language.v 13 KB
 Ralf Jung committed Nov 22, 2016 1 ``````From iris.algebra Require Export ofe. `````` Ralf Jung committed Jan 05, 2017 2 ``````Set Default Proof Using "Type". `````` Ralf Jung committed Jan 04, 2016 3 `````` `````` Robbert Krebbers committed Nov 23, 2017 4 ``````Section language_mixin. `````` 5 `````` Context {expr val state observation : Type}. `````` Robbert Krebbers committed Nov 23, 2017 6 7 `````` Context (of_val : val → expr). Context (to_val : expr → option val). `````` Ralf Jung committed Oct 05, 2018 8 9 10 `````` (** We annotate the reduction relation with observations [κ], which we will use in the definition of weakest preconditions to predict future observations and assert correctness of the predictions. *) `````` Ralf Jung committed Oct 18, 2018 11 `````` Context (prim_step : expr → state → list observation → expr → state → list expr → Prop). `````` Robbert Krebbers committed Nov 23, 2017 12 13 14 15 `````` Record LanguageMixin := { mixin_to_of_val v : to_val (of_val v) = Some v; mixin_of_to_val e v : to_val e = Some v → of_val v = e; `````` 16 `````` mixin_val_stuck e σ κ e' σ' efs : prim_step e σ κ e' σ' efs → to_val e = None `````` Robbert Krebbers committed Nov 23, 2017 17 18 19 `````` }. End language_mixin. `````` Robbert Krebbers committed Feb 01, 2016 20 21 22 23 ``````Structure language := Language { expr : Type; val : Type; state : Type; `````` 24 `````` observation : Type; `````` Robbert Krebbers committed Feb 01, 2016 25 26 `````` of_val : val → expr; to_val : expr → option val; `````` Ralf Jung committed Oct 18, 2018 27 `````` prim_step : expr → state → list observation → expr → state → list expr → Prop; `````` Robbert Krebbers committed Nov 23, 2017 28 `````` language_mixin : LanguageMixin of_val to_val prim_step `````` Ralf Jung committed Jan 04, 2016 29 ``````}. `````` Janno committed Aug 25, 2016 30 31 32 ``````Delimit Scope expr_scope with E. Delimit Scope val_scope with V. Bind Scope expr_scope with expr. `````` Jacques-Henri Jourdan committed Jan 04, 2017 33 ``````Bind Scope val_scope with val. `````` Robbert Krebbers committed Nov 23, 2017 34 `````` `````` 35 ``````Arguments Language {_ _ _ _ _ _ _} _. `````` Robbert Krebbers committed Feb 01, 2016 36 37 ``````Arguments of_val {_} _. Arguments to_val {_} _. `````` 38 ``````Arguments prim_step {_} _ _ _ _ _ _. `````` Robbert Krebbers committed Feb 01, 2016 39 `````` `````` Robbert Krebbers committed Jun 16, 2019 40 41 42 ``````Canonical Structure stateO Λ := leibnizO (state Λ). Canonical Structure valO Λ := leibnizO (val Λ). Canonical Structure exprO Λ := leibnizO (expr Λ). `````` Robbert Krebbers committed Feb 01, 2016 43 44 `````` Definition cfg (Λ : language) := (list (expr Λ) * state Λ)%type. `````` Ralf Jung committed Jan 04, 2016 45 `````` `````` Robbert Krebbers committed Nov 23, 2017 46 ``````Class LanguageCtx {Λ : language} (K : expr Λ → expr Λ) := { `````` Robbert Krebbers committed Mar 15, 2017 47 48 `````` fill_not_val e : to_val e = None → to_val (K e) = None; `````` 49 50 51 52 53 54 `````` fill_step e1 σ1 κ e2 σ2 efs : prim_step e1 σ1 κ e2 σ2 efs → prim_step (K e1) σ1 κ (K e2) σ2 efs; fill_step_inv e1' σ1 κ e2 σ2 efs : to_val e1' = None → prim_step (K e1') σ1 κ e2 σ2 efs → ∃ e2', e2 = K e2' ∧ prim_step e1' σ1 κ e2' σ2 efs `````` Robbert Krebbers committed Mar 15, 2017 55 56 ``````}. `````` Ralf Jung committed Mar 06, 2019 57 ``````Instance language_ctx_id Λ : LanguageCtx (@id (expr Λ)). `````` Robbert Krebbers committed Sep 21, 2017 58 59 ``````Proof. constructor; naive_solver. Qed. `````` Ralf Jung committed Dec 07, 2017 60 ``````Inductive atomicity := StronglyAtomic | WeaklyAtomic. `````` David Swasey committed Nov 09, 2017 61 `````` `````` Robbert Krebbers committed Jan 16, 2016 62 ``````Section language. `````` Robbert Krebbers committed Feb 01, 2016 63 64 `````` Context {Λ : language}. Implicit Types v : val Λ. `````` Robbert Krebbers committed Nov 23, 2017 65 66 67 68 69 70 `````` Implicit Types e : expr Λ. Lemma to_of_val v : to_val (of_val v) = Some v. Proof. apply language_mixin. Qed. Lemma of_to_val e v : to_val e = Some v → of_val v = e. Proof. apply language_mixin. Qed. `````` 71 `````` Lemma val_stuck e σ κ e' σ' efs : prim_step e σ κ e' σ' efs → to_val e = None. `````` Robbert Krebbers committed Nov 23, 2017 72 `````` Proof. apply language_mixin. Qed. `````` Ralf Jung committed Jan 04, 2016 73 `````` `````` Robbert Krebbers committed Feb 01, 2016 74 `````` Definition reducible (e : expr Λ) (σ : state Λ) := `````` 75 `````` ∃ κ e' σ' efs, prim_step e σ κ e' σ' efs. `````` Ralf Jung committed Oct 05, 2018 76 77 `````` (* Total WP only permits reductions without observations *) Definition reducible_no_obs (e : expr Λ) (σ : state Λ) := `````` Ralf Jung committed Oct 18, 2018 78 `````` ∃ e' σ' efs, prim_step e σ [] e' σ' efs. `````` 79 `````` Definition irreducible (e : expr Λ) (σ : state Λ) := `````` 80 `````` ∀ κ e' σ' efs, ¬prim_step e σ κ e' σ' efs. `````` David Swasey committed Nov 26, 2017 81 82 `````` Definition stuck (e : expr Λ) (σ : state Λ) := to_val e = None ∧ irreducible e σ. `````` Amin Timany committed Nov 06, 2019 83 84 `````` Definition not_stuck (e : expr Λ) (σ : state Λ) := is_Some (to_val e) ∨ reducible e σ. `````` Robbert Krebbers committed Mar 15, 2017 85 `````` `````` Ralf Jung committed Dec 07, 2017 86 `````` (* [Atomic WeaklyAtomic]: This (weak) form of atomicity is enough to open `````` Ralf Jung committed Dec 05, 2017 87 88 89 90 `````` invariants when WP ensures safety, i.e., programs never can get stuck. We have an example in lambdaRust of an expression that is atomic in this sense, but not in the stronger sense defined below, and we have to be able to open invariants around that expression. See `CasStuckS` in `````` David Swasey committed Nov 09, 2017 91 `````` [lambdaRust](https://gitlab.mpi-sws.org/FP/LambdaRust-coq/blob/master/theories/lang/lang.v). `````` Robbert Krebbers committed Mar 15, 2017 92 `````` `````` Ralf Jung committed Dec 07, 2017 93 `````` [Atomic StronglyAtomic]: To open invariants with a WP that does not ensure `````` Ralf Jung committed Dec 05, 2017 94 95 96 97 `````` safety, we need a stronger form of atomicity. With the above definition, in case `e` reduces to a stuck non-value, there is no proof that the invariants have been established again. *) Class Atomic (a : atomicity) (e : expr Λ) : Prop := `````` 98 99 `````` atomic σ e' κ σ' efs : prim_step e σ κ e' σ' efs → `````` Ralf Jung committed Dec 07, 