primitive.v 25 KB
 Robbert Krebbers committed Oct 25, 2016 1 2 ``````From iris.base_logic Require Export upred. From iris.algebra Require Export updates. `````` Ralf Jung committed Jan 05, 2017 3 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Oct 25, 2016 4 5 6 7 8 9 10 11 ``````Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|]. Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption]. Local Hint Extern 10 (_ ≤ _) => omega. (** logical connectives *) Program Definition uPred_pure_def {M} (φ : Prop) : uPred M := {| uPred_holds n x := φ |}. Solve Obligations with done. `````` Ralf Jung committed Jan 11, 2017 12 13 ``````Definition uPred_pure_aux : seal (@uPred_pure_def). by eexists. Qed. Definition uPred_pure {M} := unseal uPred_pure_aux M. `````` Robbert Krebbers committed Oct 25, 2016 14 ``````Definition uPred_pure_eq : `````` Ralf Jung committed Jan 11, 2017 15 `````` @uPred_pure = @uPred_pure_def := seal_eq uPred_pure_aux. `````` Robbert Krebbers committed Oct 25, 2016 16 17 18 19 20 21 `````` Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_pure True). Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∧ Q n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. `````` Ralf Jung committed Jan 11, 2017 22 23 24 ``````Definition uPred_and_aux : seal (@uPred_and_def). by eexists. Qed. Definition uPred_and {M} := unseal uPred_and_aux M. Definition uPred_and_eq: @uPred_and = @uPred_and_def := seal_eq uPred_and_aux. `````` Robbert Krebbers committed Oct 25, 2016 25 26 27 28 `````` Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∨ Q n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. `````` Ralf Jung committed Jan 11, 2017 29 30 31 ``````Definition uPred_or_aux : seal (@uPred_or_def). by eexists. Qed. Definition uPred_or {M} := unseal uPred_or_aux M. Definition uPred_or_eq: @uPred_or = @uPred_or_def := seal_eq uPred_or_aux. `````` Robbert Krebbers committed Oct 25, 2016 32 33 34 35 36 37 38 39 40 41 `````` Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}. Next Obligation. intros M P Q n1 x1 x1' HPQ [x2 Hx1'] n2 x3 [x4 Hx3] ?; simpl in *. rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??. eapply HPQ; auto. exists (x2 ⋅ x4); by rewrite assoc. Qed. Next Obligation. intros M P Q [|n1] [|n2] x; auto with lia. Qed. `````` Ralf Jung committed Jan 11, 2017 42 43 ``````Definition uPred_impl_aux : seal (@uPred_impl_def). by eexists. Qed. Definition uPred_impl {M} := unseal uPred_impl_aux M. `````` Robbert Krebbers committed Oct 25, 2016 44 ``````Definition uPred_impl_eq : `````` Ralf Jung committed Jan 11, 2017 45 `````` @uPred_impl = @uPred_impl_def := seal_eq uPred_impl_aux. `````` Robbert Krebbers committed Oct 25, 2016 46 47 48 49 `````` Program Definition uPred_forall_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∀ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. `````` Ralf Jung committed Jan 11, 2017 50 51 ``````Definition uPred_forall_aux : seal (@uPred_forall_def). by eexists. Qed. Definition uPred_forall {M A} := unseal uPred_forall_aux M A. `````` Robbert Krebbers committed Oct 25, 2016 52 ``````Definition uPred_forall_eq : `````` Ralf Jung committed Jan 11, 2017 53 `````` @uPred_forall = @uPred_forall_def := seal_eq uPred_forall_aux. `````` Robbert Krebbers committed Oct 25, 2016 54 55 56 57 `````` Program Definition uPred_exist_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∃ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. `````` Ralf Jung committed Jan 11, 2017 58 59 60 ``````Definition uPred_exist_aux : seal (@uPred_exist_def). by eexists. Qed. Definition uPred_exist {M A} := unseal uPred_exist_aux M A. Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := seal_eq uPred_exist_aux. `````` Robbert Krebbers committed Oct 25, 2016 61 `````` `````` Ralf Jung committed Nov 22, 2016 62 ``````Program Definition uPred_internal_eq_def {M} {A : ofeT} (a1 a2 : A) : uPred M := `````` Robbert Krebbers committed Oct 25, 2016 63 64 `````` {| uPred_holds n x := a1 ≡{n}≡ a2 |}. Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)). `````` Ralf Jung committed Jan 11, 2017 65 66 ``````Definition uPred_internal_eq_aux : seal (@uPred_internal_eq_def). by eexists. Qed. Definition uPred_internal_eq {M A} := unseal uPred_internal_eq_aux M A. `````` Robbert Krebbers committed Oct 25, 2016 67 ``````Definition uPred_internal_eq_eq: `````` Ralf Jung committed Jan 11, 2017 68 `````` @uPred_internal_eq = @uPred_internal_eq_def := seal_eq uPred_internal_eq_aux. `````` Robbert Krebbers committed Oct 25, 2016 69 70 71 72 73 74 75 76 77 78 `````` Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. Next Obligation. intros M P Q n x y (x1&x2&Hx&?