From iris.proofmode Require Import tactics. From stdpp Require Import gmap. Set Default Proof Using "Type". Section tests. Context {PROP : sbi}. Implicit Types P Q R : PROP. Lemma demo_0 P Q : □ (P ∨ Q) -∗ (∀ x, ⌜x = 0⌝ ∨ ⌜x = 1⌝) → (Q ∨ P). Proof. iIntros "H #H2". iDestruct "H" as "###H". (* should remove the disjunction "H" *) iDestruct "H" as "[#?|#?]"; last by iLeft. (* should keep the disjunction "H" because it is instantiated *) iDestruct ("H2" \$! 10) as "[%|%]". done. done. Qed. Lemma demo_2 P1 P2 P3 P4 Q (P5 : nat → PROP) `{!Affine P4, !Absorbing P2} : P2 ∗ (P3 ∗ Q) ∗ True ∗ P1 ∗ P2 ∗ (P4 ∗ (∃ x:nat, P5 x ∨ P3)) ∗ emp -∗ P1 -∗ (True ∗ True) -∗ (((P2 ∧ False ∨ P2 ∧ ⌜0 = 0⌝) ∗ P3) ∗ Q ∗ P1 ∗ True) ∧ (P2 ∨ False) ∧ (False → P5 0). Proof. (* Intro-patterns do something :) *) iIntros "[H2 ([H3 HQ]&?&H1&H2'&foo&_)] ? [??]". (* To test destruct: can also be part of the intro-pattern *) iDestruct "foo" as "[_ meh]". repeat iSplit; [|by iLeft|iIntros "#[]"]. iFrame "H2". (* split takes a list of hypotheses just for the LHS *) iSplitL "H3". - iFrame "H3". iRight. auto. - iSplitL "HQ". iAssumption. by iSplitL "H1". Qed. Lemma demo_3 P1 P2 P3 : P1 ∗ P2 ∗ P3 -∗ P1 ∗ ▷ (P2 ∗ ∃ x, (P3 ∧ ⌜x = 0⌝) ∨ P3). Proof. iIntros "(\$ & \$ & \$)". iNext. by iExists 0. Qed. Definition foo (P : PROP) := (P -∗ P)%I. Definition bar : PROP := (∀ P, foo P)%I. Lemma test_unfold_constants : bar. Proof. iIntros (P) "HP //". Qed. Lemma test_iRewrite {A : ofeT} (x y : A) P : □ (∀ z, P -∗ bi_bare (z ≡ y)) -∗ (P -∗ P ∧ (x,x) ≡ (y,x)). Proof. iIntros "#H1 H2". iRewrite (bi.internal_eq_sym x x with "[# //]"). iRewrite -("H1" \$! _ with "[- //]"). auto. Qed. Lemma test_iDestruct_and_emp P Q `{!Persistent P, !Persistent Q} : P ∧ emp -∗ emp ∧ Q -∗ bi_bare (P ∗ Q). Proof. iIntros "[#? _] [_ #?]". auto. Qed. Lemma test_iIntros_persistent P Q `{!Persistent Q} : (P → Q → P ∧ Q)%I. Proof. iIntros "H1 #H2". by iFrame. Qed. Lemma test_iIntros_pure (ψ φ : Prop) P : ψ → (⌜ φ ⌝ → P → ⌜ φ ∧ ψ ⌝ ∧ P)%I. Proof. iIntros (??) "H". auto. Qed. Lemma test_fast_iIntros P Q : (∀ x y z : nat, ⌜x = plus 0 x⌝ → ⌜y = 0⌝ → ⌜z = 0⌝ → P → □ Q → foo (x ≡ x))%I. Proof. iIntros (a) "*". iIntros "#Hfoo **". iIntros "_ //". Qed. Lemma test_very_fast_iIntros P : ∀ x y : nat, (⌜ x = y ⌝ → P -∗ P)%I. Proof. by iIntros. Qed. Lemma test_iAssumption_affine P Q R `{!Affine P, !Affine R} : P -∗ Q -∗ R -∗ Q. Proof. iIntros "H1 H2 H3". iAssumption. Qed. Lemma test_iDestruct_spatial_and