2017 100 `````` if a is WeaklyAtomic then irreducible e' σ' else is_Some (to_val e'). `````` Ralf Jung committed Oct 29, 2017 101 `````` `````` Ralf Jung committed Oct 18, 2018 102 `````` Inductive step (ρ1 : cfg Λ) (κ : list (observation Λ)) (ρ2 : cfg Λ) : Prop := `````` Robbert Krebbers committed Aug 08, 2016 103 `````` | step_atomic e1 σ1 e2 σ2 efs t1 t2 : `````` Robbert Krebbers committed Feb 01, 2016 104 `````` ρ1 = (t1 ++ e1 :: t2, σ1) → `````` Robbert Krebbers committed Aug 08, 2016 105 `````` ρ2 = (t1 ++ e2 :: t2 ++ efs, σ2) → `````` 106 107 `````` prim_step e1 σ1 κ e2 σ2 efs → step ρ1 κ ρ2. `````` Tej Chajed committed Nov 29, 2018 108 `````` Hint Constructors step : core. `````` 109 `````` `````` Marianna Rapoport committed Oct 05, 2018 110 `````` Inductive nsteps : nat → cfg Λ → list (observation Λ) → cfg Λ → Prop := `````` Ralf Jung committed Oct 18, 2018 111 112 113 114 115 `````` | nsteps_refl ρ : nsteps 0 ρ [] ρ | nsteps_l n ρ1 ρ2 ρ3 κ κs : step ρ1 κ ρ2 → nsteps n ρ2 κs ρ3 → `````` Ralf Jung committed Oct 18, 2018 116 `````` nsteps (S n) ρ1 (κ ++ κs) ρ3. `````` Tej Chajed committed Nov 29, 2018 117 `````` Hint Constructors nsteps : core. `````` 118 `````` `````` Ralf Jung committed Oct 05, 2018 119 `````` Definition erased_step (ρ1 ρ2 : cfg Λ) := ∃ κ, step ρ1 κ ρ2. `````` 120 `````` `````` Ralf Jung committed Jun 07, 2019 121 122 `````` (** [rtc erased_step] and [nsteps] encode the same thing, just packaged in a different way. *) `````` 123 `````` Lemma erased_steps_nsteps ρ1 ρ2 : `````` Robbert Krebbers committed Jun 07, 2019 124 `````` rtc erased_step ρ1 ρ2 ↔ ∃ n κs, nsteps n ρ1 κs ρ2. `````` 125 `````` Proof. `````` Ralf Jung committed Jun 07, 2019 126 `````` split. `````` Robbert Krebbers committed Jun 07, 2019 127 128 129 `````` - induction 1; firstorder eauto. (* FIXME: [naive_solver eauto] should be able to handle this *) - intros (n & κs & Hsteps). unfold erased_step. induction Hsteps; eauto using rtc_refl, rtc_l. `````` 130 `````` Qed. `````` Robbert Krebbers committed Feb 01, 2016 131 `````` `````` Robbert Krebbers committed Apr 19, 2016 132 133 `````` Lemma of_to_val_flip v e : of_val v = e → to_val e = Some v. Proof. intros <-. by rewrite to_of_val. Qed. `````` Robbert Krebbers committed Mar 15, 2017 134 135 136 `````` Lemma not_reducible e σ : ¬reducible e σ ↔ irreducible e σ. Proof. unfold reducible, irreducible. naive_solver. Qed. `````` Robbert Krebbers committed Jan 19, 2016 137 `````` Lemma reducible_not_val e σ : reducible e σ → to_val e = None. `````` 138 `````` Proof. intros (?&?&?&?&?); eauto using val_stuck. Qed. `````` Ralf Jung committed Oct 05, 2018 139 140 `````` Lemma reducible_no_obs_reducible e σ : reducible_no_obs e σ → reducible e σ. Proof. intros (?&?&?&?); eexists; eauto. Qed. `````` David Swasey committed Nov 08, 2017 141 `````` Lemma val_irreducible e σ : is_Some (to_val e) → irreducible e σ. `````` 142 `````` Proof. intros [??] ???? ?