&?) [z Hy]. exists x1, (x2 ⋅ z); split_and?; eauto using uPred_mono, cmra_includedN_l. by rewrite Hy Hx assoc. Qed. Next Obligation. intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?; rewrite {1}(dist_le _ _ _ _ Hx) // =>?. `````` Robbert Krebbers committed Feb 09, 2017 79 `````` exists x1, x2; ofe_subst; split_and!; `````` Robbert Krebbers committed Oct 25, 2016 80 81 `````` eauto using dist_le, uPred_closed, cmra_validN_op_l, cmra_validN_op_r. Qed. `````` Ralf Jung committed Jan 11, 2017 82 83 84 ``````Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed. Definition uPred_sep {M} := unseal uPred_sep_aux M. Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := seal_eq uPred_sep_aux. `````` Robbert Krebbers committed Oct 25, 2016 85 86 87 88 89 90 91 92 93 94 `````` Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}. Next Obligation. intros M P Q n x1 x1' HPQ ? n3 x3 ???; simpl in *. apply uPred_mono with (x1 ⋅ x3); eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le. Qed. Next Obligation. naive_solver. Qed. `````` Ralf Jung committed Jan 11, 2017 95 96 ``````Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed. Definition uPred_wand {M} := unseal uPred_wand_aux M. `````` Robbert Krebbers committed Oct 25, 2016 97 ``````Definition uPred_wand_eq : `````` Ralf Jung committed Jan 11, 2017 98 `````` @uPred_wand = @uPred_wand_def := seal_eq uPred_wand_aux. `````` Robbert Krebbers committed Oct 25, 2016 99 100 101 102 103 104 105 `````` Program Definition uPred_always_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := P n (core x) |}. Next Obligation. intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN. Qed. Next Obligation. naive_solver eauto using uPred_closed, @cmra_core_validN. Qed. `````` Ralf Jung committed Jan 11, 2017 106 107 ``````Definition uPred_always_aux : seal (@uPred_always_def). by eexists. Qed. Definition uPred_always {M} := unseal uPred_always_aux M. `````` Robbert Krebbers committed Oct 25, 2016 108 ``````Definition uPred_always_eq : `````` Ralf Jung committed Jan 11, 2017 109 `````` @uPred_always = @uPred_always_def := seal_eq uPred_always_aux. `````` Robbert Krebbers committed Oct 25, 2016 110 111 112 113 114 115 116 117 118 `````` Program Definition uPred_later_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}. Next Obligation. intros M P [|n] x1 x2; eauto using uPred_mono, cmra_includedN_S. Qed. Next Obligation. intros M P [|n1] [|n2] x; eauto using uPred_closed, cmra_validN_S with lia. Qed. `````` Ralf Jung committed Jan 11, 2017 119 120 ``````Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed. Definition uPred_later {M} := unseal uPred_later_aux M. `````` Robbert Krebbers committed Oct 25, 2016 121 ``````Definition uPred_later_eq : `````` Ralf Jung committed Jan 11, 2017 122 `````` @uPred_later = @uPred_later_def := seal_eq uPred_later_aux. `````` Robbert Krebbers committed Oct 25, 2016 123 124 125 126 127 128 129 130 `````` Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M := {| uPred_holds n x := a ≼{n} x |}. Next Obligation. intros M a n x1 x [a' Hx1] [x2 ->]. exists (a' ⋅ x2). by rewrite (assoc op) Hx1. Qed. Next Obligation. naive_solver eauto using cmra_includedN_le. Qed. `````` Ralf Jung committed Jan 11, 2017 131 132 ``````Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed. Definition uPred_ownM {M} := unseal uPred_ownM_aux M. `````` Robbert Krebbers committed Oct 25, 2016 133 ``````Definition uPred_ownM_eq : `````` Ralf Jung committed Jan 11, 2017 134 `````` @uPred_ownM = @uPred_ownM_def := seal_eq uPred_ownM_aux. `````` Robbert Krebbers committed Oct 25, 2016 135 136 137 138 `````` Program Definition uPred_cmra_valid_def {M} {A : cmraT} (a : A) : uPred M := {| uPred_holds n x := ✓{n} a |}. Solve Obligations with naive_solver eauto 2 using cmra_validN_le. `````` Ralf Jung committed Jan 11, 2017 139 140 ``````Definition uPred_cmra_valid_aux : seal (@uPred_cmra_valid_def). by eexists. Qed. Definition uPred_cmra_valid {M A} := unseal uPred_cmra_valid_aux M A. `````` Robbert Krebbers committed Oct 25, 2016 141 ``````Definition uPred_cmra_valid_eq : `````` Ralf Jung committed Jan 11, 2017 142 `````` @uPred_cmra_valid = @uPred_cmra_valid_def := seal_eq uPred_cmra_valid_aux. `````` Robbert Krebbers committed Oct 25, 2016 143 144 145 146 147 148 149 150 151 152 153 154 `````` Program Definition uPred_bupd_def {M} (Q : uPred M) : uPred M := {| uPred_holds n x := ∀ k yf, k ≤ n → ✓{k} (x ⋅ yf) → ∃ x', ✓{k} (x' ⋅ yf) ∧ Q k x' |}. Next Obligation. intros M Q n x1 x2 HQ [x3 Hx] k yf Hk. rewrite (dist_le _ _ _ _ Hx); last lia. intros Hxy. destruct (HQ k (x3 ⋅ yf)) as (x'&?