interface.v 20.6 KB
 1 ``````From iris.bi Require Export notation. `````` Jacques-Henri Jourdan committed Sep 13, 2019 2 ``````From iris.algebra Require Export ofe. `````` Robbert Krebbers committed Nov 14, 2017 3 ``````Set Primitive Projections. `````` Robbert Krebbers committed Oct 30, 2017 4 5 `````` Section bi_mixin. `````` Robbert Krebbers committed Mar 03, 2018 6 `````` Context {PROP : Type} `{Dist PROP, Equiv PROP}. `````` Robbert Krebbers committed Oct 30, 2017 7 8 9 10 11 12 13 14 15 16 17 `````` Context (bi_entails : PROP → PROP → Prop). Context (bi_emp : PROP). Context (bi_pure : Prop → PROP). Context (bi_and : PROP → PROP → PROP). Context (bi_or : PROP → PROP → PROP). Context (bi_impl : PROP → PROP → PROP). Context (bi_forall : ∀ A, (A → PROP) → PROP). Context (bi_exist : ∀ A, (A → PROP) → PROP). Context (bi_sep : PROP → PROP → PROP). Context (bi_wand : PROP → PROP → PROP). Context (bi_persistently : PROP → PROP). `````` Jacques-Henri Jourdan committed Feb 02, 2018 18 `````` Context (sbi_internal_eq : ∀ A : ofeT, A → A → PROP). `````` Jacques-Henri Jourdan committed Dec 04, 2017 19 `````` Context (sbi_later : PROP → PROP). `````` Robbert Krebbers committed Oct 30, 2017 20 21 22 23 24 `````` Local Infix "⊢" := bi_entails. Local Notation "'emp'" := bi_emp. Local Notation "'True'" := (bi_pure True). Local Notation "'False'" := (bi_pure False). `````` Robbert Krebbers committed Nov 14, 2017 25 `````` Local Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp). `````` Robbert Krebbers committed Oct 30, 2017 26 27 28 29 30 31 32 33 34 `````` Local Infix "∧" := bi_and. Local Infix "∨" := bi_or. Local Infix "→" := bi_impl. Local Notation "∀ x .. y , P" := (bi_forall _ (λ x, .. (bi_forall _ (λ y, P)) ..)). Local Notation "∃ x .. y , P" := (bi_exist _ (λ x, .. (bi_exist _ (λ y, P)) ..)). Local Infix "∗" := bi_sep. Local Infix "-∗" := bi_wand. `````` Robbert Krebbers committed Mar 04, 2018 35 `````` Local Notation "'' P" := (bi_persistently P). `````` Jacques-Henri Jourdan committed Feb 02, 2018 36 `````` Local Notation "x ≡ y" := (sbi_internal_eq _ x y). `````` Jacques-Henri Jourdan committed Dec 04, 2017 37 `````` Local Notation "▷ P" := (sbi_later P). `````` Robbert Krebbers committed Oct 30, 2017 38 `````` `````` Ralf Jung committed Feb 12, 2018 39 40 41 42 43 44 `````` (** * Axioms for a general BI (logic of bunched implications) *) (** The following axioms are satisifed by both affine and linear BIs, and BIs that combine both kinds of resources. In particular, we have an "ordered RA" model satisfying all these axioms. For this model, we extend RAs with an arbitrary partial order, and up-close resources wrt. that order (instead of `````` Ralf Jung committed Feb 21, 2018 45 `````` extension order). We demand composition to be monotone wrt. the order: [x1 ≼ `````` Ralf Jung committed Feb 21, 2018 46 47 48 `````` x2 → x1 ⋅ y ≼ x2 ⋅ y]. We define [emp := λ r, ε ≼ r]; persistently is still defined with the core: [persistently P := λ r, P (core r)]. This is uplcosed because the core is monotone. *) `````` Ralf Jung committed Feb 12, 2018 49 `````` `````` Ralf Jung committed Dec 20, 2017 50 `````` Record BiMixin := { `````` Robbert Krebbers committed Oct 30, 2017 51 52 53 `````` bi_mixin_entails_po : PreOrder bi_entails; bi_mixin_equiv_spec P Q : equiv P Q ↔ (P ⊢ Q) ∧ (Q ⊢ P); `````` Ralf Jung committed Mar 12, 2018 54 `````` (** Non-expansiveness *) `````` Robbert Krebbers committed Oct 30, 2017 55 56 57 58 59 60 61 62 63 64 65 66 `````` bi_mixin_pure_ne n : Proper (iff ==> dist n) bi_pure; bi_mixin_and_ne : NonExpansive2 bi_and; bi_mixin_or_ne : NonExpansive2 bi_or; bi_mixin_impl_ne : NonExpansive2 