%val_stuck. by destruct (to_val e). Qed. `````` Robbert Krebbers committed Aug 08, 2016 143 `````` Global Instance of_val_inj : Inj (=) (=) (@of_val Λ). `````` Robbert Krebbers committed Feb 11, 2016 144 `````` Proof. by intros v v' Hv; apply (inj Some); rewrite -!to_of_val Hv. Qed. `````` Amin Timany committed Nov 06, 2019 145 146 147 148 149 `````` Lemma not_not_stuck e σ : ¬not_stuck e σ ↔ stuck e σ. Proof. rewrite /stuck /not_stuck -not_eq_None_Some -not_reducible. destruct (decide (to_val e = None)); naive_solver. Qed. `````` Ralf Jung committed Jan 05, 2016 150 `````` `````` Ralf Jung committed Dec 07, 2017 151 152 153 `````` Lemma strongly_atomic_atomic e a : Atomic StronglyAtomic e → Atomic a e. Proof. unfold Atomic. destruct a; eauto using val_irreducible. Qed. `````` Ralf Jung committed Oct 29, 2017 154 `````` `````` Ralf Jung committed Mar 06, 2019 155 `````` Lemma reducible_fill `{!@LanguageCtx Λ K} e σ : `````` Robbert Krebbers committed Mar 15, 2017 156 157 `````` to_val e = None → reducible (K e) σ → reducible e σ. Proof. `````` 158 `````` intros ? (e'&σ'&k&efs&Hstep); unfold reducible. `````` Robbert Krebbers committed Mar 15, 2017 159 160 `````` apply fill_step_inv in Hstep as (e2' & _ & Hstep); eauto. Qed. `````` Ralf Jung committed Mar 06, 2019 161 `````` Lemma reducible_no_obs_fill `{!@LanguageCtx Λ K} e σ : `````` Ralf Jung committed Oct 05, 2018 162 163 164 165 166 `````` to_val e = None → reducible_no_obs (K e) σ → reducible_no_obs e σ. Proof. intros ? (e'&σ'&efs&Hstep); unfold reducible_no_obs. apply fill_step_inv in Hstep as (e2' & _ & Hstep); eauto. Qed. `````` Ralf Jung committed Mar 06, 2019 167 `````` Lemma irreducible_fill `{!@LanguageCtx Λ K} e σ : `````` Robbert Krebbers committed Mar 15, 2017 168 169 `````` to_val e = None → irreducible e σ → irreducible (K e) σ. Proof. rewrite -!not_reducible. naive_solver eauto using reducible_fill. Qed. `````` Robbert Krebbers committed Sep 24, 2017 170 `````` `````` Hai Dang committed Jun 20, 2019 171 172 `````` Lemma stuck_fill `{!@LanguageCtx Λ K} e σ : stuck e σ → stuck (K e) σ. `````` Hai Dang committed Jun 20, 2019 173 `````` Proof. intros [??]. split. by apply fill_not_val. by apply irreducible_fill. Qed. `````` Hai Dang committed Jun 20, 2019 174 `````` `````` 175 176 `````` Lemma step_Permutation (t1 t1' t2 : list (expr Λ)) κ σ1 σ2 : t1 ≡ₚ t1' → step (t1,σ1) κ (t2,σ2) → ∃ t2', t2 ≡ₚ t2' ∧ step (t1',σ1) κ (t2',σ2). `````` Robbert Krebbers committed Sep 24, 2017 177 178 179 180 181 182 183 `````` Proof. intros Ht [e1 σ1' e2 σ2' efs tl tr ?? Hstep]; simplify_eq/=. move: Ht; rewrite -Permutation_middle (symmetry_iff (≡ₚ)). intros (tl'&tr'&->&Ht)%Permutation_cons_inv. exists (tl' ++ e2 :: tr' ++ efs); split; [|by econstructor]. by rewrite -!Permutation_middle !assoc_L Ht. Qed. `````` Dan Frumin committed Sep 25, 2017 184 `````` `````` Amin Timany committed Nov 06, 2019 185 186 187 188 189 190 191 192 193 194 195 196 `````` Lemma step_insert i t2 σ2 e κ e' σ3 efs : t2 !! i = Some e → prim_step e σ2 κ e' σ3 efs → step (t2, σ2) κ (<[i:=e']>t2 ++ efs, σ3). Proof. intros. edestruct (elem_of_list_split_length t2) as (t21&t22&?