&?); [auto|by rewrite assoc|]. exists (x' ⋅ x3); split; first by rewrite -assoc. apply uPred_mono with x'; eauto using cmra_includedN_l. Qed. Next Obligation. naive_solver. Qed. `````` Ralf Jung committed Jan 11, 2017 155 156 157 ``````Definition uPred_bupd_aux : seal (@uPred_bupd_def). by eexists. Qed. Definition uPred_bupd {M} := unseal uPred_bupd_aux M. Definition uPred_bupd_eq : @uPred_bupd = @uPred_bupd_def := seal_eq uPred_bupd_aux. `````` Robbert Krebbers committed Oct 25, 2016 158 `````` `````` Ralf Jung committed Nov 22, 2016 159 160 161 ``````(* Latest notation *) Notation "'⌜' φ '⌝'" := (uPred_pure φ%C%type) (at level 1, φ at level 200, format "⌜ φ ⌝") : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 162 163 164 165 166 167 168 ``````Notation "'False'" := (uPred_pure False) : uPred_scope. Notation "'True'" := (uPred_pure True) : uPred_scope. Infix "∧" := uPred_and : uPred_scope. Notation "(∧)" := uPred_and (only parsing) : uPred_scope. Infix "∨" := uPred_or : uPred_scope. Notation "(∨)" := uPred_or (only parsing) : uPred_scope. Infix "→" := uPred_impl : uPred_scope. `````` Robbert Krebbers committed Nov 03, 2016 169 170 171 ``````Infix "∗" := uPred_sep (at level 80, right associativity) : uPred_scope. Notation "(∗)" := uPred_sep (only parsing) : uPred_scope. Notation "P -∗ Q" := (uPred_wand P Q) `````` Robbert Krebbers committed Oct 25, 2016 172 173 `````` (at level 99, Q at level 200, right associativity) : uPred_scope. Notation "∀ x .. y , P" := `````` Ralf Jung committed Oct 27, 2016 174 175 `````` (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) (at level 200, x binder, y binder, right associativity) : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 176 ``````Notation "∃ x .. y , P" := `````` Ralf Jung committed Oct 27, 2016 177 178 `````` (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) (at level 200, x binder, y binder, right associativity) : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 179 180 181 182 ``````Notation "□ P" := (uPred_always P) (at level 20, right associativity) : uPred_scope. Notation "▷ P" := (uPred_later P) (at level 20, right associativity) : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 183 ``````Infix "≡" := uPred_internal_eq : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 184 185 186 ``````Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : uPred_scope. Notation "|==> Q" := (uPred_bupd Q) (at level 99, Q at level 200, format "|==> Q") : uPred_scope. `````` Robbert Krebbers committed Nov 03, 2016 187 ``````Notation "P ==∗ Q" := (P ⊢ |==> Q) `````` Robbert Krebbers committed Oct 25, 2016 188 `````` (at level 99, Q at level 200, only parsing) : C_scope. `````` Robbert Krebbers committed Nov 03, 2016 189 190 ``````Notation "P ==∗ Q" := (P -∗ |==> Q)%I (at level 99, Q at level 200, format "P ==∗ Q") : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 191 `````` `````` 192 ``````Coercion uPred_valid {M} (P : uPred M) : Prop := True%I ⊢ P. `````` Robbert Krebbers committed Sep 26, 2017 193 194 ``````Typeclasses Opaque uPred_valid. `````` 195 ``````Notation "P -∗ Q" := (P ⊢ Q) `````` Ralf Jung committed Nov 25, 2016 196 `````` (at level 99, Q at level 200, right associativity) : C_scope. `````` 197 `````` `````` Robbert Krebbers committed Dec 13, 2016 198 ``````Module uPred. `````` Ralf Jung committed Jan 11, 2017 199 ``````Definition unseal_eqs := `````` Robbert Krebbers committed Oct 25, 2016 200 `````` (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq, `````` Robbert Krebbers committed Oct 25, 2016 201 `````` uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq, `````` Robbert Krebbers committed Oct 25, 2016 202 `````` uPred_later_eq, uPred_ownM_eq, uPred_cmra_valid_eq, uPred_bupd_eq). `````` Ralf Jung committed Jan 11, 2017 203 ``````Ltac unseal := rewrite !unseal_eqs /=. `````` Robbert Krebbers committed Oct 25, 2016 204 205 206 207 208 209 210 211 212 213 214 215 `````` Section primitive. Context {M : ucmraT}. Implicit Types φ : Prop. Implicit Types P Q : uPred M. Implicit Types A : Type. Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *) Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *) Arguments uPred_holds {_} !