Commit c70cc173 authored by Ralf Jung's avatar Ralf Jung

show that having saved propositions without a \later is inconsistent

parent 9ba4ad2b
......@@ -85,6 +85,7 @@ program_logic/auth.v
program_logic/sts.v
program_logic/namespaces.v
program_logic/boxes.v
program_logic/counter_examples.v
heap_lang/lang.v
heap_lang/tactics.v
heap_lang/wp_tactics.v
......
......@@ -345,11 +345,18 @@ Proof.
Qed.
Global Instance: AntiSymm () (@uPred_entails M).
Proof. intros P Q HPQ HQP; split=> x n; by split; [apply HPQ|apply HQP]. Qed.
Lemma sound : ¬ (True False).
Proof.
unseal. intros [H]. apply (H 0 ); last done. apply ucmra_unit_validN.
Qed.
Lemma equiv_spec P Q : (P Q) (P Q) (Q P).
Proof.
split; [|by intros [??]; apply (anti_symm ())].
intros HPQ; split; split=> x i; apply HPQ.
Qed.
Lemma equiv_entails P Q : (P Q) (P Q).
Proof. apply equiv_spec. Qed.
Lemma equiv_entails_sym P Q : (Q P) (P Q).
......
From iris.algebra Require Import upred.
From iris.proofmode Require Import tactics.
(** This proves that we need the ▷ in a "Saved Proposition" construction with
name-dependend allocation. *)
(** We fork in [uPred M] for any M, but the proof would work in any BI. *)
Section savedprop.
Context (M : ucmraT).
Notation iProp := (uPred M).
Notation "¬ P" := ((P False))%I : uPred_scope.
(* Saved Propositions. *)
Context (sprop : Type) (saved : sprop iProp iProp).
Hypothesis sprop_persistent : forall i P, PersistentP (saved i P).
Hypothesis sprop_alloc_dep :
forall (P : sprop iProp), True i, saved i (P i).
Hypothesis sprop_agree :
forall i P Q, saved i P saved i Q P Q.
(* Self-contradicting assertions are inconsistent *)
Lemma no_self_contradiction (P : iProp) `{!PersistentP P} :
(P ¬ P) False.
Proof. (* FIXME: Cannot destruct the <-> as two implications. iApply with <-> also does not work. *)
rewrite /uPred_iff. iIntros "#[H1 H2]". (* FIXME: Cannot iApply "H1". *)
iAssert P as "#HP".
{ iApply "H2". iIntros "!#HP". by iApply ("H1" with "HP"). }
by iApply ("H1" with "HP HP").
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).
Proof.
rewrite /uPred_iff. iIntros "#HS !". iSplit.
- iIntros "H". iDestruct "H" as (Q) "[#HSQ HQ]".
iPoseProof (sprop_agree i P Q with "[]") as "Heq"; first by eauto.
rewrite /uPred_iff. by iApply "Heq".
- iIntros "#HP". iExists P. by iSplit.
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". iApply (@saved_is_A i (¬ A i)%I _). (* FIXME: If we already introduced the box, this iApply does not work. *)
done.
Qed.
(* We can obtain such a [Q i]. *)
Lemma make_Q :
True i, Q i.
Proof.
apply sprop_alloc_dep.
Qed.
(* Put together all the pieces to derive a contradiction. *)
Lemma contradiction : False.
Proof.
apply (@uPred.sound M). iIntros "".
iPoseProof make_Q as "HQ". iDestruct "HQ" as (i) "HQ".
iApply (@no_self_contradiction (A i) _).
by iApply Q_self_contradiction.
Qed.
End savedprop.
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