From iris.proofmode Require Import tactics monpred. From iris.base_logic Require Import base_logic. From iris.base_logic.lib Require Import invariants cancelable_invariants na_invariants. Section base_logic_tests. Context {M : ucmraT}. Implicit Types P Q R : uPred M. Lemma test_random_stuff (P1 P2 P3 : nat → uPred M) : (∀ (x y : nat) a b, x ≡ y → □ (uPred_ownM (a ⋅ b) -∗ (∃ y1 y2 c, P1 ((x + y1) + y2) ∧ True ∧ □ uPred_ownM c) -∗ □ ▷ (∀ z, P2 z ∨ True → P2 z) -∗ ▷ (∀ n m : nat, P1 n → □ ((True ∧ P2 n) → □ (⌜n = n⌝ ↔ P3 n))) -∗ ▷ ⌜x = 0⌝ ∨ ∃ x z, ▷ P3 (x + z) ∗ uPred_ownM b ∗ uPred_ownM (core b)))%I. Proof. iIntros (i [|j] a b ?) "!# [Ha Hb] H1 #H2 H3"; setoid_subst. { iLeft. by iNext. } iRight. iDestruct "H1" as (z1 z2 c) "(H1&_&#Hc)". iPoseProof "Hc" as "foo". iRevert (a b) "Ha Hb". iIntros (b a) "Hb {foo} Ha". iAssert (uPred_ownM (a ⋅ core a)) with "[Ha]" as "[Ha #Hac]". { by rewrite cmra_core_r. } iIntros "{\$Hac \$Ha}". iExists (S j + z1), z2. iNext. iApply ("H3" \$! _ 0 with "[\$]"). - iSplit. done. iApply "H2". iLeft. iApply "H2". by iRight. - done. Qed. Lemma test_iFrame_pure (x y z : M) : ✓ x → ⌜y ≡ z⌝ -∗ (✓ x ∧ ✓ x ∧ y ≡ z : uPred M). Proof. iIntros (Hv) "Hxy". by iFrame (Hv) "Hxy". Qed. Lemma test_iAssert_modality P : (|==> False) -∗ |==> P. Proof. iIntros. iAssert False%I with "[> - //]" as %[]. Qed. Lemma test_iStartProof_1 P : P -∗ P. Proof. iStartProof. iStartProof. iIntros "\$". Qed. Lemma test_iStartProof_2 P : P -∗ P. Proof. iStartProof (uPred _). iStartProof (uPredI _). iIntros "\$". Qed. Lemma test_iStartProof_3 P : P -∗ P. Proof. iStartProof (uPredI _). iStartProof (uPredSI _). iIntros "\$". Qed. Lemma test_iStartProof_4 P : P -∗ P. Proof. iStartProof (uPredSI _). iStartProof (uPred _). iIntros "\$". Qed. End base_logic_tests. Section iris_tests. Context `{invG Σ, cinvG Σ, na_invG Σ}. Implicit Types P Q R : iProp Σ. Lemma test_masks N E P Q R : ↑N ⊆ E → (True -∗ P -∗ inv N Q -∗ True -∗ R) -∗ P -∗ ▷ Q ={E}=∗ R. Proof. iIntros (?) "H HP HQ". iApply ("H" with "[% //] [\$] [> HQ] [> //]"). by iApply inv_alloc. Qed. Lemma test_iInv_0 N P: inv N ( P) ={⊤}=∗ ▷ P. Proof. iIntros "#H". iInv N as "#H2". Show. iModIntro. iSplit; auto. Qed. Lemma test_iInv_0_with_close N P: inv N ( P) ={⊤}=∗ ▷ P. Proof. iIntros "#H". iInv N as "#H2" "Hclose". Show. iMod ("Hclose" with "H2"). iModIntro. by iNext. Qed. Lemma test_iInv_1 N E P: ↑N ⊆ E → inv N ( P) ={E}=∗ ▷ P. Proof. iIntros (?) "#H". iInv N as "#H2". iModIntro. iSplit; auto. Qed. Lemma test_iInv_2 γ p N P: cinv N γ ( P) ∗ cinv_own γ p ={⊤}=∗ cinv_own γ p ∗ ▷ P. Proof. iIntros "(#?&?)". iInv N as "(#HP&Hown)". Show. iModIntro. iSplit; auto with iFrame. Qed. Lemma test_iInv_2_with_close γ p N P: cinv N γ ( P) ∗ cinv_own γ p ={⊤}=∗ cinv_own γ p ∗ ▷ P. Proof. iIntros "(#?&?)". iInv N as "(#HP&Hown)" "Hclose". Show. iMod ("Hclose" with "HP"). iModIntro. iFrame. by iNext. Qed. Lemma test_iInv_3 γ p1 p2 N P: cinv N γ ( P) ∗ cinv_own γ p1 ∗ cinv_own γ p2 ={⊤}=∗ cinv_own γ p1 ∗ cinv_own γ p2 ∗ ▷ P. Proof. iIntros "(#?