bi_impl; bi_mixin_forall_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (bi_forall A); bi_mixin_exist_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (bi_exist A); bi_mixin_sep_ne : NonExpansive2 bi_sep; bi_mixin_wand_ne : NonExpansive2 bi_wand; bi_mixin_persistently_ne : NonExpansive bi_persistently; `````` Ralf Jung committed Mar 12, 2018 67 `````` (** Higher-order logic *) `````` 68 `````` bi_mixin_pure_intro (φ : Prop) P : φ → P ⊢ ⌜ φ ⌝; `````` Robbert Krebbers committed Oct 30, 2017 69 `````` bi_mixin_pure_elim' (φ : Prop) P : (φ → True ⊢ P) → ⌜ φ ⌝ ⊢ P; `````` Ralf Jung committed Mar 12, 2018 70 71 `````` (* This is actually derivable if we assume excluded middle in Coq, via [(∀ a, φ a) ∨ (∃ a, ¬φ a)]. *) `````` Robbert Krebbers committed Oct 30, 2017 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 `````` bi_mixin_pure_forall_2 {A} (φ : A → Prop) : (∀ a, ⌜ φ a ⌝) ⊢ ⌜ ∀ a, φ a ⌝; bi_mixin_and_elim_l P Q : P ∧ Q ⊢ P; bi_mixin_and_elim_r P Q : P ∧ Q ⊢ Q; bi_mixin_and_intro P Q R : (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R; bi_mixin_or_intro_l P Q : P ⊢ P ∨ Q; bi_mixin_or_intro_r P Q : Q ⊢ P ∨ Q; bi_mixin_or_elim P Q R : (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R; bi_mixin_impl_intro_r P Q R : (P ∧ Q ⊢ R) → P ⊢ Q → R; bi_mixin_impl_elim_l' P Q R : (P ⊢ Q → R) → P ∧ Q ⊢ R; bi_mixin_forall_intro {A} P (Ψ : A → PROP) : (∀ a, P ⊢ Ψ a) → P ⊢ ∀ a, Ψ a; bi_mixin_forall_elim {A} {Ψ : A → PROP} a : (∀ a, Ψ a) ⊢ Ψ a; bi_mixin_exist_intro {A} {Ψ : A → PROP} a : Ψ a ⊢ ∃ a, Ψ a; bi_mixin_exist_elim {A} (Φ : A → PROP) Q : (∀ a, Φ a ⊢ Q) → (∃ a, Φ a) ⊢ Q; `````` Ralf Jung committed Mar 12, 2018 91 `````` (** BI connectives *) `````` Robbert Krebbers committed Oct 30, 2017 92 93 94 95 96 97 98 99 `````` bi_mixin_sep_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q'; bi_mixin_emp_sep_1 P : P ⊢ emp ∗ P; bi_mixin_emp_sep_2 P : emp ∗ P ⊢ P; bi_mixin_sep_comm' P Q : P ∗ Q ⊢ Q ∗ P; bi_mixin_sep_assoc' P Q R : (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R); bi_mixin_wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R; bi_mixin_wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R; `````` Ralf Jung committed Mar 12, 2018 100 `````` (** Persistently *) `````` Ralf Jung committed Feb 21, 2018 101 `````` (* In the ordered RA model: Holds without further assumptions. *) `````` Robbert Krebbers committed Mar 04, 2018 102 `````` bi_mixin_persistently_mono P Q : (P ⊢ Q) → P ⊢ Q; `````` Robbert Krebbers committed Feb 08, 2018 103 `````` (* In the ordered RA model: `core` is idempotent *) `````` Robbert Krebbers committed Mar 04, 2018 104 `````` bi_mixin_persistently_idemp_2 P : P ⊢ P; `````` Robbert Krebbers committed Oct 30, 2017 105 `````` `````` Ralf Jung committed Mar 14, 2018 106 `````` (* In the ordered RA model: [ε ≼ core x]. *) `````` Robbert Krebbers committed Mar 16, 2018 107 `````` bi_mixin_persistently_emp_2 : emp ⊢ emp; `````` Robbert Krebbers committed Feb 08, 2018 108 `````` `````` Robbert Krebbers committed Oct 30, 2017 109 `````` bi_mixin_persistently_forall_2 {A} (Ψ : A → PROP) : `````` Robbert Krebbers committed Mar 04, 2018 110 `````` (∀ a, (Ψ a)) ⊢ (∀ a, Ψ a); `````` Robbert Krebbers committed Oct 30, 2017 111 `````` bi_mixin_persistently_exist_1 {A} (Ψ : A → PROP) : `````` Robbert Krebbers committed Mar 04, 2018 112 `````` (∃ a, Ψ a) ⊢ ∃ a, (Ψ a); `````` Robbert Krebbers committed Oct 30, 2017 113 `````` `````` Ralf Jung committed Mar 14, 2018 114 `````` (* In the ordered RA model: [core x ≼ core (x ⋅ y)]. *) `````` Robbert Krebbers committed Mar 04, 2018 115 `````` bi_mixin_persistently_absorbing P Q : P ∗ Q ⊢ P; `````` Ralf Jung committed Mar 14, 2018 116 `````` (* In the ordered RA model: [x ⋅ core x = x]. *) `````` Robbert