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 `````` `````` 192 193 ``````Coercion uPred_valid {M} (P : uPred M) : Prop := True%I ⊢ P. Notation "P -∗ Q" := (P ⊢ Q) `````` Ralf Jung committed Nov 25, 2016 194 `````` (at level 99, Q at level 200, right associativity) : C_scope. `````` 195 `````` `````` Robbert Krebbers committed Dec 13, 2016 196 ``````Module uPred. `````` Ralf Jung committed Jan 11, 2017 197 ``````Definition unseal_eqs := `````` Robbert Krebbers committed Oct 25, 2016 198 `````` (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq, `````` Robbert Krebbers committed Oct 25, 2016 199 `````` uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq, `````` Robbert Krebbers committed Oct 25, 2016 200 `````` uPred_later_eq, uPred_ownM_eq, uPred_cmra_valid_eq, uPred_bupd_eq). `````` Ralf Jung committed Jan 11, 2017 201 ``````Ltac unseal := rewrite !unseal_eqs /=. `````` Robbert Krebbers committed Oct 25, 2016 202 203 204 205 206 207 208 209 210 211 212 213 `````` 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 214 ``````Global Instance pure_proper : Proper (iff ==> (⊣⊢)) (@uPred_pure M) | 0. `````` Robbert Krebbers committed Oct 25, 2016 215 ``````Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed. `````` Robbert Krebbers committed Mar 09, 2017 216 217 218 ``````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 219 ``````Global Instance and_ne : NonExpansive2 (@uPred_and M). `````` Robbert Krebbers committed Oct 25, 2016 220 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 221 `````` intros n P P' HP Q Q' HQ; unseal; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 222 223 224 225 `````` 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 226 ``````Global Instance or_ne : NonExpansive2 (@uPred_or M). `````` Robbert Krebbers committed Oct 25, 2016 227 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 228 `````` intros n P P' HP Q Q' HQ; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 229 230 231 232 `````` 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 233 234 ``````Global Instance impl_ne : NonExpansive2 (@uPred_impl M). `````` Robbert Krebbers committed Oct 25, 2016 235 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 236 `````` intros n P P' HP Q Q' HQ; split=> x n' ??. `````` Robbert Krebbers committed Oct 25, 2016 237 238 239 240 `````` 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 241 ``````Global Instance sep_ne : NonExpansive2 (@uPred_sep M). `````` Robbert Krebbers committed Oct 25, 2016 242 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 243 `````` intros n P P' HP Q Q' HQ; split=> n' x ??. `````` Robbert Krebbers committed Feb 09, 2017 244 `````` unseal; split; intros (x1&x2&?&?&?); ofe_subst x; `````` Robbert Krebbers committed Oct 25, 2016 245 246 247 248 249 `````` 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 250 251 ``````Global Instance wand_ne : NonExpansive2 (@uPred_wand M). `````` Robbert Krebbers committed Oct 25, 2016 252 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 253 `````` intros n P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???; `````` Robbert Krebbers committed Oct 25, 2016 254 255 256 257 `````` 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 258 259 ``````Global Instance internal_eq_ne (A : ofeT) : NonExpansive2 (@uPred_internal_eq M A). `````` Robbert Krebbers committed Oct 25, 2016 260 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 261 `````` intros n x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *. `````` Robbert Krebbers committed Oct 25, 2016 262 263 264 `````` - 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 265 ``````Global Instance internal_eq_proper (A : ofeT) : `````` Robbert Krebbers committed Oct 25, 2016 266 `````` Proper ((≡) ==> (≡) ==> (⊣⊢)) (@uPred_internal_eq M A) := ne_proper_2 _. `````` Robbert Krebbers committed Oct 25, 2016 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 ``````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 291 292 `````` unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega. apply HPQ; eauto using cmra_validN_S. `````` Robbert Krebbers committed Oct 25, 2016 293 294 295 ``````Qed. Global Instance later_proper' : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_later M) := ne_proper _. `````` Ralf Jung committed Jan 27, 2017 296 ``````Global Instance always_ne : NonExpansive (@uPred_always M). `````` Robbert Krebbers committed Oct 25, 2016 297 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 298 `````` intros n P1 P2 HP. `````` Robbert Krebbers committed Oct 25, 2016 299 300 301 302 `````` 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 303 ``````Global Instance