Zmod_small;auto using to_Z_bounded. - rewrite ltb_spec;omega. - generalize (H2 (leb_refl _) a). - replace (min + (max - min)) with max;trivial. - apply to_Z_inj;rewrite add_spec, sub_spec, Zplus_mod_idemp_r. - ring_simplify ([|min|] + ([|max|] - [|min|]))%Z. - rewrite Zmod_small;auto using to_Z_bounded. -Qed. - -Definition foldi_ntr A f min max (a:A) := - foldi_cont (fun i cont _ => f i (cont tt)) min max (fun _ => a) tt. - -Lemma foldi_ntr_foldi_down : forall A f min max (a:A), - max < max_int = true -> - foldi_down f max min a = foldi_ntr _ f min max a. -Proof. - intros;unfold foldi_ntr. - apply foldi_cont_Ind;trivial. - intros;apply foldi_down_lt;trivial. - intros i cont Hmin Hmax Heq;rewrite <- Heq;clear Heq. - apply foldi_down_min;trivial. - apply leb_ltb_trans with (1:= Hmax);trivial. -Qed. - - -(** Two iterators *) - -Lemma foldi_cont_ZInd2 : forall A B C D (P: Z -> (A -> B) -> (C -> D) -> Prop) (f1 : int -> (A -> B) -> (A -> B)) (f2 : int -> (C -> D) -> (C -> D)) min max cont1 cont2, - (forall z, ([|max|] < z)%Z -> P z cont1 cont2) -> - (forall i cont1 cont2, min <= i = true -> i <= max = true -> P ([|i|] + 1)%Z cont1 cont2 -> - P [|i|] (f1 i cont1) (f2 i cont2)) -> - P [|min|] (foldi_cont f1 min max cont1) (foldi_cont f2 min max cont2). -Proof. - intros. - set (P' z cont := - if Zlt_bool [|max|] z then cont = cont1 - else P z cont (foldi_cont f2 (of_Z z) max cont2)). - assert (P' [|min|] (foldi_cont f1 min max cont1)). - apply foldi_cont_ZInd;unfold P';intros. - rewrite Zlt_is_lt_bool in H1;rewrite H1;trivial. - case_eq (Zlt_bool [|max|] [|i|]);intros. - rewrite <- Zlt_is_lt_bool, <- ltb_spec in H4. - elim (not_ltb_refl max);apply ltb_leb_trans with i;trivial. - rewrite of_to_Z;generalize H2;rewrite leb_ltb_eqb, orb_true_iff;intros [Hlt | Heq]. - rewrite foldi_cont_lt;[apply H0 | ];trivial. - revert H3;case_eq (Zlt_bool [|max|] ([|i|] + 1)). - rewrite <- Zlt_is_lt_bool;rewrite ltb_spec in Hlt;intros;elimtype False;omega. - rewrite <- (to_Z_add_1 _ _ Hlt), of_to_Z; intros _ W;exact W. - rewrite eqb_spec in Heq;subst. - rewrite foldi_cont_eq;[apply H0 | ];trivial. - assert ([|max|] < [|max|] + 1)%Z by auto with zarith. - rewrite Zlt_is_lt_bool in H5;rewrite H5 in H3;rewrite H3. - apply H;rewrite Zlt_is_lt_bool;trivial. - revert H1;unfold P';case_eq (Zlt_bool [|max|] [|min|]). - rewrite <- Zlt_is_lt_bool;intros. - rewrite H2;rewrite foldi_cont_gt;[ | rewrite ltb_spec];auto. - rewrite of_to_Z;auto. -Qed. - - -Lemma foldi_cont_ind2 : forall A B C D (P: (A -> B) -> (C -> D) -> Prop) (f:int -> (A -> B) -> (A -> B)) (g:int -> (C -> D) -> (C -> D)) min max cont1 cont2, - P cont1 cont2 -> - (forall i cont1 cont2, min <= i = true -> i <= max = true -> P cont1 cont2 -> P (f i cont1) (g i cont2)) -> - P (foldi_cont f min max cont1) (foldi_cont g min max cont2). -Proof. - intros A B C D P f g min max cont1 cont2 Ha Hf. - set (P2 := fun (z:Z) b c => P b c);change (P2 [|min|] (foldi_cont f min max cont1) (foldi_cont g min max cont2)). - apply foldi_cont_ZInd2;trivial. -Qed. - - -Lemma foldi_ZInd2 : forall A B (P : Z -> A -> B -> Prop) f g min max a b, - (max < min = true -> P ([|max|] + 1)%Z a b) -> - P [|min|] a b -> - (forall i a b, min <= i = true -> i <= max = true -> - P [|i|] a b -> P ([|i|] + 1)%Z (f i a) (g i b)) -> - P ([|max|]+1)%Z (foldi f min max a) (foldi g min max b). -Proof. - unfold foldi;intros A B P f g min max a b Hlt;intros. - set (P' z cont1 cont2 := - if Zlt_bool [|max|] z then cont1 = (fun a : A => a) /\ cont2 = (fun b : B => b) - else forall a b, P z a b -> P ([|max|]+1)%Z (cont1 a) (cont2 b)). - assert (P' [|min|] (foldi_cont (fun (i : int) (cont : A -> A) (a : A) => cont (f i a)) min - max (fun a : A => a)) - (foldi_cont (fun (i : int) (cont : B -> B) (b : B) => cont (g i b)) min - max (fun b : B => b))). - apply foldi_cont_ZInd2;intros;red. - rewrite Zlt_is_lt_bool in H1;rewrite H1;auto. - case_eq (Zlt_bool [|max|] [|i|]);intros. - rewrite <- Zlt_is_lt_bool in H4;rewrite leb_spec in H2;elimtype False;omega. - clear H4; revert H3;unfold P'. - case_eq (Zlt_bool [|max|] ([|i|] + 1));intros;auto. - rewrite <- Zlt_is_lt_bool in H3; assert ([|i|] = [|max|]) by (rewrite leb_spec in H2;omega). - destruct H4;subst;rewrite <- H6;apply H0;trivial. - revert H1;unfold P'. - case_eq (Zlt_bool [|max|] [|min|]);auto. - rewrite <- Zlt_is_lt_bool, <- ltb_spec;intros;rewrite !foldi_cont_gt;auto. -Qed. - - -Lemma foldi_Ind2 : forall A B (P : int -> A -> B -> Prop) f g min max a b, - (max < max_int = true) -> - (max < min = true -> P (max + 1) a b) -> - P min a b -> - (forall i a b, min <= i = true -> i <= max = true -> - P i a b -> P (i + 1) (f i a) (g i b)) -> - P (max+1) (foldi f min max a) (foldi g min max b). -Proof. - intros. - set (P' z a b := (0 <= z < wB)%Z -> P (of_Z z) a b). - assert (W:= to_Z_add_1 _ _ H). - assert (P' ([|max|]+1)%Z (foldi f min max a) (foldi g min max b)). - apply foldi_ZInd2;unfold P';intros. - rewrite <- W, of_to_Z;auto. - rewrite of_to_Z;trivial. - assert (i < max_int = true). - apply leb_ltb_trans with max;trivial. - rewrite <- (to_Z_add_1 _ _ H7), of_to_Z;apply H2;trivial. - rewrite of_to_Z in H5;apply H5;apply to_Z_bounded. - unfold P' in H3;rewrite <- W, of_to_Z in H3;apply H3;apply to_Z_bounded. -Qed. - - -Lemma foldi_ind2 : forall A B (P: A -> B -> Prop) (f:int -> A -> A) (g:int -> B -> B) min max a b, - P a b -> - (forall i a b, min <= i = true -> i <= max = true -> P a b -> P (f i a) (g i b)) -> - P (foldi f min max a) (foldi g min max b). -Proof. - unfold foldi;intros A B P f g min max a b Ha Hr; revert a b Ha. - apply (foldi_cont_ind2 _ _ _ _ (fun cont1 cont2 => forall a b, P a b -> P (cont1 a) (cont2 b))); auto. -Qed. - - -Lemma fold_ind2 : forall A B (P: A -> B -> Prop) (f: A -> A) (g: B -> B) min max a b, - P a b -> (forall a b, P a b -> P (f a) (g b)) -> P (fold f min max a) (fold g min max b). -Proof. - unfold fold;intros A B P f g min max a b Ha Hr;revert a b Ha. - apply (foldi_cont_ind2 _ _ _ _ (fun cont1 cont2 => forall a b, P a b -> P (cont1 a) (cont2 b)));auto. -Qed. - -Lemma foldi_eq_compat : forall A (f1 f2:int -> A -> A) min max a, - (forall i a, min <= i = true -> i <= max = true -> f1 i a = f2 i a) -> - foldi f1 min max a = foldi f2 min max a. -Proof. - intros; set (P' (z:Z) (a1 a2:A) := a1 = a2). - assert (P' ([|max|] + 1)%Z (foldi f1 min max a) (foldi f2 min max a)). - apply foldi_ZInd2;unfold P';intros;subst;auto. - apply H0. -Qed. - -Lemma foldi_down_cont_ZInd2 : - forall A B C D (P: Z -> (A -> B) -> (C -> D) -> Prop) (f1:int -> (A -> B) -> (A -> B)) (f2:int -> (C -> D) -> (C -> D)) max min cont1 