Krebbers committed Mar 04, 2018 117 `````` bi_mixin_persistently_and_sep_elim P Q : P ∧ Q ⊢ P ∗ Q; `````` Robbert Krebbers committed Oct 30, 2017 118 119 `````` }. `````` Ralf Jung committed Dec 20, 2017 120 `````` Record SbiMixin := { `````` Jacques-Henri Jourdan committed Dec 04, 2017 121 `````` sbi_mixin_later_contractive : Contractive sbi_later; `````` Jacques-Henri Jourdan committed Feb 02, 2018 122 123 124 125 126 127 `````` sbi_mixin_internal_eq_ne (A : ofeT) : NonExpansive2 (sbi_internal_eq A); (* Equality *) sbi_mixin_internal_eq_refl {A : ofeT} P (a : A) : P ⊢ a ≡ a; sbi_mixin_internal_eq_rewrite {A : ofeT} a b (Ψ : A → PROP) : NonExpansive Ψ → a ≡ b ⊢ Ψ a → Ψ b; `````` Robbert Krebbers committed Jun 16, 2019 128 `````` sbi_mixin_fun_ext {A} {B : A → ofeT} (f g : discrete_fun B) : (∀ x, f x ≡ g x) ⊢ f ≡ g; `````` Jacques-Henri Jourdan committed Feb 02, 2018 129 130 `````` sbi_mixin_sig_eq {A : ofeT} (P : A → Prop) (x y : sig P) : `x ≡ `y ⊢ x ≡ y; sbi_mixin_discrete_eq_1 {A : ofeT} (a b : A) : Discrete a → a ≡ b ⊢ ⌜a ≡ b⌝; `````` Robbert Krebbers committed Oct 30, 2017 131 `````` `````` Jacques-Henri Jourdan committed Feb 02, 2018 132 `````` (* Later *) `````` Robbert Krebbers committed Oct 30, 2017 133 134 135 136 `````` sbi_mixin_later_eq_1 {A : ofeT} (x y : A) : Next x ≡ Next y ⊢ ▷ (x ≡ y); sbi_mixin_later_eq_2 {A : ofeT} (x y : A) : ▷ (x ≡ y) ⊢ Next x ≡ Next y; sbi_mixin_later_mono P Q : (P ⊢ Q) → ▷ P ⊢ ▷ Q; `````` Ralf Jung committed May 03, 2018 137 `````` sbi_mixin_later_intro P : P ⊢ ▷ P; `````` Robbert Krebbers committed Oct 30, 2017 138 139 140 141 142 143 `````` sbi_mixin_later_forall_2 {A} (Φ : A → PROP) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a; sbi_mixin_later_exist_false {A} (Φ : A → PROP) : (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a); sbi_mixin_later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q; sbi_mixin_later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q); `````` Robbert Krebbers committed Mar 04, 2018 144 145 `````` sbi_mixin_later_persistently_1 P : ▷ P ⊢ ▷ P; sbi_mixin_later_persistently_2 P : ▷ P ⊢ ▷ P; `````` Robbert Krebbers committed Oct 30, 2017 146 147 148 149 150 `````` sbi_mixin_later_false_em P : ▷ P ⊢ ▷ False ∨ (▷ False → P); }. End bi_mixin. `````` Ralf Jung committed Dec 20, 2017 151 ``````Structure bi := Bi { `````` Robbert Krebbers committed Oct 30, 2017 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 `````` bi_car :> Type; bi_dist : Dist bi_car; bi_equiv : Equiv bi_car; bi_entails : bi_car → bi_car → Prop; bi_emp : bi_car; bi_pure : Prop → bi_car; bi_and : bi_car → bi_car → bi_car; bi_or : bi_car → bi_car → bi_car; bi_impl : bi_car → bi_car → bi_car; bi_forall : ∀ A, (A → bi_car) → bi_car; bi_exist : ∀ A, (A → bi_car) → bi_car; bi_sep : bi_car → bi_car → bi_car; bi_wand : bi_car → bi_car → bi_car; bi_persistently : bi_car → bi_car; bi_ofe_mixin : OfeMixin bi_car; `````` Jacques-Henri Jourdan committed Feb 02, 2018 167 `````` bi_bi_mixin : BiMixin bi_entails bi_emp bi_pure bi_and bi_or bi_impl bi_forall `````` Robbert Krebbers committed Mar 03, 2018 168 `````` bi_exist bi_sep bi_wand bi_persistently; `````` Robbert Krebbers committed Oct 30, 2017 169 170 ``````}. `````` Robbert Krebbers committed Jun 16, 2019 171 172 ``````Coercion bi_ofeO (PROP : bi) : ofeT := OfeT PROP (bi_ofe_mixin PROP). Canonical Structure bi_ofeO. `````` Robbert Krebbers committed Oct 30, 2017 173 `````` `````` Maxime Dénès committed Jan 24, 2019 174 175 176 177 178 179 180 181 182 183 184 ``````Instance: Params (@bi_entails) 1 := {}. Instance: Params (@bi_emp) 1 := {}. Instance: Params (@bi_pure) 1 := {}. Instance: Params (@bi_and) 1 := {}. Instance: Params (@bi_or) 1 := {}. Instance: Params (@bi_impl) 1 := {}. Instance: Params (@bi_forall) 2 := {}. Instance: Params (@bi_exist) 2 := {}. Instance: Params (@bi_sep) 1 := {}. Instance: Params (@bi_wand) 1 := {}. Instance: Params (@bi_persistently) 1 := {}. `````` Robbert Krebbers committed Oct 30, 2017 185 186 187 188 189 190 `````` Arguments bi_car : simpl never. Arguments bi_dist : simpl never. Arguments bi_equiv : simpl never. Arguments bi_entails {PROP} _%I _%I : simpl never, rename. Arguments bi_emp {PROP} : simpl never, rename. `````` Robbert Krebbers committed Nov 14, 2017 191 ``````Arguments bi_pure {PROP} _%stdpp : simpl never, rename. `````` Robbert Krebbers committed Oct 30, 2017 192 193 194 195 196 197 198 199 200 ``````Arguments bi_and {PROP} _%I _%I : simpl never, rename. Arguments bi_or {PROP} _%I _%I : simpl never, rename. Arguments bi_impl {PROP} _%I _%I : simpl never, rename. Arguments bi_forall {PROP _} _%I : simpl never, rename. Arguments bi_exist {PROP _} _%I : simpl never, rename. Arguments bi_sep {PROP} _%I _%I : simpl never, rename. Arguments bi_wand {PROP} _%I _%I : simpl never, rename. Arguments bi_persistently {PROP} _%I : simpl never, rename. `````` Ralf Jung committed Dec 20, 2017 201 ``````Structure sbi := Sbi { `````` Robbert Krebbers committed Oct 30, 2017 202 203 204 205 206 207 208 209 210 211 212 213 214 215 `````` sbi_car :> Type; sbi_dist : Dist sbi_car; sbi_equiv : Equiv sbi_car; sbi_entails : sbi_car → sbi_car → Prop; sbi_emp : sbi_car; sbi_pure : Prop → sbi_car; sbi_and : sbi_car → sbi_car → sbi_car; sbi_or : sbi_car → sbi_car → sbi_car; sbi_impl : sbi_car → sbi_car → sbi_car; sbi_forall : ∀ A, (A → sbi_car) → sbi_car; sbi_exist : ∀ A, (A → sbi_car) → sbi_car; sbi_sep : sbi_car → sbi_car → sbi_car; sbi_wand : sbi_car → sbi_car → sbi_car; sbi_persistently : sbi_car → sbi_car; `````` Jacques-Henri Jourdan committed Feb 02, 2018 216 `````` sbi_internal_eq : ∀ A : ofeT, A → A → sbi_car; `````` Jacques-Henri Jourdan committed Dec 04, 2017 217 `````` sbi_later : sbi_car → sbi_car; `````` Robbert Krebbers committed Oct 30, 2017 218 `````` sbi_ofe_mixin : OfeMixin sbi_car; `````` Ralf Jung committed May 04, 2018 219 `````` sbi_cofe : Cofe (OfeT sbi_car sbi_ofe_mixin); `````` Jacques-Henri Jourdan committed Feb 02, 2018 220 `````` sbi_bi_mixin : BiMixin sbi_entails sbi_emp sbi_pure sbi_and sbi_or sbi_impl `````` Robbert Krebbers committed Mar 03, 2018 221 222 223 224 `````` sbi_forall sbi_exist sbi_sep sbi_wand sbi_persistently; sbi_sbi_mixin : SbiMixin sbi_entails sbi_pure sbi_or sbi_impl sbi_forall sbi_exist sbi_sep sbi_persistently sbi_internal_eq sbi_later; `````` Robbert Krebbers committed Oct 30, 2017 225 226 ``````}. `````` Maxime Dénès committed Jan 24, 2019 227 228 ``````Instance: Params (@sbi_later) 1 := {}. Instance: Params (@sbi_internal_eq) 1 := {}. `````` Jacques-Henri Jourdan committed Feb 02, 2018 229 230 231 `````` Arguments sbi_later {PROP} _%I : simpl never, rename. Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename. `````` Robbert Krebbers committed Oct 30, 2017 232 `````` `````` Robbert Krebbers committed Jun 16, 2019 233 234 ``````Coercion sbi_ofeO (PROP : sbi) : ofeT := OfeT PROP (sbi_ofe_mixin PROP). Canonical Structure sbi_ofeO. `````` Robbert Krebbers committed Oct 30, 2017 235 236 237 ``````Coercion sbi_bi (PROP : sbi) : bi := {| bi_ofe_mixin := sbi_ofe_mixin PROP; bi_bi_mixin := sbi_bi_mixin PROP |}. Canonical Structure sbi_bi. `````` Ralf Jung committed May 04, 2018 238 239 ``````Global Instance sbi_cofe' (PROP : sbi) : Cofe PROP. Proof. apply sbi_cofe. Qed. `````` Robbert Krebbers committed Oct 30, 2017 240 241 242 243 244 245 `````` Arguments sbi_car : simpl never. Arguments sbi_dist : simpl