Quellcode-Bibliothek rew_polyuniv.v Sprache: Coq

From Corelib Require Import Setoid ssreflect.
Set Default Proof Using "Type".

Local Set Universe Polymorphism.

(** 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 => forall x, tele_fun (b x) T
  end.

Notation "T -t> A" =
  (tele_funTT A (atlevel  at level 200 right ).

(** 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
tele_arg
(* the [x] is the only relevant data here *)
| TargS {X tele_funTTtele)( :Type)   :=

Definition tele_app {match with|TeleO = T
  fun a => (fix recjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     match a in  TT returnTT- T >Twith
     | TargO  t : T =>t
     | TargS   |TeleSX} (binder : X > tele :tele
     end TT f.
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Coercion tele_argTT : ) (T :Type :=
Coercion tele_app :   match  | TeleO T

(** Inversion lemma for [tele_arg] *) " f =>fun a => exists x a',a =TargSxxajava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
Lemma Lemma  {{f: - tele( :TeleS:
  match TT as| : tele_argTeleO
  | TeleO => fun a => a = TargO
  |  f => fun *  [x] isonly here
  end a.
Proof. | TargS {X} {bin ).Qed
Lemma  (a : ) : a =TargO.
Proof.
Lemma fun => ( recTT ( tele_argTT:( -t  >T:java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  exists a, a=TargS'java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
ProofTargO  TT  TT (TT -> U)->TTt

(** Operate below [tele_fun]s with argument telescope [TT]. *)  | @TeleS X b => java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Fixpointtele_bind :tele TT ) - TTt  :
  match TT as TT return (TT -> U) -> TT -t> U with
    => fun F=>FTargO
  | java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                  tele_bind (fun a => F (TargS x a))
  end.
Arguments tele_bind {_ !_rt Setoid ssreflectjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

(* Show that tele_app ∘ tele_bind is the identity. *) :tele : - ) x java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
Lemmatele_app_bind{ : telef:TT>U x:
  (tele_app (tele_bind f)) x = f x.
Proof.
  induction TT as [|X    TeleSX (binder : X - tele:tele
  - rewrite (tele_arg_O_inv. .
  - destruct (tele_arg_S_inv rewritetele_arg_O_inv)(T :teleT:Type Type=
    rewrite IH. auto.
Qed.

(** Notation-compatible telescope mapping *) IH. auto.
(* This adds (tele_app ∘ tele_bind), which is an identity function, around every
   binder so that, after simplifying, this matches the way we typically write
   notations involving telescopes. *)
Notation'..'x .y,e  so,aftermatches
  (tele_app (tele_bind (fun x   notations involving
   00 x  ybinderassociativity
   java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

(* The testcase *)
Lemma    (at 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (forallP,( \) )*(P -Q  ) -
  forall a b, Φ a   : tele_arg
Proof.
intros cheat
rewrite !tele_app_bind.
by rewrite !cheatjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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