&?); first (by eauto using elem_of_list_lookup_2); simplify_eq. econstructor; eauto. by rewrite insert_app_r_alt // Nat.sub_diag /= -assoc_L. Qed. `````` 197 198 199 `````` Lemma erased_step_Permutation (t1 t1' t2 : list (expr Λ)) σ1 σ2 : t1 ≡ₚ t1' → erased_step (t1,σ1) (t2,σ2) → ∃ t2', t2 ≡ₚ t2' ∧ erased_step (t1',σ1) (t2',σ2). Proof. `````` Ralf Jung committed Oct 18, 2018 200 `````` intros Heq [? Hs]. pose proof (step_Permutation _ _ _ _ _ _ Heq Hs). firstorder. `````` Ralf Jung committed Oct 18, 2018 201 `````` (* FIXME: [naive_solver] should be able to handle this *) `````` 202 203 `````` Qed. `````` Robbert Krebbers committed Oct 05, 2018 204 `````` Record pure_step (e1 e2 : expr Λ) := { `````` Ralf Jung committed Oct 18, 2018 205 206 `````` pure_step_safe σ1 : reducible_no_obs e1 σ1; pure_step_det σ1 κ e2' σ2 efs : `````` 207 `````` prim_step e1 σ1 κ e2' σ2 efs → κ = [] ∧ σ2 = σ1 ∧ e2' = e2 ∧ efs = [] `````` Dan Frumin committed Sep 25, 2017 208 209 `````` }. `````` Amin Timany committed Nov 06, 2019 210 211 `````` Notation pure_steps_tp := (Forall2 (rtc pure_step)). `````` Robbert Krebbers committed Oct 05, 2018 212 213 214 `````` (* TODO: Exclude the case of [n=0], either here, or in [wp_pure] to avoid it succeeding when it did not actually do anything. *) Class PureExec (φ : Prop) (n : nat) (e1 e2 : expr Λ) := `````` Ralf Jung committed Oct 18, 2018 215 `````` pure_exec : φ → relations.nsteps pure_step n e1 e2. `````` Dan Frumin committed Sep 25, 2017 216 `````` `````` Ralf Jung committed Mar 06, 2019 217 `````` Lemma pure_step_ctx K `{!@LanguageCtx Λ K} e1 e2 : `````` Robbert Krebbers committed Oct 05, 2018 218 219 `````` pure_step e1 e2 → pure_step (K e1) (K e2). `````` Ralf Jung committed Nov 24, 2017 220 221 `````` Proof. intros [Hred Hstep]. split. `````` Ralf Jung committed Oct 05, 2018 222 `````` - unfold reducible_no_obs in *. naive_solver eauto using fill_step. `````` Ralf Jung committed Oct 18, 2018 223 `````` - intros σ1 κ e2' σ2 efs Hpstep. `````` 224 `````` destruct (fill_step_inv e1 σ1 κ e2' σ2 efs) as (e2'' & -> & ?); [|exact Hpstep|]. `````` Ralf Jung committed Oct 05, 2018 225 `````` + destruct (Hred σ1) as (? & ? & ? & ?); eauto using val_stuck. `````` 226 `````` + edestruct (Hstep σ1 κ e2'' σ2 efs) as (? & -> & -> & ->); auto. `````` Ralf Jung committed Nov 24, 2017 227 228 `````` Qed. `````` Ralf Jung committed Mar 06, 2019 229 `````` Lemma pure_step_nsteps_ctx K `{!@LanguageCtx Λ K} n e1 e2 : `````` Ralf Jung committed Oct 18, 2018 230 231 `````` relations.nsteps pure_step n e1 e2 → relations.nsteps pure_step n (K e1) (K e2). `````` Amin Timany committed Nov 06, 2019 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 `````` Proof. eauto using nsteps_congruence, pure_step_ctx. Qed. Lemma rtc_pure_step_ctx K `{!@LanguageCtx Λ K} e1 e2 : rtc pure_step e1 e2 → rtc pure_step (K e1) (K e2). Proof. eauto using rtc_congruence, pure_step_ctx. Qed. Lemma not_stuck_under_ectx K `{!@LanguageCtx Λ K} e σ : not_stuck (K e) σ → not_stuck e σ. Proof. rewrite /not_stuck /reducible /=. intros [[? HK]|(?&?&?&?&Hstp)]; simpl in *. - left. apply not_eq_None_Some; intros ?%fill_not_val; simplify_eq. - destruct (to_val e) eqn:?; first by left; eauto. apply fill_step_inv in Hstp; naive_solver. Qed. `````` Robbert Krebbers committed Oct 05, 2018 247 248 `````` (* We do not make this an instance because it is awfully general. *) `````` Ralf Jung committed Mar 06, 2019 249 `````` Lemma pure_exec_ctx K `{!