never. Arguments sbi_equiv : simpl never. Arguments sbi_entails {PROP} _%I _%I : simpl never, rename. Arguments sbi_emp {PROP} : simpl never, rename. `````` Robbert Krebbers committed Nov 14, 2017 246 ``````Arguments sbi_pure {PROP} _%stdpp : simpl never, rename. `````` Robbert Krebbers committed Oct 30, 2017 247 248 249 250 251 252 253 254 ``````Arguments sbi_and {PROP} _%I _%I : simpl never, rename. Arguments sbi_or {PROP} _%I _%I : simpl never, rename. Arguments sbi_impl {PROP} _%I _%I : simpl never, rename. Arguments sbi_forall {PROP _} _%I : simpl never, rename. Arguments sbi_exist {PROP _} _%I : simpl never, rename. Arguments sbi_sep {PROP} _%I _%I : simpl never, rename. Arguments sbi_wand {PROP} _%I _%I : simpl never, rename. Arguments sbi_persistently {PROP} _%I : simpl never, rename. `````` Jacques-Henri Jourdan committed Feb 02, 2018 255 ``````Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename. `````` Jacques-Henri Jourdan committed Dec 04, 2017 256 ``````Arguments sbi_later {PROP} _%I : simpl never, rename. `````` Robbert Krebbers committed Oct 30, 2017 257 `````` `````` Tej Chajed committed Nov 29, 2018 258 ``````Hint Extern 0 (bi_entails _ _) => reflexivity : core. `````` Maxime Dénès committed Jan 24, 2019 259 ``````Instance bi_rewrite_relation (PROP : bi) : RewriteRelation (@bi_entails PROP) := {}. `````` Robbert Krebbers committed Oct 30, 2017 260 261 ``````Instance bi_inhabited {PROP : bi} : Inhabited PROP := populate (bi_pure True). `````` Robbert Krebbers committed Nov 14, 2017 262 ``````Notation "P ⊢ Q" := (bi_entails P%I Q%I) : stdpp_scope. `````` 263 ``````Notation "P ⊢@{ PROP } Q" := (bi_entails (PROP:=PROP) P%I Q%I) (only parsing) : stdpp_scope. `````` Robbert Krebbers committed Nov 14, 2017 264 ``````Notation "(⊢)" := bi_entails (only parsing) : stdpp_scope. `````` Ralf Jung committed Apr 05, 2018 265 ``````Notation "(⊢@{ PROP } )" := (bi_entails (PROP:=PROP)) (only parsing) : stdpp_scope. `````` Robbert Krebbers committed Oct 30, 2017 266 `````` `````` Ralf Jung committed Apr 05, 2018 267 ``````Notation "P ⊣⊢ Q" := (equiv (A:=bi_car _) P%I Q%I) : stdpp_scope. `````` 268 ``````Notation "P ⊣⊢@{ PROP } Q" := (equiv (A:=bi_car PROP) P%I Q%I) (only parsing) : stdpp_scope. `````` Robbert Krebbers committed Nov 14, 2017 269 ``````Notation "(⊣⊢)" := (equiv (A:=bi_car _)) (only parsing) : stdpp_scope. `````` Ralf Jung committed Apr 05, 2018 270 ``````Notation "(⊣⊢@{ PROP } )" := (equiv (A:=bi_car PROP)) (only parsing) : stdpp_scope. `````` Robbert Krebbers committed Oct 30, 2017 271 `````` `````` Robbert Krebbers committed Nov 14, 2017 272 ``````Notation "P -∗ Q" := (P ⊢ Q) : stdpp_scope. `````` Robbert Krebbers committed Oct 30, 2017 273 274 `````` Notation "'emp'" := (bi_emp) : bi_scope. `````` Robbert Krebbers committed Nov 14, 2017 275 ``````Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp) : bi_scope. `````` Robbert Krebbers committed Oct 30, 2017 276 277 278 279 280 281 282 283 284 285 286 287 288 289 ``````Notation "'True'" := (bi_pure True) : bi_scope. Notation "'False'" := (bi_pure False) : bi_scope. Infix "∧" := bi_and : bi_scope. Notation "(∧)" := bi_and (only parsing) : bi_scope. Infix "∨" := bi_or : bi_scope. Notation "(∨)" := bi_or (only parsing) : bi_scope. Infix "→" := bi_impl : bi_scope. Infix "∗" := bi_sep : bi_scope. Notation "(∗)" := bi_sep (only parsing) : bi_scope. Notation "P -∗ Q" := (bi_wand P Q) : bi_scope. Notation "∀ x .. y , P" := (bi_forall (λ x, .. (bi_forall (λ y, P)) ..)%I) : bi_scope. Notation "∃ x .. y , P" := (bi_exist (λ x, .. (bi_exist (λ y, P)) ..)