From 1cd3c4c9b7372ca8de128a9ce60ed00210fd0e28 Mon Sep 17 00:00:00 2001 From: David Monniaux Date: Mon, 3 Feb 2020 21:35:34 +0100 Subject: CSE2 with NOTRAP --- backend/CSE2proof.v | 275 +++++++++++++++++++++++++++++++++++----------------- 1 file changed, 184 insertions(+), 91 deletions(-) (limited to 'backend/CSE2proof.v') diff --git a/backend/CSE2proof.v b/backend/CSE2proof.v index 3b28cf84..254cc4ce 100644 --- a/backend/CSE2proof.v +++ b/backend/CSE2proof.v @@ -475,7 +475,6 @@ Proof. pose proof (REL r) as RELr. rewrite RELatr in RELr. simpl in RELr. - Show. destruct eval_addressing. { destruct Mem.loadv. congruence. @@ -512,6 +511,46 @@ Proof. discriminate. Qed. +Lemma find_load_notrap1_sound' : + forall rel : RELATION.t, + forall chunk : memory_chunk, + forall addr : addressing, + forall src : reg, + forall args: list reg, + forall rs : regset, + sem_rel rel rs -> + find_load rel chunk addr args = Some src -> + eval_addressing genv sp addr rs##args = None -> + rs # src = (default_notrap_load_value chunk). +Proof. + intros until rs. intros REL FINDLOAD ADDR. + pose proof (find_load_sound rel chunk addr src args rs REL FINDLOAD) as Z. + rewrite ADDR in Z. + assumption. +Qed. + +Lemma find_load_notrap2_sound' : + forall rel : RELATION.t, + forall chunk : memory_chunk, + forall addr : addressing, + forall src : reg, + forall args: list reg, + forall rs : regset, + forall a, + sem_rel rel rs -> + find_load rel chunk addr args = Some src -> + eval_addressing genv sp addr rs##args = Some a -> + Mem.loadv chunk m a = None -> + rs # src = (default_notrap_load_value chunk). +Proof. + intros until a. intros REL FINDLOAD ADDR LOAD. + pose proof (find_load_sound rel chunk addr src args rs REL FINDLOAD) as Z. + rewrite ADDR in Z. + destruct Mem.loadv. + discriminate. + assumption. +Qed. + Lemma forward_move_map: forall rel args rs, sem_rel rel rs -> @@ -1186,82 +1225,49 @@ Proof. (* load *) - unfold transf_instr in *. - destruct trap. - { (* TRAP *) - destruct find_load_in_fmap eqn:FIND_LOAD. + destruct find_load_in_fmap eqn:FIND_LOAD. + { + unfold find_load_in_fmap, fmap_sem', fmap_sem in *. + destruct (forward_map f) as [map |] eqn:MAP. + 2: discriminate. + change (@PMap.get (option RELATION.t) pc map) with (map # pc) in *. + destruct (map # pc) as [mpc | ] eqn:MPC. + 2: discriminate. + econstructor; split. { - unfold find_load_in_fmap, fmap_sem', fmap_sem in *. - destruct (forward_map f) as [map |] eqn:MAP. - 2: discriminate. - change (@PMap.get (option RELATION.t) pc map) with (map # pc) in *. - destruct (map # pc) as [mpc | ] eqn:MPC. - 2: discriminate. - econstructor; split. - { - eapply exec_Iop with (v := v); eauto. - simpl. - rewrite <- subst_args_ok with (genv := ge) (f := f) (pc := pc) (sp := sp) (m := m) in H0. - { - f_equal. - symmetry. - apply find_load_sound' with (chunk := chunk) (m := m) (a := a) (genv := ge) (sp := sp) (addr := addr) (args := subst_args (Some map) pc args) (rel := mpc) (src := r) (rs := rs); trivial. - rewrite MAP in H0. - assumption. - } - unfold fmap_sem. rewrite MAP. rewrite MPC. assumption. - } - constructor; eauto. - unfold fmap_sem', fmap_sem in *. - rewrite MAP. - apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + eapply exec_Iop with (v := v); eauto. + simpl. + rewrite <- subst_args_ok with (genv := ge) (f := f) (pc := pc) (sp := sp) (m := m) in H0. { - replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). - { - eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. - 2: apply apply_instr'_bot. - simpl. tauto. - } - unfold apply_instr'. - rewrite H. - rewrite MPC. - simpl. - reflexivity. + f_equal. + symmetry. + apply find_load_sound' with (chunk := chunk) (m := m) (a := a) (genv := ge) (sp := sp) (addr := addr) (args := subst_args (Some map) pc args) (rel := mpc) (src := r) (rs := rs); trivial. + rewrite MAP in H0. + assumption. } - unfold sem_rel_b', sem_rel_b. - apply load_sound with (a := a); auto. + unfold fmap_sem. rewrite MAP. rewrite MPC. assumption. } - { - econstructor; split. - assert (eval_addressing tge sp addr rs ## args = Some a). - rewrite <- H0. - apply eval_addressing_preserved. exact symbols_preserved. - eapply exec_Iload; eauto. - rewrite (subst_args_ok' sp m); assumption. - constructor; auto. - - simpl in *. - unfold fmap_sem', fmap_sem in *. - destruct (forward_map _) as [map |] eqn:MAP in *; trivial. - destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction. - apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + constructor; eauto. + unfold fmap_sem', fmap_sem in *. + rewrite MAP. + apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + { + replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). { - replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). - { - eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. - 2: apply apply_instr'_bot. - simpl. tauto. - } - unfold apply_instr'. - rewrite H. - rewrite MPC. - simpl. - reflexivity. + eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. + 2: apply