cont2, - (forall z, (z < [|min|])%Z -> P z cont1 cont2) -> - (forall i cont1 cont2, min <= i = true -> i <= max = true -> P ([|i|] - 1)%Z cont1 cont2 -> - P [|i|] (f1 i cont1) (f2 i cont2)) -> - P [|max|] (foldi_down_cont f1 max min cont1) (foldi_down_cont f2 max min cont2). -Proof. - intros. - set (P' z cont := - if Zlt_bool z [|min|] then cont = cont1 - else P z cont (foldi_down_cont f2 (of_Z z) min cont2)). - assert (P' [|max|] (foldi_down_cont f1 max min cont1)). - apply foldi_down_cont_ZInd;unfold P';intros. - rewrite Zlt_is_lt_bool in H1;rewrite H1;trivial. - case_eq (Zlt_bool [|i|] [|min|]);intros. - rewrite <- Zlt_is_lt_bool, <- ltb_spec in H4. - elim (not_ltb_refl min);apply leb_ltb_trans with i;trivial. - rewrite of_to_Z;generalize H1;rewrite leb_ltb_eqb, orb_true_iff;intros [Hlt | Heq]. - rewrite foldi_down_cont_gt;[apply H0 | ];trivial. - revert H3;case_eq (Zlt_bool ([|i|] - 1) [|min|]). - rewrite <- Zlt_is_lt_bool;rewrite ltb_spec in Hlt;intros;elimtype False;omega. - rewrite <- (to_Z_sub_1 _ _ Hlt), of_to_Z; intros _ W;exact W. - rewrite eqb_spec in Heq;subst. - rewrite foldi_down_cont_eq;[apply H0 | ];trivial. - assert ([|i|] - 1 < [|i|])%Z by auto with zarith. - rewrite Zlt_is_lt_bool in H5;rewrite H5 in H3;rewrite H3. - apply H;rewrite Zlt_is_lt_bool;trivial. - revert H1;unfold P';case_eq (Zlt_bool [|max|] [|min|]). - rewrite <- Zlt_is_lt_bool;intros. - rewrite H2;rewrite foldi_down_cont_lt;[ | rewrite ltb_spec];auto. - rewrite of_to_Z;auto. -Qed. - - -Lemma foldi_down_cont_ind2 : forall A B C D (P: (A -> B) -> (C -> D) -> Prop) (f:int -> (A -> B) -> (A -> B)) (g:int -> (C -> D) -> (C -> D)) max min cont1 cont2, - P cont1 cont2 -> - (forall i cont1 cont2, min <= i = true -> i <= max = true -> P cont1 cont2 -> P (f i cont1) (g i cont2)) -> - P (foldi_down_cont f max min cont1) (foldi_down_cont g max min cont2). -Proof. - intros A B C D P f g max min cont1 cont2 Ha Hf. - set (P2 := fun (z:Z) b c => P b c);change (P2 [|max|] (foldi_down_cont f max min cont1) (foldi_down_cont g max min cont2)). - apply foldi_down_cont_ZInd2;trivial. -Qed. - - -Lemma foldi_down_ZInd2 : - forall A B (P: Z -> A -> B -> Prop) (f1:int -> A -> A) (f2:int -> B -> B) max min a1 a2, - (max < min = true -> P ([|min|] - 1)%Z a1 a2) -> - (P [|max|] a1 a2) -> - (forall z, (z < [|min|])%Z -> P z a1 a2) -> - (forall i a1 a2, min <= i = true -> i <= max = true -> P [|i|] a1 a2 -> - P ([|i|] - 1)%Z (f1 i a1) (f2 i a2)) -> - P ([|min|] - 1)%Z (foldi_down f1 max min a1) (foldi_down f2 max min a2). -Proof. - unfold foldi_down;intros A B P f1 f2 max min a1 a2 Hlt;intros. - set (P' z cont1 cont2 := - if Zlt_bool z [|min|] then cont1 = (fun a0 : A => a0) /\ cont2 = (fun a0 : B => a0) - else forall a1 a2, P z a1 a2 -> P ([|min|] - 1)%Z (cont1 a1) (cont2 a2)). - assert (P' [|max|] (foldi_down_cont (fun (i : int) (cont : A -> A) (a0 : A) => cont (f1 i a0)) max - min (fun a0 : A => a0)) - (foldi_down_cont (fun (i : int) (cont : B -> B) (a0 : B) => cont (f2 i a0)) max - min (fun a0 : B => a0))). - apply foldi_down_cont_ZInd2;intros;red. - rewrite Zlt_is_lt_bool in H2;rewrite H2;auto. - case_eq (Zlt_bool [|i|] [|min|]);intros. - rewrite <- Zlt_is_lt_bool in H5;rewrite leb_spec in H2;elimtype False;omega. - clear H5;revert H4;unfold P'. - case_eq (Zlt_bool ([|i|] - 