%I) : bi_scope. `````` Robbert Krebbers committed Mar 04, 2018 290 ``````Notation "'' P" := (bi_persistently P) : bi_scope. `````` Robbert Krebbers committed Oct 30, 2017 291 `````` `````` Jacques-Henri Jourdan committed Feb 02, 2018 292 ``````Infix "≡" := sbi_internal_eq : bi_scope. `````` Jacques-Henri Jourdan committed Dec 04, 2017 293 ``````Notation "▷ P" := (sbi_later P) : bi_scope. `````` Robbert Krebbers committed Oct 30, 2017 294 `````` `````` Ralf Jung committed Mar 19, 2018 295 296 ``````Coercion bi_emp_valid {PROP : bi} (P : PROP) : Prop := emp ⊢ P. Coercion sbi_emp_valid {PROP : sbi} : PROP → Prop := bi_emp_valid. `````` Robbert Krebbers committed Oct 30, 2017 297 `````` `````` Ralf Jung committed Mar 19, 2018 298 299 ``````Arguments bi_emp_valid {_} _%I : simpl never. Typeclasses Opaque bi_emp_valid. `````` Robbert Krebbers committed Oct 30, 2017 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 `````` Module bi. Section bi_laws. Context {PROP : bi}. Implicit Types φ : Prop. Implicit Types P Q R : PROP. Implicit Types A : Type. (* About the entailment *) Global Instance entails_po : PreOrder (@bi_entails PROP). Proof. eapply bi_mixin_entails_po, bi_bi_mixin. Qed. Lemma equiv_spec P Q : P ≡ Q ↔ (P ⊢ Q) ∧ (Q ⊢ P). Proof. eapply bi_mixin_equiv_spec, bi_bi_mixin. Qed. (* Non-expansiveness *) Global Instance pure_ne n : Proper (iff ==> dist n) (@bi_pure PROP). Proof. eapply bi_mixin_pure_ne, bi_bi_mixin. Qed. Global Instance and_ne : NonExpansive2 (@bi_and PROP). Proof. eapply bi_mixin_and_ne, bi_bi_mixin. Qed. Global Instance or_ne : NonExpansive2 (@bi_or PROP). Proof. eapply bi_mixin_or_ne, bi_bi_mixin. Qed. Global Instance impl_ne : NonExpansive2 (@bi_impl PROP). Proof. eapply bi_mixin_impl_ne, bi_bi_mixin. Qed. Global Instance forall_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@bi_forall PROP A). Proof. eapply bi_mixin_forall_ne, bi_bi_mixin. Qed. Global Instance exist_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@bi_exist PROP A). Proof. eapply bi_mixin_exist_ne, bi_bi_mixin. Qed. Global Instance sep_ne : NonExpansive2 (@bi_sep PROP). Proof. eapply bi_mixin_sep_ne, bi_bi_mixin. Qed. Global Instance wand_ne : NonExpansive2 (@bi_wand PROP). Proof. eapply bi_mixin_wand_ne, bi_bi_mixin. Qed. Global Instance persistently_ne : NonExpansive (@bi_persistently PROP). Proof. eapply bi_mixin_persistently_ne, bi_bi_mixin. Qed. (* Higher-order logic *) `````` 337 ``````Lemma pure_intro (φ : Prop) P : φ → P ⊢ ⌜ φ ⌝. `````` Robbert Krebbers committed Oct 30, 2017 338 339 340 ``````Proof. eapply bi_mixin_pure_intro, bi_bi_mixin. Qed. Lemma pure_elim' (φ : Prop) P : (φ → True ⊢ P) → ⌜ φ ⌝ ⊢ P. Proof. eapply bi_mixin_pure_elim', bi_bi_mixin. Qed. `````` Ralf Jung committed Apr 05, 2018 341 ``````Lemma pure_forall_2 {A} (φ : A → Prop) : (∀ a, ⌜ φ a ⌝) ⊢@{PROP} ⌜ ∀ a, φ a ⌝. `````` Robbert Krebbers committed Oct 30, 2017 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 ``````Proof. eapply bi_mixin_pure_forall_2, bi_bi_mixin. Qed. Lemma and_elim_l P Q : P ∧ Q ⊢ P. Proof. eapply bi_mixin_and_elim_l, bi_bi_mixin. Qed. Lemma and_elim_r P Q : P ∧ Q ⊢ Q. Proof. eapply bi_mixin_and_elim_r, bi_bi_mixin. Qed. Lemma and_intro P Q R : (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R. Proof. eapply bi_mixin_and_intro, bi_bi_mixin. Qed. Lemma or_intro_l P Q : P ⊢ P ∨ Q. Proof. eapply bi_mixin_or_intro_l, bi_bi_mixin. Qed. Lemma or_intro_r P Q : Q ⊢ P ∨ Q. Proof. eapply bi_mixin_or_intro_r, bi_bi_mixin. Qed. Lemma or_elim P Q R : (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R. Proof. eapply bi_mixin_or_elim, bi_bi_mixin. Qed. Lemma impl_intro_r P Q R : (P ∧ Q ⊢ R) → P ⊢ Q → R. Proof. eapply bi_mixin_impl_intro_r, bi_bi_mixin. Qed. Lemma impl_elim_l' P Q R : (P ⊢ Q → R) → P ∧ Q ⊢ R. Proof. eapply bi_mixin_impl_elim_l', bi_bi_mixin. Qed. Lemma