### 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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