ownM_ne : NonExpansive (@uPred_ownM M). `````` Robbert Krebbers committed Oct 25, 2016 304 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 305 `````` intros n a b Ha. `````` Robbert Krebbers committed Oct 25, 2016 306 307 308 `````` 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 309 310 ``````Global Instance cmra_valid_ne {A : cmraT} : NonExpansive (@uPred_cmra_valid M A). `````` Robbert Krebbers committed Oct 25, 2016 311 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 312 `````` intros n a b Ha; unseal; split=> n' x ? /=. `````` Robbert Krebbers committed Oct 25, 2016 313 314 315 316 `````` 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 317 ``````Global Instance bupd_ne : NonExpansive (@uPred_bupd M). `````` Robbert Krebbers committed Oct 25, 2016 318 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 319 `````` intros n P Q HPQ. `````` Robbert Krebbers committed Oct 25, 2016 320 321 322 323 324 `````` 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 _. `````` 325 326 327 328 ``````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 329 330 331 ``````Global Instance uPred_valid_flip_mono : Proper (flip (⊢) ==> flip impl) (@uPred_valid M). Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 332 333 `````` (** Introduction and elimination rules *) `````` Ralf Jung committed Nov 22, 2016 334 ``````Lemma pure_intro φ P : φ → P ⊢ ⌜φ⌝. `````` Robbert Krebbers committed Oct 25, 2016 335 ``````Proof. by intros ?; unseal; split. Qed. `````` Ralf Jung committed Nov 22, 2016 336 ``````Lemma pure_elim' φ P : (φ → True ⊢ P) → ⌜φ⌝ ⊢ P. `````` Robbert Krebbers committed Nov 22, 2016 337 ``````Proof. unseal; intros HP; split=> n x ??. by apply HP. Qed. `````` Ralf Jung committed Nov 22, 2016 338 ``````Lemma pure_forall_2 {A} (φ : A → Prop) : (∀ x : A, ⌜φ x⌝) ⊢ ⌜∀ x : A, φ x⌝. `````` Robbert Krebbers committed Oct 25, 2016 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 ``````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. `````` 373 ``````Lemma internal_eq_refl {A : ofeT} (a : A) : uPred_valid (M:=M) (a ≡ a). `````` Robbert Krebbers committed Oct 25, 2016 374 ``````Proof. unseal; by split=> n x ??; simpl. Qed. `````` Ralf Jung committed Nov 22, 2016 375 ``````Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A → uPred M) P `````` Ralf Jung committed Jan 27, 2017 376 `````` {HΨ : NonExpansive Ψ} : (P ⊢ a ≡ b) → (P ⊢ Ψ a) → P ⊢ Ψ b. `````` Robbert Krebbers committed Oct 25, 2016 377 378 379 380 381 382 383 ``````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 384 ``````Lemma sep_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q'. `````` Robbert Krebbers committed Oct 25, 2016 385 386 ``````Proof. intros HQ HQ'; unseal. `````` Robbert Krebbers committed Feb 09, 2017 387 `````` split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; ofe_subst x; `````` Robbert Krebbers committed Oct 25, 2016 388 389 `````` eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails. Qed. `````` Robbert Krebbers committed Nov 03, 2016 390 ``````Lemma True_sep_1 P : P ⊢ True ∗ P. `````` Robbert Krebbers committed Oct 25, 2016 391 392 393 ``````Proof. unseal; split; intros n x ??. exists (core x), x. by rewrite cmra_core_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 394 ``````Lemma True_sep_2 P : True ∗ P ⊢ P. `````` Robbert Krebbers committed Oct 25, 2016 395 ``````Proof. `````` Robbert Krebbers committed Feb 09, 2017 396 `````` unseal; split; intros n x ? (x1&x2&?&_&?); ofe_subst; `````` Robbert Krebbers committed Oct 25, 2016 397 398 `````` eauto using uPred_mono, cmra_includedN_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 399 ``````Lemma sep_comm' P Q : P ∗ Q ⊢ Q ∗ P. `````` Robbert Krebbers committed Oct 25, 2016 400 401 402 ``````Proof. unseal; split; intros n x ? (x1&x2&?&?&?); exists x2, x1; by rewrite (comm op). Qed. `````` Robbert Krebbers committed Nov 03, 2016 403 ``````Lemma sep_assoc' P Q R : (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 404 405 406 407 408 409 ``````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 410 ``````Lemma wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R. `````` Robbert Krebbers committed Oct 25, 2016 411 412 413 414 415 ``````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 416 ``````Lemma wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 417 ``````Proof. `````` Robbert Krebbers committed Feb 09, 2017 418 `````` unseal =>HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst. `````` Robbert Krebbers committed Oct 25, 2016 419 420 421 422 423 424 425 426 427 428 429 `````` 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 430 ``````Lemma always_idemp_2 P : □ P ⊢ □ □ P. `````` Robbert Krebbers committed Oct 25, 2016 431 432 433 434 435 436 437 ``````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 438 ``````Lemma always_and_sep_l_1 P Q : □ P ∧ Q ⊢ □ P ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 ``````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 459 ``````Lemma later_sep P Q : ▷ (P ∗ Q) ⊣⊢ ▷ P ∗ ▷ Q. `````` Robbert Krebbers committed Oct 25, 2016 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 ``````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 481 `````` uPred_ownM (a1 ⋅ a2) ⊣⊢ uPred_ownM a1 ∗ uPred_ownM a2. `````` Robbert Krebbers committed Oct 25, 2016 482 483 484 485 486 487 488 489 490 491 492 493 ``````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. `````` 494 ``````Lemma ownM_empty : uPred_valid (M:=M) (uPred_ownM ∅). `````` Robbert Krebbers committed Oct 25, 2016 495 496 497 498 499 500 501 502 503 504 505 506 ``````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 507 `````` unseal; split=> n x Hv [a' ?]; ofe_subst; eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Oct 25, 2016 508 ``````Qed. `````` 509 ``````Lemma cmra_valid_intro {A : cmraT} (a : A) : ✓ a → uPred_valid (M:=M) (✓ a). `````` Robbert Krebbers committed Oct 25, 2016 510 511 512 513 514 515 516 517 518 ``````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 519 ``````Lemma bupd_intro P : P ==∗ P. `````` Robbert Krebbers committed Oct 25, 2016 520 521 522 523 ``````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 524 ``````Lemma bupd_mono P Q : (P ⊢ Q) → (|==> P) ==∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 525 526 527 528 529 ``````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 530 ``````Lemma bupd_trans P : (|==> |==> P) ==∗ P. `````` Robbert Krebbers committed Oct 25, 2016 531 ``````Proof. unseal; split; naive_solver. Qed. `````` Robbert Krebbers committed Nov 03, 2016 532 ``````Lemma bupd_frame_r P R : (|==> P) ∗ R ==∗ P ∗ R. `````` Robbert Krebbers committed Oct 25, 2016 533 534 535 536 537 538 539 540 541 ``````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 542 `````` x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y. `````` Robbert Krebbers committed Oct 25, 2016 543 544 545 546 547 548 549 550 551 ``````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 552 ``````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 553 554 555 556 ``````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 557 ``````(* Type-level Later *) `````` Ralf Jung committed Nov 22, 2016 558 ``````Lemma later_equivI {A : ofeT} (x y : A) : Next x ≡ Next y ⊣⊢ ▷ (x ≡ y). `````` Robbert Krebbers committed Oct 25, 2016 559 560 561 ``````Proof. by unseal. Qed. (* Discrete *) `````` Ralf Jung committed Nov 22, 2016 562 ``````Lemma discrete_valid {A : cmraT} `{!CMRADiscrete A} (a : A) : ✓ a ⊣⊢ ⌜✓ a⌝. `````` Robbert Krebbers committed Oct 25, 2016 563 ``````Proof. unseal; split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed. `````` Ralf Jung committed Nov 22, 2016 564 ``````Lemma timeless_eq {A : ofeT} (a b : A) : Timeless a → a ≡ b ⊣⊢ ⌜a ≡ b⌝. `````` Robbert Krebbers committed Oct 25, 2016 565 566 567 568 569 ``````Proof. unseal=> ?. apply (anti_symm (⊢)); split=> n x ?; by apply (timeless_iff n). Qed. (* Option *) `````` Ralf Jung committed Nov 22, 2016 570 ``````Lemma option_equivI {A : ofeT} (mx my : option A) : `````` Robbert Krebbers committed Oct 25, 2016 571 572 573 574 575 576 577 578 579 580 `````` 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 581 582 ``````(* Contractive functions *) Lemma contractiveI {A B : ofeT} (f : A → B) : `````` Robbert Krebbers committed Dec 05, 2016 583 `````` Contractive f ↔ (∀ a b, ▷ (a ≡ b) ⊢ f a ≡ f b). `````` Robbert Krebbers committed Dec 05, 2016 584 585 ``````Proof. split; unseal; intros Hf. `````` Robbert Krebbers committed Dec 05, 2016 586 587 `````` - intros a b; split=> n x _; apply Hf. - intros i a b; eapply Hf, ucmra_unit_validN. `````` Robbert Krebbers committed Dec 05, 2016 588 589 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2016 590 ``````(* Functions *) `````` Jacques-Henri Jourdan committed Aug 07, 2017 591 ``````Lemma ofe_funC_equivI {A B} (f g : A -c> B) : f ≡ g ⊣⊢ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Oct 25, 2016 592 ``````Proof. by unseal. Qed. `````` Jacques-Henri Jourdan committed Aug 07, 2017 593 ``````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 594 ``````Proof. by unseal. Qed. `````` Jacques-Henri Jourdan committed Aug 07, 2017 595 596 597 598 599 `````` (* 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 600 ``````End primitive. `````` Robbert Krebbers committed Dec 13, 2016 601 ``End uPred.``