Library iris.base_logic.bi

From iris.bi Require Export derived_connectives updates internal_eq plainly.
From iris.base_logic Require Export upred.
Import uPred_primitive.

BI instances for uPred, and re-stating the remaining primitive laws in terms of the BI interface. This file does *not* unseal.

Definition uPred_emp {M} : uPred M := uPred_pure True.

Local Existing Instance entails_po.

Lemma uPred_bi_mixin (M : ucmraT) :
  BiMixin
    uPred_entails uPred_emp uPred_pure uPred_and uPred_or uPred_impl
    (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand
    uPred_persistently.
Proof.
  split.
  - exact: entails_po.
  - exact: equiv_spec.
  - exact: pure_ne.
  - exact: and_ne.
  - exact: or_ne.
  - exact: impl_ne.
  - exact: forall_ne.
  - exact: exist_ne.
  - exact: sep_ne.
  - exact: wand_ne.
  - exact: persistently_ne.
  - exact: pure_intro.
  - exact: pure_elim'.
  - exact: @pure_forall_2.
  - exact: and_elim_l.
  - exact: and_elim_r.
  - exact: and_intro.
  - exact: or_intro_l.
  - exact: or_intro_r.
  - exact: or_elim.
  - exact: impl_intro_r.
  - exact: impl_elim_l'.
  - exact: @forall_intro.
  - exact: @forall_elim.
  - exact: @exist_intro.
  - exact: @exist_elim.
  - exact: sep_mono.
  - exact: True_sep_1.
  - exact: True_sep_2.
  - exact: sep_comm'.
  - exact: sep_assoc'.
  - exact: wand_intro_r.
  - exact: wand_elim_l'.
  - exact: persistently_mono.
  - exact: persistently_idemp_2.
  -
    trans (uPred_forall (M:=M) (λ _ : False, uPred_persistently uPred_emp)).
    + apply forall_intro=>[[]].
    + etrans; first exact: persistently_forall_2.
      apply persistently_mono. exact: pure_intro.
  - exact: @persistently_forall_2.
  - exact: @persistently_exist_1.
  -
    intros. etrans; first exact: sep_comm'.
    etrans; last exact: True_sep_2.
    apply sep_mono; last done.
    exact: pure_intro.
  - exact: persistently_and_sep_l_1.
Qed.

Lemma uPred_bi_later_mixin (M : ucmraT) :
  BiLaterMixin
    uPred_entails uPred_pure uPred_or uPred_impl
    (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_persistently uPred_later.
Proof.
  split.
  - apply contractive_ne, later_contractive.
  - exact: later_mono.
  - exact: later_intro.
  - exact: @later_forall_2.
  - exact: @later_exist_false.
  - exact: later_sep_1.
  - exact: later_sep_2.
  - exact: later_persistently_1.
  - exact: later_persistently_2.
  - exact: later_false_em.
Qed.

Canonical Structure uPredI (M : ucmraT) : bi :=
  {| bi_ofe_mixin := ofe_mixin_of (uPred M);
     bi_bi_mixin := uPred_bi_mixin M;
     bi_bi_later_mixin := uPred_bi_later_mixin M |}.

Instance uPred_later_contractive {M} : BiLaterContractive (uPredI M).
Proof. apply later_contractive. Qed.

Lemma uPred_internal_eq_mixin M : BiInternalEqMixin (uPredI M) (@uPred_internal_eq M).
Proof.
  split.
  - exact: internal_eq_ne.
  - exact: @internal_eq_refl.
  - exact: @internal_eq_rewrite.
  - exact: @fun_ext.
  - exact: @sig_eq.
  - exact: @discrete_eq_1.
  - exact: @later_eq_1.
  - exact: @later_eq_2.
Qed.
Global Instance uPred_internal_eq M : BiInternalEq (uPredI M) :=
  {| bi_internal_eq_mixin := uPred_internal_eq_mixin M |}.

Lemma uPred_plainly_mixin M : BiPlainlyMixin (uPredI M) uPred_plainly.
Proof.
  split.
  - exact: plainly_ne.
  - exact: plainly_mono.
  - exact: plainly_elim_persistently.
  - exact: plainly_idemp_2.
  - exact: @plainly_forall_2.
  - exact: persistently_impl_plainly.
  - exact: plainly_impl_plainly.
  -
    intros P.
    trans (uPred_forall (M:=M) (λ _ : False , uPred_plainly uPred_emp)).
    + apply forall_intro=>[[]].
    + etrans; first exact: plainly_forall_2.
      apply plainly_mono. exact: pure_intro.
  -
    intros P Q. etrans; last exact: True_sep_2.
    etrans; first exact: sep_comm'.
    apply sep_mono; last done.
    exact: pure_intro.
  - exact: later_plainly_1.
  - exact: later_plainly_2.
Qed.
Global Instance uPred_plainly M : BiPlainly (uPredI M) :=
  {| bi_plainly_mixin := uPred_plainly_mixin M |}.

Global Instance uPred_prop_ext M : BiPropExt (uPredI M).
Proof. exact: prop_ext_2. Qed.