forall_intro {A} P (Ψ : A → PROP) : (∀ a, P ⊢ Ψ a) → P ⊢ ∀ a, Ψ a. Proof. eapply bi_mixin_forall_intro, bi_bi_mixin. Qed. Lemma forall_elim {A} {Ψ : A → PROP} a : (∀ a, Ψ a) ⊢ Ψ a. `````` Jacques-Henri Jourdan committed Feb 02, 2018 366 ``````Proof. eapply (bi_mixin_forall_elim bi_entails), bi_bi_mixin. Qed. `````` Robbert Krebbers committed Oct 30, 2017 367 368 369 370 371 372 373 374 375 376 377 378 379 380 `````` Lemma exist_intro {A} {Ψ : A → PROP} a : Ψ a ⊢ ∃ a, Ψ a. Proof. eapply bi_mixin_exist_intro, bi_bi_mixin. Qed. Lemma exist_elim {A} (Φ : A → PROP) Q : (∀ a, Φ a ⊢ Q) → (∃ a, Φ a) ⊢ Q. Proof. eapply bi_mixin_exist_elim, bi_bi_mixin. Qed. (* BI connectives *) Lemma sep_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q'. Proof. eapply bi_mixin_sep_mono, bi_bi_mixin. Qed. Lemma emp_sep_1 P : P ⊢ emp ∗ P. Proof. eapply bi_mixin_emp_sep_1, bi_bi_mixin. Qed. Lemma emp_sep_2 P : emp ∗ P ⊢ P. Proof. eapply bi_mixin_emp_sep_2, bi_bi_mixin. Qed. Lemma sep_comm' P Q : P ∗ Q ⊢ Q ∗ P. `````` Jacques-Henri Jourdan committed Feb 02, 2018 381 ``````Proof. eapply (bi_mixin_sep_comm' bi_entails), bi_bi_mixin. Qed. `````` Robbert Krebbers committed Oct 30, 2017 382 383 384 385 386 387 388 389 ``````Lemma sep_assoc' P Q R : (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R). Proof. eapply bi_mixin_sep_assoc', bi_bi_mixin. Qed. Lemma wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R. Proof. eapply bi_mixin_wand_intro_r, bi_bi_mixin. Qed. Lemma wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R. Proof. eapply bi_mixin_wand_elim_l', bi_bi_mixin. Qed. (* Persistently *) `````` Robbert Krebbers committed Mar 04, 2018 390 ``````Lemma persistently_mono P Q : (P ⊢ Q) → P ⊢ Q. `````` Robbert Krebbers committed Oct 30, 2017 391 ``````Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed. `````` Robbert Krebbers committed Mar 04, 2018 392 ``````Lemma persistently_idemp_2 P : P ⊢ P. `````` Robbert Krebbers committed Oct 30, 2017 393 394 ``````Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed. `````` Ralf Jung committed Apr 05, 2018 395 ``````Lemma persistently_emp_2 : emp ⊢@{PROP} emp. `````` Robbert Krebbers committed Mar 16, 2018 396 ``````Proof. eapply bi_mixin_persistently_emp_2, bi_bi_mixin. Qed. `````` Robbert Krebbers committed Mar 03, 2018 397 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 398 ``````Lemma persistently_forall_2 {A} (Ψ : A → PROP) : `````` Robbert Krebbers committed Mar 04, 2018 399 `````` (∀ a, (Ψ a)) ⊢ (∀ a, Ψ a). `````` Robbert Krebbers committed Oct 30, 2017 400 ``````Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 401 ``````Lemma persistently_exist_1 {A} (Ψ : A → PROP) : `````` Robbert Krebbers committed Mar 04, 2018 402 `````` (∃ a, Ψ a) ⊢ ∃ a, (Ψ a). `````` Robbert Krebbers committed Oct 30, 2017 403 404 ``````Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed. `````` Robbert Krebbers committed Mar 04, 2018 405 ``````Lemma persistently_absorbing P Q : P ∗ Q ⊢ P. `````` Jacques-Henri Jourdan committed Feb 02, 2018 406 ``````Proof. eapply (bi_mixin_persistently_absorbing bi_entails), bi_bi_mixin. Qed. `````` Robbert Krebbers committed Mar 04, 2018 407 ``````Lemma persistently_and_sep_elim P Q : P ∧ Q ⊢ P ∗ Q. `````` Ralf Jung committed Feb 21, 2018 408 ``````Proof. eapply (bi_mixin_persistently_and_sep_elim bi_entails), bi_bi_mixin. Qed. `````` Robbert Krebbers committed Oct 30, 2017 409 410 411 412 413 414 415 ``````End bi_laws. Section sbi_laws. Context {PROP : sbi}. Implicit Types φ : Prop. Implicit Types P Q R : PROP. `````` Jacques-Henri Jourdan committed Feb 02, 2018 416 417 418 419 420 421 422 423 424 425 ``````(* Equality *) Global Instance internal_eq_ne (A : ofeT) : NonExpansive2 (@sbi_internal_eq PROP A). Proof. eapply sbi_mixin_internal_eq_ne, sbi_sbi_mixin. Qed. Lemma internal_eq_refl {A : ofeT} P (a : A) : P ⊢ a ≡ a. Proof. eapply sbi_mixin_internal_eq_refl, sbi_sbi_mixin. Qed. Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A → PROP) : NonExpansive Ψ → a ≡ b ⊢ Ψ a → Ψ b. Proof. eapply sbi_mixin_internal_eq_rewrite, sbi_sbi_mixin. Qed. `````` Robbert Krebbers committed Jun 16, 2019 426 ``````Lemma fun_ext {A} {B : A → ofeT} (f g : discrete_fun B) : `````` Ralf Jung committed Apr 05, 2018 427 `````` (∀ x, f x ≡ g x) ⊢@{PROP} f ≡ g. `````` Jacques-Henri Jourdan committed Feb 02, 2018 428 ``````Proof. eapply sbi_mixin_fun_ext, sbi_sbi_mixin. Qed. `````` Ralf Jung committed Apr 05, 2018 429 ``````Lemma sig_eq {A : ofeT} (P : A → Prop) (x y : sig P) : `````` Ralf Jung committed Apr 05, 2018 430 `````` `x ≡ `y ⊢@{PROP} x ≡ y. `````` Jacques-Henri Jourdan committed Feb 02, 2018 431 432 ``````Proof. eapply sbi_mixin_sig_eq, sbi_sbi_mixin. Qed. Lemma discrete_eq_1 {A : ofeT} (a b : A) : `````` Ralf Jung committed Apr 05, 2018 433 `````` Discrete a → a ≡ b ⊢@{PROP} ⌜a ≡ b⌝. `````` Jacques-Henri Jourdan committed Feb 02, 2018 434 435 436 ``````Proof. eapply sbi_mixin_discrete_eq_1, sbi_sbi_mixin. Qed. (* Later *) `````` Jacques-Henri Jourdan committed Dec 04, 2017 437 ``````Global Instance later_contractive : Contractive (@sbi_later PROP). `````` Robbert Krebbers committed Oct 30, 2017 438 439 ``````Proof. eapply sbi_mixin_later_contractive, sbi_sbi_mixin. Qed. `````` Ralf Jung committed Apr 05, 2018 440 ``````Lemma later_eq_1 {A : ofeT} (x y : A) : Next x ≡ Next y ⊢@{PROP} ▷ (x ≡ y). `````` Robbert Krebbers committed Oct 30, 2017 441 ``````Proof. eapply sbi_mixin_later_eq_1, sbi_sbi_mixin. Qed. `````` Ralf Jung committed Apr 05, 2018 442 ``````Lemma later_eq_2 {A : ofeT} (x y : A) : ▷ (x ≡ y) ⊢@{PROP} Next x ≡ Next y. `````` Robbert Krebbers committed Oct 30, 2017 443 444 445 446 ``````Proof. eapply sbi_mixin_later_eq_2, sbi_sbi_mixin. Qed. Lemma later_mono P Q : (P ⊢ Q) → ▷ P ⊢ ▷ Q. Proof. eapply sbi_mixin_later_mono, sbi_sbi_mixin. Qed. `````` Ralf Jung committed May 03, 2018 447 448 ``````Lemma later_intro P : P ⊢ ▷ P. Proof. eapply sbi_mixin_later_intro, sbi_sbi_mixin. Qed. `````` Robbert Krebbers committed Oct 30, 2017 449 450 451 452 453 454 455 456 457 458 `````` Lemma later_forall_2 {A} (Φ : A → PROP) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a. Proof. eapply sbi_mixin_later_forall_2, sbi_sbi_mixin. Qed. Lemma later_exist_false {A} (Φ : A → PROP) : (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a). Proof. eapply sbi_mixin_later_exist_false, sbi_sbi_mixin. Qed. Lemma later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q. Proof. eapply sbi_mixin_later_sep_1, sbi_sbi_mixin. Qed. Lemma later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q). Proof. eapply sbi_mixin_later_sep_2, sbi_sbi_mixin. Qed. `````` Robbert Krebbers committed Mar 04, 2018 459 ``````Lemma later_persistently_1 P : ▷ P ⊢ ▷ P. `````` Robbert Krebbers committed Mar 03, 2018 460 ``````Proof. eapply (sbi_mixin_later_persistently_1 bi_entails), sbi_sbi_mixin. Qed. `````` Robbert Krebbers committed Mar 04, 2018 461 ``````Lemma later_persistently_2 P : ▷ P ⊢ ▷ P. `````` Robbert Krebbers committed Mar 03, 2018 462 ``````Proof. eapply (sbi_mixin_later_persistently_2 bi_entails), sbi_sbi_mixin. Qed. `````` Robbert Krebbers committed Oct 30, 2017 463 464 465 466 `````` Lemma later_false_em P : ▷ P ⊢ ▷ False ∨ (▷ False → P). Proof. eapply sbi_mixin_later_false_em, sbi_sbi_mixin. Qed. End sbi_laws. `````` Jacques-Henri Jourdan committed Dec 11, 2017 467 `````` `````` Robbert Krebbers committed Oct 30, 2017 468 ``End bi.``