Commit 49b04e85 by Ralf Jung

parent d8755b5e
 ... ... @@ -42,4 +42,4 @@ theories/sorting.v theories/infinite.v theories/nat_cancel.v theories/namespaces.v theories/telescopes.v
 From stdpp Require Import base tactics. Set Default Proof Using "Type". (** Telescopes *) Inductive tele : Type := | TeleO : tele | TeleS {X} (binder : X → tele) : tele. Arguments TeleS {_} _. (** The telescope version of Coq's function type *) Fixpoint tele_fun (TT : tele) (T : Type) : Type := match TT with | TeleO => T | TeleS b => ∀ x, tele_fun (b x) T end. Notation "TT -t> A" := (tele_fun TT A) (at level 99, A at level 200, right associativity). (** An eliminator for elements of [tele_fun]. We use a [fix] because, for some reason, that makes stuff print nicer in the proofs in iris:bi/lib/telescopes.v *) Definition tele_fold {X Y} {TT : tele} (step : ∀ {A : Type}, (A → Y) → Y) (base : X → Y) : (TT -t> X) → Y := (fix rec {TT} : (TT -t> X) → Y := match TT as TT return (TT -t> X) → Y with | TeleO => λ x : X, base x | TeleS b => λ f, step (λ x, rec (f x)) end) TT. Arguments tele_fold {_ _ !_} _ _ _ /. (** A sigma-like type for an "element" of a telescope, i.e. the data it takes to get a [T] from a [TT -t> T]. *) Inductive tele_arg : tele → Type := | TargO : tele_arg TeleO (* the [x] is the only relevant data here *) | TargS {X} {binder} (x : X) : tele_arg (binder x) → tele_arg (TeleS binder). Definition tele_app {TT : tele} {T} (f : TT -t> T) : tele_arg TT → T := λ a, (fix rec {TT} (a : tele_arg TT) : (TT -t> T) → T := match a in tele_arg TT return (TT -t> T) → T with | TargO => λ t : T, t | TargS x a => λ f, rec a (f x) end) TT a f. Arguments tele_app {!_ _} _ !_ /. Coercion tele_arg : tele >-> Sortclass. Coercion tele_app : tele_fun >-> Funclass. (** Inversion lemma for [tele_arg] *) Lemma tele_arg_inv {TT : tele} (a : TT) : match TT as TT return TT → Prop with | TeleO => λ a, a = TargO | TeleS f => λ a, ∃ x a', a = TargS x a' end a. Proof. induction a; eauto. Qed. Lemma tele_arg_O_inv (a : TeleO) : a = TargO. Proof. exact (tele_arg_inv a). Qed. Lemma tele_arg_S_inv {X} {f : X → tele} (a : TeleS f) : ∃ x a', a = TargS x a'. Proof. exact (tele_arg_inv a). Qed. (** Map below a tele_fun *) Fixpoint tele_map {T U} {TT : tele} : (T → U) → (TT -t> T) → TT -t> U := match TT as TT return (T → U) → (TT -t> T) → TT -t> U with | TeleO => λ F : T → U, F | @TeleS X b => λ (F : T → U) (f : TeleS b -t> T) (x : X), tele_map F (f x) end. Arguments tele_map {_ _ !_} _ _ /. Lemma tele_map_app {T U} {TT : tele} (F : T → U) (t : TT -t> T) (x : TT) : (tele_map F t) x = F (t x). Proof. induction TT as [|X f IH]; simpl in *. - rewrite (tele_arg_O_inv x). done. - destruct (tele_arg_S_inv x) as [x' [a' ->]]. simpl. rewrite <-IH. done. Qed. Global Instance tele_fmap {TT : tele} : FMap (tele_fun TT) := λ T U, tele_map. Lemma tele_fmap_app {T U} {TT : tele} (F : T → U) (t : TT -t> T) (x : TT) : (F <\$> t) x = F (t x). Proof. apply tele_map_app. Qed. Global Instance tele_fmap2 {TT1 TT2 : tele} : FMap (tele_fun TT1 ∘ tele_fun TT2) := λ T U, tele_map ∘ tele_map. Lemma tele_fmap2_app {T U} {TT1 TT2 : tele} (F : T → U) (t : TT1 -t> TT2 -t> T) (x : TT1) (y : TT2) : (F <\$> t) x y = F (t x y). Proof. unfold fmap, tele_fmap2. simpl. rewrite !tele_map_app. done. Qed. (** Operate below [tele_fun]s with argument telescope [TT]. *) Fixpoint tele_bind {U} {TT : tele} : (TT → U) → TT -t> U := match TT as TT return (TT → U) → TT -t> U with | TeleO => λ F, F TargO | @TeleS X b => λ (F : TeleS b → U) (x : X), (* b x -t> U *) tele_bind (λ a, F (TargS x a)) end. Arguments tele_bind {_ !_} _ /. (** A function that looks funny. *) Definition tele_arg_id (TT : tele) : TT -t> TT := tele_bind id. (** Notation *) Notation "'[tele' x .. z ]" := (TeleS (fun x => .. (TeleS (fun z => TeleO)) ..)) (x binder, z binder, format "[tele '[hv' x .. z ']' ]"). Notation "'[tele' ]" := (TeleO) (format "[tele ]"). Notation "'[tele_arg' x ; .. ; z ]" := (TargS x ( .. (TargS z TargO) ..)) (format "[tele_arg '[hv' x ; .. ; z ']' ]"). Notation "'[tele_arg' ]" := (TargO) (format "[tele_arg ]"). (** Notation-compatible telescope mapping *) Notation "'λ..' x .. y , e" := (tele_app \$ tele_bind (λ x, .. (tele_app \$ tele_bind (λ y, e)) .. )) (at level 200, x binder, y binder, right associativity, format "'[ ' 'λ..' x .. y ']' , e").
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