_ _ _ /. Hint Immediate uPred_in_entails. (** Non-expansiveness and setoid morphisms *) `````` Robbert Krebbers committed Mar 09, 2017 216 ``````Global Instance pure_proper : Proper (iff ==> (⊣⊢)) (@uPred_pure M) | 0. `````` Robbert Krebbers committed Oct 25, 2016 217 ``````Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed. `````` Robbert Krebbers committed Mar 09, 2017 218 219 220 ``````Global Instance pure_ne n : Proper (iff ==> dist n) (@uPred_pure M) | 1. Proof. by intros φ1 φ2 ->. Qed. `````` Ralf Jung committed Jan 27, 2017 221 ``````Global Instance and_ne : NonExpansive2 (@uPred_and M). `````` Robbert Krebbers committed Oct 25, 2016 222 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 223 `````` intros n P P' HP Q Q' HQ; unseal; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 224 225 226 227 `````` split; (intros [??]; split; [by apply HP|by apply HQ]). Qed. Global Instance and_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_and M) := ne_proper_2 _. `````` Ralf Jung committed Jan 27, 2017 228 ``````Global Instance or_ne : NonExpansive2 (@uPred_or M). `````` Robbert Krebbers committed Oct 25, 2016 229 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 230 `````` intros n P P' HP Q Q' HQ; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 231 232 233 234 `````` unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]). Qed. Global Instance or_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_or M) := ne_proper_2 _. `````` Ralf Jung committed Jan 27, 2017 235 236 ``````Global Instance impl_ne : NonExpansive2 (@uPred_impl M). `````` Robbert Krebbers committed Oct 25, 2016 237 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 238 `````` intros n P P' HP Q Q' HQ; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 239 240 241 242 `````` unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto. Qed. Global Instance impl_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_impl M) := ne_proper_2 _. `````` Ralf Jung committed Jan 27, 2017 243 ``````Global Instance sep_ne : NonExpansive2 (@uPred_sep M). `````` Robbert Krebbers committed Oct 25, 2016 244 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 245 `````` intros n P P' HP Q Q' HQ; split=> n' x ??. `````` Robbert Krebbers committed Feb 09, 2017 246 `````` unseal; split; intros (x1&x2&?&?&?); ofe_subst x; `````` Robbert Krebbers committed Oct 25, 2016 247 248 249 250 251 `````` exists x1, x2; split_and!; try (apply HP || apply HQ); eauto using cmra_validN_op_l, cmra_validN_op_r. Qed. Global Instance sep_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_sep M) := ne_proper_2 _. `````` Ralf Jung committed Jan 27, 2017 252 253 ``````Global Instance wand_ne : NonExpansive2 (@uPred_wand M). `````` Robbert Krebbers committed Oct 25, 2016 254 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 255 `````` intros n P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???; `````` Robbert Krebbers committed Oct 25, 2016 256 257 258 259 `````` apply HQ, HPQ, HP; eauto using cmra_validN_op_r. Qed. Global Instance wand_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_wand M) := ne_proper_2 _. `````` Ralf Jung committed Jan 27, 2017 260 261 ``````Global Instance internal_eq_ne (A : ofeT) : NonExpansive2 (@uPred_internal_eq M A). `````` Robbert Krebbers committed Oct 25, 2016 262 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 263 `````` intros n x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *. `````` Robbert Krebbers committed Oct 25, 2016 264 265 266 `````` - by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto. - by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto. Qed. `````` Ralf Jung committed Nov 22, 2016 267 ``````Global Instance internal_eq_proper (A : ofeT) : `````` Robbert Krebbers committed Oct 25, 2016 268 `````` Proper ((≡) ==> (≡) ==> (⊣⊢)) (@uPred_internal_eq M A) := ne_proper_2 _. `````` Robbert Krebbers committed Oct 25, 2016 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 ``````Global Instance forall_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A). Proof. by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ. Qed. Global Instance forall_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@uPred_forall M A). Proof. by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ. Qed. Global Instance exist_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A). Proof. intros Ψ1 Ψ2 