@LanguageCtx Λ K} φ n e1 e2 : `````` Robbert Krebbers committed Oct 05, 2018 250 251 252 253 `````` PureExec φ n e1 e2 → PureExec φ n (K e1) (K e2). Proof. rewrite /PureExec; eauto using pure_step_nsteps_ctx. Qed. `````` Dan Frumin committed Sep 25, 2017 254 255 `````` (* This is a family of frequent assumptions for PureExec *) Class IntoVal (e : expr Λ) (v : val Λ) := `````` Ralf Jung committed Jun 18, 2018 256 `````` into_val : of_val v = e. `````` Robbert Krebbers committed Nov 02, 2017 257 `````` `````` Ralf Jung committed Jun 18, 2018 258 `````` Class AsVal (e : expr Λ) := as_val : ∃ v, of_val v = e. `````` Robbert Krebbers committed Nov 02, 2017 259 260 261 262 263 `````` (* There is no instance [IntoVal → AsVal] as often one can solve [AsVal] more efficiently since no witness has to be computed. *) Global Instance as_vals_of_val vs : TCForall AsVal (of_val <\$> vs). Proof. apply TCForall_Forall, Forall_fmap, Forall_true=> v. `````` Ralf Jung committed Jun 18, 2018 264 `````` rewrite /AsVal /=; eauto. `````` Robbert Krebbers committed Nov 02, 2017 265 `````` Qed. `````` Amin Timany committed Nov 06, 2019 266 `````` `````` Ralf Jung committed Jun 21, 2018 267 268 269 `````` Lemma as_val_is_Some e : (∃ v, of_val v = e) → is_Some (to_val e). Proof. intros [v <-]. rewrite to_of_val. eauto. Qed. `````` Amin Timany committed Nov 06, 2019 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 `````` Lemma prim_step_not_stuck e σ κ e' σ' efs : prim_step e σ κ e' σ' efs → not_stuck e σ. Proof. rewrite /not_stuck /reducible. eauto 10. Qed. Lemma rtc_pure_step_val `{!Inhabited (state Λ)} v e : rtc pure_step (of_val v) e → to_val e = Some v. Proof. intros ?; rewrite <- to_of_val. f_equal; symmetry; eapply rtc_nf; first done. intros [e' [Hstep _]]. destruct (Hstep inhabitant) as (?&?&?&Hval%val_stuck). by rewrite to_of_val in Hval. Qed. (** Let thread pools [t1] and [t3] be such that each thread in [t1] makes (zero or more) pure steps to the corresponding thread in [t3]. Furthermore, let [t2] be a thread pool such that [t1] under state [σ1] makes a (single) step to thread pool [t2] and state [σ2]. In this situation, either the step from [t1] to [t2] corresponds to one of the pure steps between [t1] and [t3], or, there is an [i] such that [i]th thread does not participate in the pure steps between [t1] and [t3] and [t2] corresponds to taking a step in the [i]th thread starting from [t1]. *) Lemma erased_step_pure_step_tp t1 σ1 t2 σ2 t3 : erased_step (t1, σ1) (t2, σ2) → pure_steps_tp t1 t3 → (σ1 = σ2 ∧ pure_steps_tp t2 t3) ∨ (∃ i e efs e' κ, t1 !! i = Some e ∧ t3 !! i = Some e ∧ t2 = <[i:=e']>t1 ++ efs ∧ prim_step e σ1 κ e' σ2 efs). Proof. intros [κ [e σ e' σ' ? t11 t12 ?? Hstep]] Hps; simplify_eq/=. apply Forall2_app_inv_l in Hps as (t31&?&Hpsteps&(e''&t32&Hps&?&->)%Forall2_cons_inv_l&->). destruct Hps as [e|e1 e2 e3 [_ Hprs]]. - right. exists (length t11), e, efs, e', κ; split_and!; last done. + by rewrite lookup_app_r // Nat.sub_diag. + apply Forall2_length in Hpsteps. by rewrite lookup_app_r Hpsteps // Nat.sub_diag. + by rewrite insert_app_r_alt // Nat.sub_diag /= -assoc_L. - edestruct Hprs as (?&?&?&?); first done; simplify_eq. left; split; first done. rewrite right_id_L. eauto using Forall2_app. Qed. `````` Robbert Krebbers committed Mar 15, 2017 318 ``````End language. `````` Amin Timany committed Nov 06, 2019 319 320 `````` Notation pure_steps_tp := (Forall2 (rtc pure_step)).``````