Lemma uPred_bupd_mixin M : BiBUpdMixin (uPredI M) uPred_bupd.
Proof.
  split.
  - exact: bupd_ne.
  - exact: bupd_intro.
  - exact: bupd_mono.
  - exact: bupd_trans.
  - exact: bupd_frame_r.
Qed.
Global Instance uPred_bi_bupd M : BiBUpd (uPredI M) := {| bi_bupd_mixin := uPred_bupd_mixin M |}.

Global Instance uPred_bi_bupd_plainly M : BiBUpdPlainly (uPredI M).
Proof. exact: bupd_plainly. Qed.

extra BI instances

Global Instance uPred_affine M : BiAffine (uPredI M) | 0.
Proof. intros P. exact: pure_intro. Qed.
Hint Immediate uPred_affine : core.

Global Instance uPred_plainly_exist_1 M : BiPlainlyExist (uPredI M).
Proof. exact: @plainly_exist_1. Qed.

Re-state/export lemmas about Iris-specific primitive connectives (own, valid)

Module uPred.

Section restate.
Context {M : ucmraT}.
Implicit Types φ : Prop.
Implicit Types P Q : uPred M.
Implicit Types A : Type.

Notation "P ⊢ Q" := (bi_entails (PROP:=uPredI M) P%I Q%I).
Notation "P ⊣⊢ Q" := (equiv (A:=uPredI M) P%I Q%I).

Global Instance ownM_ne : NonExpansive (@uPred_ownM M) := uPred_primitive.ownM_ne.
Global Instance cmra_valid_ne {A : cmraT} : NonExpansive (@uPred_cmra_valid M A)
  := uPred_primitive.cmra_valid_ne.

Re-exporting primitive Own and valid lemmas
Lemma ownM_op (a1 a2 : M) :
  uPred_ownM (a1 a2) ⊣⊢ uPred_ownM a1 uPred_ownM a2.
Proof. exact: uPred_primitive.ownM_op. Qed.
Lemma persistently_ownM_core (a : M) : uPred_ownM a <pers> uPred_ownM (core a).
Proof. exact: uPred_primitive.persistently_ownM_core. Qed.
Lemma ownM_unit P : P (uPred_ownM ε).
Proof. exact: uPred_primitive.ownM_unit. Qed.
Lemma later_ownM a : uPred_ownM a b, uPred_ownM b (a b).
Proof. exact: uPred_primitive.later_ownM. Qed.
Lemma bupd_ownM_updateP x (Φ : M Prop) :
  x ~~>: Φ uPred_ownM x |==> y, Φ y uPred_ownM y.
Proof. exact: uPred_primitive.bupd_ownM_updateP. Qed.

Lemma ownM_valid (a : M) : uPred_ownM a a.
Proof. exact: uPred_primitive.ownM_valid. Qed.
Lemma cmra_valid_intro {A : cmraT} P (a : A) : a P ( a).
Proof. exact: uPred_primitive.cmra_valid_intro. Qed.
Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a a False.
Proof. exact: uPred_primitive.cmra_valid_elim. Qed.
Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : a a.
Proof. exact: uPred_primitive.plainly_cmra_valid_1. Qed.
Lemma cmra_valid_weaken {A : cmraT} (a b : A) : (a b) a.
Proof. exact: uPred_primitive.cmra_valid_weaken. Qed.
Lemma prod_validI {A B : cmraT} (x : A × B) : x ⊣⊢ x.1 x.2.
Proof. exact: uPred_primitive.prod_validI. Qed.
Lemma option_validI {A : cmraT} (mx : option A) :
   mx ⊣⊢ match mx with Some x x | NoneTrue : uPred M end.
Proof. exact: uPred_primitive.option_validI. Qed.
Lemma discrete_valid {A : cmraT} `{!CmraDiscrete A} (a : A) : a ⊣⊢ a.
Proof. exact: uPred_primitive.discrete_valid. Qed.
Lemma discrete_fun_validI {A} {B : A ucmraT} (g : discrete_fun B) : g ⊣⊢ i, g i.
Proof. exact: uPred_primitive.discrete_fun_validI. Qed.

Consistency/soundness statement
Lemma pure_soundness φ : (⊢@{uPredI M} φ ) φ.
Proof. apply pure_soundness. Qed.

Lemma internal_eq_soundness {A : ofeT} (x y : A) : (⊢@{uPredI M} x y) x y.
Proof. apply internal_eq_soundness. Qed.

Lemma later_soundness P : ( P) P.
Proof. apply later_soundness. Qed.
See derived.v for a similar soundness result for basic updates.
End restate.

New unseal tactic that also unfolds the BI layer. This is used by base_logic.bupd_alt. TODO: Can we get rid of this?
Ltac unseal :=
  unfold bi_emp; simpl;
  unfold uPred_emp, bupd, bi_bupd_bupd, bi_pure,
    bi_and, bi_or, bi_impl, bi_forall, bi_exist,
    bi_sep, bi_wand, bi_persistently, bi_later; simpl;
  unfold internal_eq, bi_internal_eq_internal_eq,
    plainly, bi_plainly_plainly; simpl;
  uPred_primitive.unseal.

End uPred.