apply_instr'_bot. + simpl. tauto. } - apply load_sound with (a := a); assumption. + unfold apply_instr'. + rewrite H. + rewrite MPC. + simpl. + reflexivity. } + unfold sem_rel_b', sem_rel_b. + apply load_sound with (a := a); auto. } - - { (* NOTRAP *) + { econstructor; split. assert (eval_addressing tge sp addr rs ## args = Some a). rewrite <- H0. @@ -1291,35 +1297,121 @@ Proof. apply load_sound with (a := a); assumption. } -- (* load notrap1 *) - econstructor; split. - assert (eval_addressing tge sp addr rs ## args = None). - rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved. - eapply exec_Iload_notrap1; eauto. - rewrite subst_args_ok with (genv := ge) (sp := sp) (m := m); assumption. - constructor; auto. +- unfold transf_instr in *. + destruct find_load_in_fmap eqn:FIND_LOAD. + { + unfold find_load_in_fmap, fmap_sem', fmap_sem in *. + destruct (forward_map f) as [map |] eqn:MAP. + 2: discriminate. + change (@PMap.get (option RELATION.t) pc map) with (map # pc) in *. + destruct (map # pc) as [mpc | ] eqn:MPC. + 2: discriminate. - simpl in *. - unfold fmap_sem', fmap_sem in *. - destruct (forward_map _) as [map |] eqn:MAP in *; trivial. - destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction. - apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + econstructor; split. + { + eapply exec_Iop with (v := (default_notrap_load_value chunk)); eauto. + rewrite <- subst_args_ok with (genv := ge) (f := f) (pc := pc) (sp := sp) (m := m) in H0. + { simpl. + f_equal. + apply find_load_notrap1_sound' with (chunk := chunk) (m := m) (genv := ge) (sp := sp) (addr := addr) (args := subst_args (Some map) pc args) (rel := mpc) (src := r) (rs := rs); trivial. + rewrite MAP in H0. + assumption. + } + unfold fmap_sem. rewrite MAP. rewrite MPC. assumption. + } + constructor; eauto. + unfold fmap_sem', fmap_sem in *. + rewrite MAP. + apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + { + replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). + { + eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. + 2: apply apply_instr'_bot. + simpl. tauto. + } + unfold apply_instr'. + rewrite H. + rewrite MPC. + simpl. + reflexivity. + } + unfold sem_rel_b', sem_rel_b. + apply load_notrap1_sound; auto. + } { - replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). + econstructor; split. + assert (eval_addressing tge sp addr rs ## args = None). + rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved. + eapply exec_Iload_notrap1; eauto. + rewrite subst_args_ok with (genv := ge) (sp := sp) (m := m) ; assumption. + constructor; auto. + + simpl in *. + unfold fmap_sem', fmap_sem in *. + destruct (forward_map _) as [map |] eqn:MAP in *; trivial. + destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction. + apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). { - eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. - 2: apply apply_instr'_bot. - simpl. tauto. + replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). + { + eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. + 2: apply apply_instr'_bot. + simpl. tauto. + } + unfold apply_instr'. + rewrite H. + rewrite MPC. + simpl. + reflexivity. } - unfold apply_instr'. - rewrite H. - rewrite MPC. - simpl. - reflexivity. + apply load_notrap1_sound; trivial. } - apply load_notrap1_sound; trivial. - (* load notrap2 *) + unfold transf_instr in *. + destruct find_load_in_fmap eqn:FIND_LOAD. + { + unfold find_load_in_fmap, fmap_sem', fmap_sem in *. + destruct (forward_map f) as [map |] eqn:MAP. + 2: discriminate. + change (@PMap.get (option RELATION.t) pc map) with (map # pc) in *. + destruct (map # pc) as [mpc | ] eqn:MPC. + 2: discriminate. + econstructor; split. + { + eapply exec_Iop with (v := (default_notrap_load_value chunk)); eauto. + simpl. + rewrite <- subst_args_ok with (genv := ge) (f := f) (pc := pc) (sp := sp) (m := m) in H0. + { + f_equal. + apply find_load_notrap2_sound' with (chunk := chunk) (m := m) (a := a) (genv := ge) (sp := sp) (addr := addr) (args := subst_args (Some map) pc args) (rel := mpc) (src := r) (rs := rs); trivial. + rewrite MAP in H0. + assumption. + } + unfold fmap_sem. rewrite MAP. rewrite MPC. assumption. + } + constructor; eauto. + unfold fmap_sem', fmap_sem in *. + rewrite MAP. + apply sem_rel_b_ge with (rb2 := Some (load chunk addr dst args mpc)). + { + replace (Some (load chunk addr dst args mpc)) with (apply_instr' (fn_code f) pc (map # pc)). + { + eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption. + 2: apply apply_instr'_bot. + simpl. tauto. + } + unfold apply_instr'. + rewrite H. + rewrite MPC. + simpl. + reflexivity. + } + unfold sem_rel_b', sem_rel_b. + apply load_notrap2_sound with (a := a); auto. + } + { econstructor; split. assert (eval_addressing tge sp addr rs ## args = Some a). rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved. @@ -1346,6 +1438,7 @@ Proof. reflexivity. } apply load_notrap2_sound with (a := a); assumption. + } - (* store *) econstructor. split. -- cgit