Commit 19e70e9e authored by Ralf Jung's avatar Ralf Jung

make the two proofs of contradictions more similar to each other

parent 1c5a85f0
......@@ -14,49 +14,38 @@ Module savedprop. Section savedprop.
Hypothesis sprop_persistent : i P, PersistentP (saved i P).
Hypothesis sprop_alloc_dep :
(P : sprop iProp), True =r=> ( i, saved i (P i)).
Hypothesis sprop_agree : i P Q, saved i P saved i Q P Q.
Hypothesis sprop_agree : i P Q, saved i P saved i Q (P Q).
(* Self-contradicting assertions are inconsistent *)
Lemma no_self_contradiction P `{!PersistentP P} : (P ¬ P) False.
Proof.
iIntros "#[H1 H2]".
iAssert P as "#HP".
{ iApply "H2". iIntros "!# #HP". by iApply ("H1" with "[#]"). }
by iApply ("H1" with "[#]").
Qed.
(** A bad recursive reference: "Assertion with name [i] does not hold" *)
Definition A (i : sprop) : iProp := P, ¬ P saved i P.
Lemma A_alloc : True =r=> i, saved i (A i).
Proof. by apply sprop_alloc_dep. Qed.
(* "Assertion with name [i]" is equivalent to any assertion P s.t. [saved i P] *)
Definition A (i : sprop) : iProp := P, saved i P P.
Lemma saved_is_A i P `{!PersistentP P} : saved i P (A i P).
Lemma saved_NA i : saved i (A i) ¬ A i.
Proof.
iIntros "#HS !#". iSplit.
- iDestruct 1 as (Q) "[#HSQ HQ]".
iApply (sprop_agree i P Q with "[]"); eauto.
- iIntros "#HP". iExists P. by iSplit.
iIntros "#Hs !# #HA". iPoseProof "HA" as "HA'".
iDestruct "HA'" as (P) "[#HNP HsP]". iApply "HNP".
iDestruct (sprop_agree i P (A i) with "[]") as "#[_ HP]".
{ eauto. }
iApply "HP". done.
Qed.
(* Define [Q i] to be "negated assertion with name [i]". Show that this
implies that assertion with name [i] is equivalent to its own negation. *)
Definition Q i := saved i (¬ A i).
Lemma Q_self_contradiction i : Q i (A i ¬ A i).
Proof. iIntros "#HQ !#". by iApply (saved_is_A i (¬A i)). Qed.
(* We can obtain such a [Q i]. *)
Lemma make_Q : True =r=> i, Q i.
Proof. apply sprop_alloc_dep. Qed.
(* Put together all the pieces to derive a contradiction. *)
Lemma rvs_false : (True : uPred M) =r=> False.
Lemma saved_A i : saved i (A i) A i.
Proof.
rewrite make_Q. apply uPred.rvs_mono. iDestruct 1 as (i) "HQ".
iApply (no_self_contradiction (A i)). by iApply Q_self_contradiction.
iIntros "#Hs". iExists (A i). iFrame "#".
by iApply saved_NA.
Qed.
Lemma contradiction : False.
Proof.
apply (@uPred.adequacy M False 1); simpl.
rewrite -uPred.later_intro. apply rvs_false.
iIntros "". iVs A_alloc as (i) "#H".
iPoseProof (saved_NA with "H") as "HN".
iVsIntro. iNext.
iApply "HN". iApply saved_A. done.
Qed.
End savedprop. End savedprop.
(** This proves that we need the ▷ when opening invariants. *)
......@@ -180,26 +169,26 @@ Module inv. Section inv.
Lemma A_alloc : True pvs M1 ( i, saved i (A i)).
Proof. by apply saved_alloc. Qed.
Lemma alloc_NA i : saved i (A i) ¬A i.
Lemma saved_NA i : saved i (A i) ¬A i.
Proof.
iIntros "#Hi !# #HA". iPoseProof "HA" as "HA'".
iDestruct "HA'" as (P) "#[HNP Hi']".
iVs (saved_cast i with "[]") as "HP".
{ iSplit; first iExact "Hi". by iFrame "#". }
iVs (saved_cast i (A i) P with "[]") as "HP".
{ eauto. }
by iApply "HNP".
Qed.
Lemma alloc_A i : saved i (A i) A i.
Lemma saved_A i : saved i (A i) A i.
Proof.
iIntros "#Hi". iPoseProof (alloc_NA with "Hi") as "HNA".
iExists (A i). by iFrame "#".
iIntros "#Hi". iExists (A i). iFrame "#".
by iApply saved_NA.
Qed.
Lemma contradiction : False.
Proof.
apply soundness. iIntros "".
iVs A_alloc as (i) "#H".
iPoseProof (alloc_NA with "H") as "HN".
iApply "HN". iApply alloc_A. done.
iPoseProof (saved_NA with "H") as "HN".
iApply "HN". iApply saved_A. done.
Qed.
End inv. End inv.
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