P Q1 Q2 : P ∗ (Q1 ∧ Q2) -∗ P ∗ Q1. Proof. iIntros "[H1 [H2 _]]". iFrame. Qed. Lemma test_iAssert_persistent P Q : P -∗ Q -∗ True. Proof. iIntros "HP HQ". iAssert True%I as "#_". { by iClear "HP HQ". } iAssert True%I with "[HP]" as "#_". { Fail iClear "HQ". by iClear "HP". } iAssert True%I as %_. { by iClear "HP HQ". } iAssert True%I with "[HP]" as %_. { Fail iClear "HQ". by iClear "HP". } done. Qed. Lemma test_iSpecialize_auto_frame P Q R : (P -∗ True -∗ True -∗ Q -∗ R) -∗ P -∗ Q -∗ R. Proof. iIntros "H HP HQ". by iApply ("H" with "[\$]"). Qed. Lemma test_iEmp_intro P Q R `{!Affine P, !Persistent Q, !Affine R} : P -∗ Q → R -∗ emp. Proof. iIntros "HP #HQ HR". iEmpIntro. Qed. (* Check coercions *) Lemma test_iExist_coercion (P : Z → PROP) : (∀ x, P x) -∗ ∃ x, P x. Proof. iIntros "HP". iExists (0:nat). iApply ("HP" \$! (0:nat)). Qed. Lemma test_iExist_tc `{Collection A C} P : (∃ x1 x2 : gset positive, P -∗ P)%I. Proof. iExists {[ 1%positive ]}, ∅. auto. Qed. Lemma test_iSpecialize_tc P : (∀ x y z : gset positive, P) -∗ P. Proof. iIntros "H". iSpecialize ("H" \$! ∅ {[ 1%positive ]} ∅). done. Qed. Lemma test_iFrame_pure {A : ofeT} (φ : Prop) (y z : A) : φ → bi_bare ⌜y ≡ z⌝ -∗ (⌜ φ ⌝ ∧ ⌜ φ ⌝ ∧ y ≡ z : PROP). Proof. iIntros (Hv) "#Hxy". iFrame (Hv) "Hxy". Qed. Lemma test_iAssert_modality P : ◇ False -∗ ▷ P. Proof. iIntros "HF". iAssert (bi_bare False)%I with "[> -]" as %[]. by iMod "HF". Qed. Lemma test_iMod_bare_timeless P `{!Timeless P} : bi_bare (▷ P) -∗ ◇ bi_bare P. Proof. iIntros "H". iMod "H". done. Qed. Lemma test_iAssumption_False P : False -∗ P. Proof. iIntros "H". done. Qed. (* Check instantiation and dependent types *) Lemma test_iSpecialize_dependent_type (P : ∀ n, vec nat n → PROP) : (∀ n v, P n v) -∗ ∃ n v, P n v. Proof. iIntros "H". iExists _, [#10]. iSpecialize ("H" \$! _ [#10]). done. Qed. Lemma test_eauto_iFrame P Q R `{!Persistent R} : P -∗ Q -∗ R → R ∗ Q ∗ P ∗ R ∨ False. Proof. eauto with iFrame. Qed. Lemma test_iCombine_persistent P Q R `{!Persistent R} : P -∗ Q -∗ R → R ∗ Q ∗ P ∗ R ∨ False. Proof. iIntros "HP HQ #HR". iCombine "HR HQ HP HR" as "H". auto. Qed. Lemma test_iNext_evar P : P -∗ True. Proof. iIntros "HP". iAssert (▷ _ -∗ ▷ P)%I as "?"; last done. iIntros "?". iNext. iAssumption. Qed. Lemma test_iNext_sep1 P Q (R1 := (P ∗ Q)%I) (R2 := (▷ P ∗ ▷ Q)%I) : (▷ P ∗ ▷ Q) ∗ R1 ∗ R2 -∗ ▷ (P ∗ Q) ∗ ▷ R1 ∗ R2. Proof. iIntros "H". iNext. rewrite {1 2}(lock R1). (* check whether R1 has not been