list_reverse.v 1.59 KB
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(** Correctness of in-place list reversal *)
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From iris.proofmode Require Export tactics.
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From iris.program_logic Require Export total_weakestpre weakestpre.
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From iris.heap_lang Require Export lang.
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From iris.heap_lang Require Import proofmode notation.
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Set Default Proof Using "Type".
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Section list_reverse.
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Context `{!heapG Σ}.
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Implicit Types l : loc.

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Fixpoint is_list (hd : val) (xs : list val) : iProp Σ :=
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  match xs with
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  | [] => hd = NONEV
  | x :: xs =>  l hd', hd = SOMEV #l  l  (x,hd')  is_list hd' xs
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  end%I.

Definition rev : val :=
  rec: "rev" "hd" "acc" :=
    match: "hd" with
      NONE => "acc"
    | SOME "l" =>
       let: "tmp1" := Fst !"l" in
       let: "tmp2" := Snd !"l" in
       "l" <- ("tmp1", "acc");;
       "rev" "tmp2" "hd"
    end.

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Lemma rev_acc_wp hd acc xs ys :
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  [[{ is_list hd xs  is_list acc ys }]]
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    rev hd acc
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  [[{ w, RET w; is_list w (reverse xs ++ ys) }]].
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Proof.
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  iIntros (Φ) "[Hxs Hys] HΦ". Show.
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  iInduction xs as [|x xs] "IH" forall (hd acc ys Φ);
    iSimplifyEq; wp_rec; wp_let.
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  - Show. wp_match. by iApply "HΦ".
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  - iDestruct "Hxs" as (l hd' ->) "[Hx Hxs]".
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    wp_load. wp_load. wp_store.
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    iApply ("IH" $! hd' (SOMEV #l) (x :: ys) with "Hxs [Hx Hys]"); simpl.
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    { iExists l, acc; by iFrame. }
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    iIntros (w). rewrite cons_middle assoc -reverse_cons. iApply "HΦ".
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Qed.

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Lemma rev_wp hd xs :
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  [[{ is_list hd xs }]] rev hd NONEV [[{ w, RET w; is_list w (reverse xs) }]].
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Proof.
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  iIntros (Φ) "Hxs HΦ".
  iApply (rev_acc_wp hd NONEV xs [] with "[$Hxs //]").
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  iIntros (w). rewrite right_id_L. iApply "HΦ".
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Qed.
End list_reverse.