1) [|min|]);intros;auto. - rewrite <- Zlt_is_lt_bool in H4; assert ([|i|] = [|min|]) by (rewrite leb_spec in H2;omega). - destruct H5;subst;rewrite <- H7;apply H1;trivial. - revert H2;unfold P'. - case_eq (Zlt_bool [|max|] [|min|]);auto. - rewrite <- Zlt_is_lt_bool, <- ltb_spec;intros;rewrite foldi_down_cont_lt;auto. - destruct H3. rewrite H4;auto. -Qed. - - -Lemma foldi_down_ind2 : forall A B (P: A -> B -> Prop) (f:int -> A -> A) (g:int -> B -> B) max min a b, - P a b -> - (forall i a b, min <= i = true -> i <= max = true -> P a b -> P (f i a) (g i b)) -> - P (foldi_down f max min a) (foldi_down g max min b). -Proof. - unfold foldi_down;intros A B P f g max min a b Ha Hr;revert a b Ha. - apply (foldi_down_cont_ind2 _ _ _ _ (fun cont1 cont2 => forall a b, P a b -> P (cont1 a) (cont2 b)));auto. -Qed. - - -Lemma fold_down_ind2 : forall A B (P: A -> B -> Prop) (f: A -> A) (g: B -> B) max min a b, - P a b -> (forall a b, P a b -> P (f a) (g b)) -> P (fold_down f max min a) (fold_down g max min b). -Proof. - unfold fold_down;intros A B P f g max min a b Ha Hr;revert a b Ha. - apply (foldi_down_cont_ind2 _ _ _ _ (fun cont1 cont2 => forall a b, P a b -> P (cont1 a) (cont2 b)));auto. -Qed. - -Lemma foldi_down_eq_compat : forall A (f1 f2:int -> A -> A) max min a, - (forall i a, min <= i = true -> i <= max = true -> f1 i a = f2 i a) -> - foldi_down f1 max min a = foldi_down f2 max min a. -Proof. - intros; set (P' (z:Z) (a1 a2:A) := a1 = a2). - assert (P' ([|min|] - 1)%Z (foldi_down f1 max min a) (foldi_down f2 max min a)). - apply foldi_down_ZInd2;unfold P';intros;subst;auto. - apply H0. -Qed. - - -Lemma forallb_spec : forall f from to, - forallb f from to = true <-> - forall i, from <= i = true -> i <= to = true -> f i = true. -Proof. - unfold forallb;intros f from to. - setoid_rewrite leb_spec. - apply foldi_cont_ZInd. - intros;split;[intros;elimtype False;omega | trivial]. - intros i cont Hfr Hto Hcont. - case_eq (f i);intros Heq. - rewrite Hcont;clear Hcont;split;auto with zarith;intros. - assert (H2 : ([|i0|] = [|i|] \/ [|i|] + 1 <= [|i0|])%Z) by omega; destruct H2;auto with zarith. - apply to_Z_inj in H2;rewrite H2;trivial. - split;[discriminate | intros]. - rewrite leb_spec in Hto;rewrite <- Heq;auto with zarith. -Qed. - -Lemma forallb_eq_compat : forall f1 f2 from to, - (forall i, from <= i = true -> i <= to = true -> f1 i = f2 i) -> - forallb f1 from to = forallb f2 from to. -Proof. - unfold forallb;intros. - set (P' (z:Z) (cont1 cont2:unit -> bool) := cont1 tt = cont2 tt). - refine (foldi_cont_ZInd2 _ _ _ _ P' _ _ from to _ _ _ _);unfold P';intros;trivial. - rewrite H2, H;trivial. -Qed. - -Lemma existsb_spec : forall f from to, - existsb f from to = true <-> - exists i, ((from <= i) && (i <= to) && (f i)) = true . -Proof. - unfold existsb;intros. - repeat setoid_rewrite andb_true_iff;setoid_rewrite leb_spec. - apply foldi_cont_ZInd. - intros;split;[discriminate | intros [i [W1 W2]];elimtype False;omega]. - intros i cont Hfr Hto Hcont. - case_eq (f i);intros Heq. - split;trivial. - exists i;rewrite leb_spec in Hto;auto with zarith. - rewrite Hcont;clear Hcont;split;intros [i0 [W1 W2]]; - exists i0;split;auto with zarith. - assert (~ [|i|] = [|i0|]);[ | auto with zarith]. - intros W;apply to_Z_inj in W;rewrite W in Heq;rewrite