HΨ. unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ. Qed. Global Instance exist_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@uPred_exist M A). Proof. intros Ψ1 Ψ2 HΨ. unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ. Qed. Global Instance later_contractive : Contractive (@uPred_later M). Proof. `````` Robbert Krebbers committed Dec 05, 2016 293 294 `````` unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega. apply HPQ; eauto using cmra_validN_S. `````` Robbert Krebbers committed Oct 25, 2016 295 296 297 ``````Qed. Global Instance later_proper' : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_later M) := ne_proper _. `````` Ralf Jung committed Jan 27, 2017 298 ``````Global Instance always_ne : NonExpansive (@uPred_always M). `````` Robbert Krebbers committed Oct 25, 2016 299 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 300 `````` intros n P1 P2 HP. `````` Robbert Krebbers committed Oct 25, 2016 301 302 303 304 `````` unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN. Qed. Global Instance always_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_always M) := ne_proper _. `````` Ralf Jung committed Jan 27, 2017 305 ``````Global Instance ownM_ne : NonExpansive (@uPred_ownM M). `````` Robbert Krebbers committed Oct 25, 2016 306 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 307 `````` intros n a b Ha. `````` Robbert Krebbers committed Oct 25, 2016 308 309 310 `````` unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. Qed. Global Instance ownM_proper: Proper ((≡) ==> (⊣⊢)) (@uPred_ownM M) := ne_proper _. `````` Ralf Jung committed Jan 27, 2017 311 312 ``````Global Instance cmra_valid_ne {A : cmraT} : NonExpansive (@uPred_cmra_valid M A). `````` Robbert Krebbers committed Oct 25, 2016 313 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 314 `````` intros n a b Ha; unseal; split=> n' x ? /=. `````` Robbert Krebbers committed Oct 25, 2016 315 316 317 318 `````` by rewrite (dist_le _ _ _ _ Ha); last lia. Qed. Global Instance cmra_valid_proper {A : cmraT} : Proper ((≡) ==> (⊣⊢)) (@uPred_cmra_valid M A) := ne_proper _. `````` Ralf Jung committed Jan 27, 2017 319 ``````Global Instance bupd_ne : NonExpansive (@uPred_bupd M). `````` Robbert Krebbers committed Oct 25, 2016 320 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 321 `````` intros n P Q HPQ. `````` Robbert Krebbers committed Oct 25, 2016 322 323 324 325 326 `````` unseal; split=> n' x; split; intros HP k yf ??; destruct (HP k yf) as (x'&?&?); auto; exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l. Qed. Global Instance bupd_proper : Proper ((≡) ==> (≡)) (@uPred_bupd M) := ne_proper _. `````` 327 328 329 330 ``````Global Instance uPred_valid_proper : Proper ((⊣⊢) ==> iff) (@uPred_valid M). Proof. solve_proper. Qed. Global Instance uPred_valid_mono : Proper ((⊢) ==> impl) (@uPred_valid M). Proof. solve_proper. Qed. `````` Robbert Krebbers committed Dec 02, 2016 331 332 333 ``````Global Instance uPred_valid_flip_mono : Proper (flip (⊢) ==> flip impl) (@uPred_valid M). Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 334 335 `````` (** Introduction and elimination rules *) `````` Ralf Jung committed Nov 22, 2016 336 ``````Lemma pure_intro φ P : φ → P ⊢ ⌜φ⌝. `````` Robbert Krebbers committed Oct 25, 2016 337 ``````Proof. by intros ?; unseal; split. Qed. `````` Ralf Jung committed Nov 22, 2016 338 ``````Lemma pure_elim' φ P : (φ → True ⊢ P) → ⌜φ⌝ ⊢ P. `````` Robbert Krebbers committed Nov 22, 2016 339 ``````Proof. unseal; intros HP; split=> n x ??. by apply HP. Qed. `````` Ralf Jung committed Nov 22, 2016 340 ``````Lemma pure_forall_2 {A} (φ : A → Prop) : (∀ x : A, ⌜φ x⌝) ⊢ ⌜∀ x : A, φ x⌝. `````` Robbert Krebbers committed Oct 25, 2016 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 ``````Proof. by unseal. Qed. Lemma and_elim_l P Q : P ∧ Q ⊢ P. Proof. by unseal; split=> n x ? [??]. Qed. Lemma and_elim_r P Q : P ∧ Q ⊢ Q. Proof. by unseal; split=> n x ? [??]. Qed. Lemma and_intro P Q R : (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R. Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed. Lemma or_intro_l P Q : P ⊢ P ∨ Q. Proof. unseal; split=> n x ??; left; auto. Qed. Lemma or_intro_r P Q : Q ⊢ P ∨ Q. Proof. unseal; split=> n x ??; right; auto. Qed. Lemma or_elim P Q R : (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R. Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed. Lemma impl_intro_r P Q R : (P ∧ Q ⊢ R) → P ⊢ Q → R. Proof. unseal; intros HQ; split=> n x ?? n' x' ????. apply HQ; naive_solver eauto using uPred_mono, uPred_closed, cmra_included_includedN. Qed. Lemma impl_elim P Q R : (P ⊢ Q → R) → (P ⊢ Q) → P ⊢ R. Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed. Lemma forall_intro {A} P (Ψ : A → uPred M): (∀ a, P ⊢ Ψ a) → P ⊢ ∀ a, Ψ a. Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed. Lemma forall_elim {A} {Ψ : A → uPred M} a : (∀ a, Ψ a) ⊢ Ψ a. Proof. unseal; split=> n x ? HP; apply HP. Qed. Lemma exist_intro {A} {Ψ : A → uPred M} a : Ψ a ⊢ ∃ a, Ψ a. Proof. unseal; split=> n x ??; by exists a. Qed. Lemma exist_elim {A} (Φ : A → uPred M) Q : (∀ a, Φ a ⊢ Q) → (∃ a, Φ a) ⊢ Q. Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed. `````` 375 ``````Lemma internal_eq_refl {A : ofeT} (a : A) : uPred_valid (M:=M) (a ≡ a). `````` Robbert Krebbers committed Oct 25, 2016 376 ``````Proof. unseal; by split=> n x ??; simpl. Qed. `````` Ralf Jung committed Nov 22, 2016 377 ``````Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A → uPred M) P `````` Ralf Jung committed Jan 27, 2017 378 `````` {HΨ : NonExpansive Ψ} : (P ⊢ a ≡ b) → (P ⊢ Ψ a) → P ⊢ Ψ b. `````` Robbert Krebbers committed Oct 25, 2016 379 380 381 382 383 384 385 ``````Proof. unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto. - by symmetry; apply Hab with x. - by apply Ha. Qed. (* BI connectives *) `````` Robbert Krebbers committed Nov 03, 2016 386 ``````Lemma sep_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q'. `````` Robbert Krebbers committed Oct 25, 2016 387 388 ``````Proof. intros HQ HQ'; unseal. `````` Robbert Krebbers committed Feb 09, 2017 389 `````` split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; ofe_subst x; `````` Robbert Krebbers committed Oct 25, 2016 390 391 `````` eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails. Qed. `````` Robbert Krebbers committed Nov 03, 2016 392 ``````Lemma True_sep_1 P : P ⊢ True ∗ P. `````` Robbert Krebbers committed Oct 25, 2016 393 394 395 ``````Proof. unseal; split; intros n x ??. exists (core x), x. by rewrite cmra_core_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 396 ``````Lemma True_sep_2 P : True ∗ P ⊢ P. `````` Robbert Krebbers committed Oct 25, 2016 397 ``````Proof. `````` Robbert Krebbers committed Feb 09, 2017 398 `````` unseal; split; intros n x ? (x1&x2&?&_&?); ofe_subst; `````` Robbert Krebbers committed Oct 25, 2016 399 400 `````` eauto using uPred_mono, cmra_includedN_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 401 ``````Lemma sep_comm' P Q : P ∗ Q ⊢ Q ∗ P. `````` Robbert Krebbers committed Oct 25, 2016 402 403 404 ``````Proof. unseal; split; intros n x ? (x1&x2&?&?&?); exists x2, x1; by rewrite (comm op). Qed. `````` Robbert Krebbers committed Nov 03, 2016 405 ``````Lemma sep_assoc' P Q R : (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 406 407 408 409 410 411 ``````Proof. unseal; split; intros n x ? (x1&x2&Hx&(y1&y2&Hy&?&?)&?). exists y1, (y2 ⋅ x2); split_and?; auto. + by rewrite (assoc op) -Hy -Hx. + by exists y2, x2. Qed. `````` Robbert Krebbers committed Nov 03, 2016 412 ``````Lemma wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R. `````` Robbert Krebbers committed Oct 25, 2016 413 414 415 416 417 ``````Proof. unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto. exists x, x'; split_and?; auto. eapply uPred_closed with n; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 418 ``````Lemma wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 419 ``````Proof. `````` Robbert Krebbers committed Feb 09, 2017 420 `````` unseal =>HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst. `````` Robbert Krebbers committed Oct 25, 2016 421 422 423 424 425 426 427 428 429 430 431 `````` eapply HPQR; eauto using cmra_validN_op_l. Qed. (* Always *) Lemma always_mono P Q : (P ⊢ Q) → □ P ⊢ □ Q. Proof. intros HP; unseal; split=> n x ? /=. by apply HP, cmra_core_validN. Qed. Lemma always_elim P : □ P ⊢ P. Proof. unseal; split=> n x ? /=. eauto using uPred_mono, @cmra_included_core, cmra_included_includedN. Qed. `````` Robbert Krebbers committed Jun 13, 2017 432 ``````Lemma always_idemp_2 P : □ P ⊢ □ □ P. `````` Robbert Krebbers committed Oct 25, 2016 433 434 435 436 437 438 439 ``````Proof. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp. Qed. Lemma always_forall_2 {A} (Ψ : A → uPred M) : (∀ a, □ Ψ a) ⊢ (□ ∀ a, Ψ a). Proof. by unseal. Qed. Lemma always_exist_1 {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a) ⊢ (∃ a, □ Ψ a). Proof. by unseal. Qed. `````` Robbert Krebbers committed Nov 03, 2016 440 ``````Lemma always_and_sep_l_1 P Q : □ P ∧ Q ⊢ □ P ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 ``````Proof. unseal; split=> n x ? [??]; exists (core x), x; simpl in *. by rewrite cmra_core_l cmra_core_idemp. Qed. (* Later *) Lemma later_mono P Q : (P ⊢ Q) → ▷ P ⊢ ▷ Q. Proof. unseal=> HP; split=>-[|n] x ??; [done|apply HP; eauto using cmra_validN_S]. Qed. Lemma löb P : (▷ P → P) ⊢ P. Proof. unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|]. apply HP, IH, uPred_closed with (S n); eauto using cmra_validN_S. Qed. Lemma later_forall_2 {A} (Φ : A → uPred M) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a. Proof. unseal; by split=> -[|n] x. Qed. Lemma later_exist_false {A} (Φ : A → uPred M) : (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a). Proof. unseal; split=> -[|[|n]] x /=; eauto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 461 ``````Lemma later_sep P Q : ▷ (P ∗ Q) ⊣⊢ ▷ P ∗ ▷ Q. `````` Robbert Krebbers committed Oct 25, 2016 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 ``````Proof. unseal; split=> n x ?; split. - destruct n as [|n]; simpl. { by exists x, (core x); rewrite cmra_core_r. } intros (x1&x2&Hx&?&?); destruct (cmra_extend n x x1 x2) as (y1&y2&Hx'&Hy1&Hy2); eauto using cmra_validN_S; simpl in *. exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2]. - destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)]. exists x1, x2; eauto using dist_S. Qed. Lemma later_false_excluded_middle P : ▷ P ⊢ ▷ False ∨ (▷ False → P). Proof. unseal; split=> -[|n] x ? /= HP; [by left|right]. intros [|n'] x' ????; [|done]. eauto using uPred_closed, uPred_mono, cmra_included_includedN. Qed. Lemma always_later P : □ ▷ P ⊣⊢ ▷ □ P. Proof. by unseal. Qed. (* Own *) Lemma ownM_op (a1 a2 : M) : `````` Robbert Krebbers committed Nov 03, 2016 483 `````` uPred_ownM (a1 ⋅ a2) ⊣⊢ uPred_ownM a1 ∗ uPred_ownM a2. `````` Robbert Krebbers committed Oct 25, 2016 484 485 486 487 488 489 490 491 492 493 494 495 ``````Proof. unseal; split=> n x ?; split. - intros [z ?]; exists a1, (a2 ⋅ z); split; [by rewrite (assoc op)|]. split. by exists (core a1); rewrite cmra_core_r. by exists z. - intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1 ⋅ z2). by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1) -(assoc op _ a2) (comm op z1) -Hy1 -Hy2. Qed. Lemma always_ownM_core (a : M) : uPred_ownM a ⊢ □ uPred_ownM (core a). Proof. split=> n x /=; unseal; intros Hx. simpl. by apply cmra_core_monoN. Qed. `````` Robbert Krebbers committed Sep 17, 2017 496 ``````Lemma ownM_unit : uPred_valid (M:=M) (uPred_ownM ε). `````` Robbert Krebbers committed Oct 25, 2016 497 498 499 500 501 502 503 504 505 506 507 508 ``````Proof. unseal; split=> n x ??; by exists x; rewrite left_id. Qed. Lemma later_ownM a : ▷ uPred_ownM a ⊢ ∃ b, uPred_ownM b ∧ ▷ (a ≡ b). Proof. unseal; split=> -[|n] x /= ? Hax; first by eauto using ucmra_unit_leastN. destruct Hax as [y ?]. destruct (cmra_extend n x a y) as (a'&y'&Hx&?&?); auto using cmra_validN_S. exists a'. rewrite Hx. eauto using cmra_includedN_l. Qed. (* Valid *) Lemma ownM_valid (a : M) : uPred_ownM a ⊢ ✓ a. Proof. `````` Robbert Krebbers committed Feb 09, 2017 509 `````` unseal; split=> n x Hv [a' ?]; ofe_subst; eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Oct 25, 2016 510 ``````Qed. `````` 511 ``````Lemma cmra_valid_intro {A : cmraT} (a : A) : ✓ a → uPred_valid (M:=M) (✓ a). `````` Robbert Krebbers committed Oct 25, 2016 512 513 514 515 516 517 518 519 520 ``````Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed. Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ False. Proof. unseal=> Ha; split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed. Lemma always_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ □ ✓ a. Proof. by unseal. Qed. Lemma cmra_valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ ✓ a. Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed. (* Basic update modality *) `````` Robbert Krebbers committed Nov 03, 2016 521 ``````Lemma bupd_intro P : P ==∗ P. `````` Robbert Krebbers committed Oct 25, 2016 522 523 524 525 ``````Proof. unseal. split=> n x ? HP k yf ?; exists x; split; first done. apply uPred_closed with n; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 526 ``````Lemma bupd_mono P Q : (P ⊢ Q) → (|==> P) ==∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 527 528 529 530 531 ``````Proof. unseal. intros HPQ; split=> n x ? HP k yf ??. destruct (HP k yf) as (x'&?