&Hown1&Hown2)". iInv N with "[Hown2 //]" as "(#HP&Hown2)". iModIntro. iSplit; auto with iFrame. Qed. Lemma test_iInv_4 t N E1 E2 P: ↑N ⊆ E2 → na_inv t N ( P) ∗ na_own t E1 ∗ na_own t E2 ⊢ |={⊤}=> na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&Hown1&Hown2)". iInv N as "(#HP&Hown2)". Show. iModIntro. iSplitL "Hown2"; auto with iFrame. Qed. Lemma test_iInv_4_with_close t N E1 E2 P: ↑N ⊆ E2 → na_inv t N ( P) ∗ na_own t E1 ∗ na_own t E2 ⊢ |={⊤}=> na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&Hown1&Hown2)". iInv N as "(#HP&Hown2)" "Hclose". Show. iMod ("Hclose" with "[HP Hown2]"). { iFrame. done. } iModIntro. iFrame. by iNext. Qed. (* test named selection of which na_own to use *) Lemma test_iInv_5 t N E1 E2 P: ↑N ⊆ E2 → na_inv t N ( P) ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&Hown1&Hown2)". iInv N with "Hown2" as "(#HP&Hown2)". iModIntro. iSplitL "Hown2"; auto with iFrame. Qed. Lemma test_iInv_6 t N E1 E2 P: ↑N ⊆ E1 → na_inv t N ( P) ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&Hown1&Hown2)". iInv N with "Hown1" as "(#HP&Hown1)". iModIntro. iSplitL "Hown1"; auto with iFrame. Qed. (* test robustness in presence of other invariants *) Lemma test_iInv_7 t N1 N2 N3 E1 E2 P: ↑N3 ⊆ E1 → inv N1 P ∗ na_inv t N3 ( P) ∗ inv N2 P ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&#?&#?&Hown1&Hown2)". iInv N3 with "Hown1" as "(#HP&Hown1)". iModIntro. iSplitL "Hown1"; auto with iFrame. Qed. (* iInv should work even where we have "inv N P" in which P contains an evar *) Lemma test_iInv_8 N : ∃ P, inv N P ={⊤}=∗ P ≡ True ∧ inv N P. Proof. eexists. iIntros "#H". iInv N as "HP". iFrame "HP". auto. Qed. (* test selection by hypothesis name instead of namespace *) Lemma test_iInv_9 t N1 N2 N3 E1 E2 P: ↑N3 ⊆ E1 → inv N1 P ∗ na_inv t N3 ( P) ∗ inv N2 P ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)". iInv "HInv" with "Hown1" as "(#HP&Hown1)". iModIntro. iSplitL "Hown1"; auto with iFrame. Qed. (* test selection by hypothesis name instead of namespace *) Lemma test_iInv_10 t N1 N2 N3 E1 E2 P: ↑N3 ⊆ E1 → inv N1 P ∗ na_inv t N3 ( P) ∗ inv N2 P ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)". iInv "HInv" as "(#HP&Hown1)". iModIntro. iSplitL "Hown1"; auto with iFrame. Qed. (* test selection by ident name *) Lemma test_iInv_11 N P: inv N ( P) ={⊤}=∗ ▷ P. Proof. let H := iFresh in (iIntros H; iInv H as "#H2"). auto. Qed. (* error messages *) Lemma test_iInv_12 N P: inv N ( P) ={⊤}=∗ True. Proof. iIntros "H". Fail iInv 34 as "#H2". Fail iInv nroot as "#H2". Fail iInv "H2" as "#H2". done. Qed. (* test destruction of existentials when opening an invariant *) Lemma test_iInv_13 N: inv N (∃ (v1 v2 v3 : nat), emp ∗ emp ∗ emp) ={⊤}=∗ ▷ emp. Proof. iIntros "H"; iInv "H" as (v1 v2 v3) "(?&?&_)". eauto. Qed. End iris_tests. Section monpred_tests. Context `{invG Σ}. Context {I : biIndex}. Local Notation monPred := (monPred I (iPropI Σ)). Local Notation monPredI := (monPredI I (iPropI Σ)). Local Notation monPredSI := (monPredSI I (iPropSI Σ)). Implicit Types P Q R : monPred. Implicit Types 𝓟 𝓠 𝓡 : iProp Σ. Lemma test_iInv N E 𝓟 : ↑N ⊆ E → ⎡inv N 𝓟⎤ ⊢@{monPredI} |={E}=> emp. Proof. iIntros (?) "Hinv". iInv N as "HP". Show. iFrame "HP". auto. Qed. Lemma test_iInv_with_close N E 𝓟 : ↑N ⊆ E → ⎡inv N 𝓟⎤ ⊢@{monPredI} |={E}=> emp. Proof. iIntros (?) "Hinv". iInv N as "HP" "Hclose". Show. iMod ("Hclose" with "HP"). auto. Qed. End monpred_tests.