Heq in W2;discriminate. -Qed. - -Lemma existsb_eq_compat : forall f1 f2 from to, - (forall i, from <= i = true -> i <= to = true -> f1 i = f2 i) -> - existsb f1 from to = existsb f2 from to. -Proof. - unfold existsb;intros. - set (P' (z:Z) (cont1 cont2:unit -> bool) := cont1 tt = cont2 tt). - refine (foldi_cont_ZInd2 _ _ _ _ P' _ _ from to _ _ _ _);unfold P';intros;trivial. - rewrite H2, H;trivial. -Qed. - - -Lemma bit_max_int : forall i, (i < digits)%int = true -> bit max_int i = true. -Proof. - intros;apply (forallb_spec (bit max_int) 0 (digits - 1)). - vm_compute;trivial. - apply leb_0. - rewrite ltb_spec in H. - destruct (to_Z_bounded i);rewrite leb_spec. - change [|digits - 1 |] with ([|digits|] - 1)%Z;omega. -Qed. - -Lemma land_max_int_l : forall i, max_int land i = i. -Proof. - intros;apply bit_eq;intros. - rewrite land_spec. - destruct (reflect_leb digits i0). - rewrite <- leb_spec in l. - rewrite !bit_M;trivial. - rewrite bit_max_int;trivial. - rewrite ltb_spec;omega. -Qed. - -Lemma land_max_int_r : forall i, i land max_int = i. -Proof. - intros;rewrite land_comm;apply land_max_int_l. -Qed. - - -(* int is an OrderedType *) - -Require Import OrderedType. - -Module IntOrderedType <: OrderedType. - - Definition t := int. - - Definition eq x y := (x == y) = true. - - Definition lt x y := (x < y) = true. - - Lemma eq_refl x : eq x x. - Proof. unfold eq. rewrite eqb_spec. reflexivity. Qed. - - Lemma eq_sym x y : eq x y -> eq y x. - Proof. unfold eq. rewrite !eqb_spec. intros ->. reflexivity. Qed. - - Lemma eq_trans x y z : eq x y -> eq y z -> eq x z. - Proof. unfold eq. rewrite !eqb_spec. intros -> ->. reflexivity. Qed. - - Lemma lt_trans x y z : lt x y -> lt y z -> lt x z. - Proof. apply ltb_trans. Qed. - - Lemma lt_not_eq x y : lt x y -> ~ eq x y. - Proof. unfold lt, eq. rewrite ltb_negb_geb, eqb_spec. intros H1 H2. rewrite H2, leb_refl in H1. discriminate. Qed. - - Definition compare x y : Compare lt eq x y. - Proof. - case_eq (x < y); intro e. - exact (LT e). - case_eq (x == y); intro e2. - exact (EQ e2). apply GT. unfold lt. rewrite ltb_negb_geb, leb_ltb_eqb, e, e2. reflexivity. - Defined. - - Definition eq_dec x y : { eq x y } + { ~ eq x y }. - Proof. - case_eq (x == y); intro e. - left; exact e. - right. intro H. rewrite H in e. discriminate. - Defined. - -End IntOrderedType. - - -(* - Local Variables: - coq-load-path: ((rec "../../.." "SMTCoq")) - End: -*) diff --git a/src/versions/standard/Int63/Int63_standard.v b/src/versions/standard/Int63/Int63_standard.v deleted file mode 100644 index acee305..0000000 --- a/src/versions/standard/Int63/Int63_standard.v +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* *) -(* SMTCoq *) -(* Copyright (C) 2011 - 2021 *) -(* *) -(* See file "AUTHORS" for the list of authors *) -(* *) -(* This file is distributed under the terms of the CeCILL-C licence *) -(* *) -(**************************************************************************) - - -(** Glue with the Int31 library of standard coq, which is linked to - native integers during VM computations. - - CAUTION: The name "Int63" is given for compatibility reasons, but - int31 is used. **) - -Require Export Ring31. -Require Export Int63Native. -Require Export Int63Op. -Require Export Int63Axioms. -Require Export Int63Properties. |