unfolded *) done. Qed. Lemma test_iNext_sep2 P Q : ▷ P ∗ ▷ Q -∗ ▷ (P ∗ Q). Proof. iIntros "H". iNext. iExact "H". (* Check that the laters are all gone. *) Qed. Lemma test_iNext_quantifier {A} (Φ : A → A → PROP) : (∀ y, ∃ x, ▷ Φ x y) -∗ ▷ (∀ y, ∃ x, Φ x y). Proof. iIntros "H". iNext. done. Qed. Lemma test_iFrame_persistent (P Q : PROP) : □ P -∗ Q -∗ bi_persistently (P ∗ P) ∗ (P ∗ Q ∨ Q). Proof. iIntros "#HP". iFrame "HP". iIntros "\$". Qed. Lemma test_iSplit_persistently P Q : □ P -∗ bi_persistently (P ∗ P). Proof. iIntros "#?". by iSplit. Qed. Lemma test_iSpecialize_persistent P Q : □ P -∗ (bi_persistently P → Q) -∗ Q. Proof. iIntros "#HP HPQ". by iSpecialize ("HPQ" with "HP"). Qed. Lemma test_iDestruct_persistent P (Φ : nat → PROP) `{!∀ x, Persistent (Φ x)}: □ (P -∗ ∃ x, Φ x) -∗ P -∗ ∃ x, Φ x ∗ P. Proof. iIntros "#H HP". iDestruct ("H" with "HP") as (x) "#H2". eauto with iFrame. Qed. Lemma test_iLöb P : (∃ n, ▷^n P)%I. Proof. iLöb as "IH". iDestruct "IH" as (n) "IH". by iExists (S n). Qed. Lemma test_iInduction_wf (x : nat) P Q : □ P -∗ Q -∗ ⌜ (x + 0 = x)%nat ⌝. Proof. iIntros "#HP HQ". iInduction (lt_wf x) as [[|x] _] "IH"; simpl; first done. rewrite (inj_iff S). by iApply ("IH" with "[%]"); first omega. Qed. Lemma test_iIntros_start_proof : (True : PROP)%I. Proof. (* Make sure iIntros actually makes progress and enters the proofmode. *) progress iIntros. done. Qed. Lemma test_True_intros : (True : PROP) -∗ True. Proof. iIntros "?". done. Qed. Lemma test_iPoseProof_let P Q : (let R := True%I in R ∗ P ⊢ Q) → P ⊢ Q. Proof. iIntros (help) "HP". iPoseProof (help with "[\$HP]") as "?". done. Qed. Lemma test_iIntros_let P : ∀ Q, let R := emp%I in P -∗ R -∗ Q -∗ P ∗ Q. Proof. iIntros (Q R) "\$ _ \$". Qed. Lemma test_foo P Q : bi_bare (▷ (Q ≡ P)) -∗ bi_bare (▷ Q) -∗ bi_bare (▷ P). Proof. iIntros "#HPQ HQ !#". iNext. by iRewrite "HPQ" in "HQ". Qed. Lemma test_iIntros_modalities `(!Absorbing P) : (bi_persistently (▷ ∀ x : nat, ⌜ x = 0 ⌝ → ⌜ x = 0 ⌝ -∗ False -∗ P -∗ P))%I. Proof. iIntros (x ??). iIntros "* **". (* Test that fast intros do not work under modalities *) iIntros ([]). Qed. Lemma test_iNext_affine P Q : bi_bare (▷ (Q ≡ P)) -∗ bi_bare (▷ Q) -∗ bi_bare (▷ P). Proof. iIntros "#HPQ HQ !#". iNext. by iRewrite "HPQ" in "HQ". Qed. Lemma test_iAlways P Q R : □ P -∗ bi_persistently Q → R -∗ bi_persistently (bi_bare (bi_bare P)) ∗ □ Q. Proof. iIntros "#HP #HQ HR". iSplitL. iAlways. done. iAlways. done. Qed. End tests.