&?); eauto. exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 532 ``````Lemma bupd_trans P : (|==> |==> P) ==∗ P. `````` Robbert Krebbers committed Oct 25, 2016 533 ``````Proof. unseal; split; naive_solver. Qed. `````` Robbert Krebbers committed Nov 03, 2016 534 ``````Lemma bupd_frame_r P R : (|==> P) ∗ R ==∗ P ∗ R. `````` Robbert Krebbers committed Oct 25, 2016 535 536 537 538 539 540 541 542 543 ``````Proof. unseal; split; intros n x ? (x1&x2&Hx&HP&?) k yf ??. destruct (HP k (x2 ⋅ yf)) as (x'&?&?); eauto. { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. } exists (x' ⋅ x2); split; first by rewrite -assoc. exists x', x2; split_and?; auto. apply uPred_closed with n; eauto 3 using cmra_validN_op_l, cmra_validN_op_r. Qed. Lemma bupd_ownM_updateP x (Φ : M → Prop) : `````` Ralf Jung committed Nov 22, 2016 544 `````` x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y. `````` Robbert Krebbers committed Oct 25, 2016 545 546 547 548 549 550 551 552 553 ``````Proof. unseal=> Hup; split=> n x2 ? [x3 Hx] k yf ??. destruct (Hup k (Some (x3 ⋅ yf))) as (y&?&?); simpl in *. { rewrite /= assoc -(dist_le _ _ _ _ Hx); auto. } exists (y ⋅ x3); split; first by rewrite -assoc. exists y; eauto using cmra_includedN_l. Qed. (* Products *) `````` Ralf Jung committed Nov 22, 2016 554 ``````Lemma prod_equivI {A B : ofeT} (x y : A * B) : x ≡ y ⊣⊢ x.1 ≡ y.1 ∧ x.2 ≡ y.2. `````` Robbert Krebbers committed Oct 25, 2016 555 556 557 558 ``````Proof. by unseal. Qed. Lemma prod_validI {A B : cmraT} (x : A * B) : ✓ x ⊣⊢ ✓ x.1 ∧ ✓ x.2. Proof. by unseal. Qed. `````` Ralf Jung committed Dec 05, 2016 559 ``````(* Type-level Later *) `````` Ralf Jung committed Nov 22, 2016 560 ``````Lemma later_equivI {A : ofeT} (x y : A) : Next x ≡ Next y ⊣⊢ ▷ (x ≡ y). `````` Robbert Krebbers committed Oct 25, 2016 561 562 563 ``````Proof. by unseal. Qed. (* Discrete *) `````` Ralf Jung committed Nov 22, 2016 564 ``````Lemma discrete_valid {A : cmraT} `{!CMRADiscrete A} (a : A) : ✓ a ⊣⊢ ⌜✓ a⌝. `````` Robbert Krebbers committed Oct 25, 2016 565 ``````Proof. unseal; split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed. `````` Robbert Krebbers committed Oct 25, 2017 566 ``````Lemma discrete_eq {A : ofeT} (a b : A) : Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌝. `````` Robbert Krebbers committed Oct 25, 2016 567 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2017 568 `````` unseal=> ?. apply (anti_symm (⊢)); split=> n x ?; by apply (discrete_iff n). `````` Robbert Krebbers committed Oct 25, 2016 569 570 571 ``````Qed. (* Option *) `````` Ralf Jung committed Nov 22, 2016 572 ``````Lemma option_equivI {A : ofeT} (mx my : option A) : `````` Robbert Krebbers committed Oct 25, 2016 573 574 575 576 577 578 579 580 581 582 `````` mx ≡ my ⊣⊢ match mx, my with | Some x, Some y => x ≡ y | None, None => True | _, _ => False end. Proof. unseal. do 2 split. by destruct 1. by destruct mx, my; try constructor. Qed. Lemma option_validI {A : cmraT} (mx : option A) : ✓ mx ⊣⊢ match mx with Some x => ✓ x | None => True end. Proof. unseal. by destruct mx. Qed. `````` Robbert Krebbers committed Dec 05, 2016 583 584 ``````(* Contractive functions *) Lemma contractiveI {A B : ofeT} (f : A → B) : `````` Robbert Krebbers committed Dec 05, 2016 585 `````` Contractive f ↔ (∀ a b, ▷ (a ≡ b) ⊢ f a ≡ f b). `````` Robbert Krebbers committed Dec 05, 2016 586 587 ``````Proof. split; unseal; intros Hf. `````` Robbert Krebbers committed Dec 05, 2016 588 589 `````` - intros a b; split=> n x _; apply Hf. - intros i a b; eapply Hf, ucmra_unit_validN. `````` Robbert Krebbers committed Dec 05, 2016 590 591 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2016 592 ``````(* Functions *) `````` Jacques-Henri Jourdan committed Aug 07, 2017 593 ``````Lemma ofe_funC_equivI {A B} (f g : A -c> B) : f ≡ g ⊣⊢ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Oct 25, 2016 594 ``````Proof. by unseal. Qed. `````` Jacques-Henri Jourdan committed Aug 07, 2017 595 ``````Lemma ofe_morC_equivI {A B : ofeT} (f g : A -n> B) : f ≡ g ⊣⊢ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Oct 25, 2016 596 ``````Proof. by unseal. Qed. `````` Jacques-Henri Jourdan committed Aug 07, 2017 597 598 599 600 601 `````` (* Sig ofes *) Lemma sig_equivI {A : ofeT} (P : A → Prop) (x y : sigC P) : x ≡ y ⊣⊢ proj1_sig x ≡ proj1_sig y. Proof. by unseal. Qed. `````` Robbert Krebbers committed Oct 25, 2016 602 ``````End primitive. `````` Robbert Krebbers committed Dec 13, 2016 603 ``End uPred.``