| Require Import Coq.ZArith.ZArith. |
| Require Import Coq.Lists.List. |
| Require Import Coq.micromega.Lia. |
| From AUXLib Require Import ListLib. |
| From SimpleC.SL Require Import Mem SeparationLogic ArrayLib. |
| Require Import Logic.LogicGenerator.demo932.Interface. |
|
|
| Import ListNotations. |
| Local Open Scope Z_scope. |
| Local Open Scope list_scope. |
| Import naive_C_Rules. |
| Local Open Scope sac. |
|
|
| (** |
| [sll] is the abstract, singly-linked-list-shaped sequence used to describe |
| stack contents. It is deliberately independent of the concrete memory |
| representation: this case stores the sequence in a contiguous C array. |
| The head of an [sll] is the logical top of the stack. |
| *) |
| Require Export PVbench.Data_structures.stack.rocq.spec_lib. |
|
|
| Lemma stack_representation_push__push_state : |
| forall (before : sll) (concrete : list Z) (n x : Z), |
| stack_representation before concrete n -> |
| n < stack_capacity -> |
| stack_representation |
| (sll_cons x before) (concrete ++ [x]) (n + 1). |
| Proof. |
| intros before concrete n x Hrep Hcapacity. |
| unfold stack_representation in Hrep |- *. |
| destruct Hrep as |
| (Hnonnegative & Hbound & Habstract_length & Hconcrete_length & Hreverse). |
| unfold sll_cons. |
| repeat split. |
| - lia. |
| - lia. |
| - rewrite Zlength_cons. |
| lia. |
| - rewrite Zlength_app, Zlength_cons, Zlength_nil. |
| lia. |
| - rewrite Hreverse. |
| reflexivity. |
| Qed. |
|
|
| Lemma stack_representation_pop__pop_state : |
| forall top rest concrete size, |
| 1 <= size -> |
| stack_representation (sll_cons top rest) concrete size -> |
| concrete = rev rest ++ [top] /\ |
| Zlength (rev rest) = size - 1 /\ |
| stack_representation rest (rev rest) (size - 1) /\ |
| Znth (size - 1) concrete 0 = top. |
| Proof. |
| intros top rest concrete size Hpositive Hrepresentation. |
| unfold stack_representation in Hrepresentation. |
| unfold sll_cons in Hrepresentation. |
| destruct Hrepresentation as |
| [Hnonnegative |
| [Hcapacity |
| [Hcontents_length [Hconcrete_length Hconcrete]]]]. |
| assert (Hrest_length : Zlength rest = size - 1). |
| { |
| rewrite Zlength_cons in Hcontents_length. |
| lia. |
| } |
| assert (Hrev_rest_length : Zlength (rev rest) = size - 1). |
| { |
| rewrite Zlength_correct, length_rev, <- Zlength_correct. |
| exact Hrest_length. |
| } |
| assert (Hconcrete_pop : concrete = rev rest ++ [top]). |
| { |
| rewrite Hconcrete. |
| simpl. |
| reflexivity. |
| } |
| split. |
| - exact Hconcrete_pop. |
| - split. |
| + exact Hrev_rest_length. |
| + split. |
| * unfold stack_representation. |
| repeat split. |
| -- lia. |
| -- lia. |
| -- exact Hrest_length. |
| -- exact Hrev_rest_length. |
| * rewrite Hconcrete_pop. |
| rewrite app_Znth2 by lia. |
| replace (size - 1 - Zlength (rev rest)) with 0 by lia. |
| apply Znth0_cons. |
| Qed. |
|
|
| Lemma stack_representation_prefix__build_loop : |
| forall (input : list Z) (k : Z), |
| 0 <= k <= Zlength input -> |
| k <= stack_capacity -> |
| stack_representation |
| (sll_from_array (sublist 0 k input)) (sublist 0 k input) k. |
| Proof. |
| intros input k Hrange Hcapacity. |
| unfold stack_representation, sll_from_array. |
| repeat split; try lia. |
| - rewrite Zlength_correct, length_rev, <- Zlength_correct. |
| rewrite Zlength_sublist by lia. |
| lia. |
| - rewrite Zlength_sublist by lia. |
| lia. |
| - rewrite rev_involutive. |
| reflexivity. |
| Qed. |
|
|
| Lemma build_stack_prefix_succ__build_loop : |
| forall (prefix : sll) (input : list Z) (i x : Z), |
| 0 <= i < Zlength input -> |
| BuildStackPrefix prefix input i -> |
| x = Znth i input 0 -> |
| BuildStackPrefix (sll_cons x prefix) input (i + 1). |
| Proof. |
| intros prefix input i x Hrange Hprefix Hx. |
| unfold BuildStackPrefix, sll_cons, sll_from_array in *. |
| subst prefix x. |
| rewrite (sublist_split 0 (i + 1) i input) by lia. |
| rewrite (sublist_single 0 i input) by lia. |
| rewrite rev_app_distr. |
| reflexivity. |
| Qed. |
|
|
| Lemma build_stack_prefix_complete__build_completion : |
| forall prefix input processed size, |
| processed >= size -> |
| processed <= size -> |
| Zlength input = size -> |
| BuildStackPrefix prefix input processed -> |
| processed = size /\ prefix = sll_from_array input. |
| Proof. |
| intros prefix input processed size Hge Hle Hlength Hprefix. |
| assert (Hprocessed : processed = size) by lia. |
| subst processed. |
| split. |
| - reflexivity. |
| - unfold BuildStackPrefix in Hprefix. |
| rewrite sublist_self in Hprefix by lia. |
| exact Hprefix. |
| Qed. |
|
|
|
|
