SSL rew_polyuniv.v Sprache: Coq

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Set

Local Set }} ( :X : (binder (TeleS

(** Telescopes *)   TTTTtT > =
java.lang.StringIndexOutOfBoundsException: Range [11, 9) out of bounds for length 24
  | TeleO : tele =>fun
    {X binder -)  .

Arguments TeleS)  a.

(** The telescope version of Coq's function type *)
(TTtele:) : Type
  match TT with
=
  |  |TeleO a
  end

Notation TeleSfun> x'  = a
  e a.

(** A sigma-like type for an "element" of a telescope, i.e. the data it  ) : a  .
  takes to get a [T] from a [TT -t> T]. *)
InductiveLemma tele_arg_invtele_arg_S_inv}f  X-tele (   ) :
|TargO java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
(* the [x] is the only relevant data here *)   data
le_arg_inva..

Definition tele_app {Lemmatele_arg_O_inv:TeleO=TargO
   a= fix {TT(:tele_arg  TTT > =
     match a in tele_arg TT return (TT -t> T)    x '    x a.
     |  =>   matchas return ->U   -t> U with
     | TargS x a => fun f => rec a (f                   tele_bind ( a = F(TargS tele_bind TT} :( ->U)-  ->U:
     java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Arguments tele_app {!_ _} _ !_ /.

Coercion tele_arg : tele >-> Sortclass.
Coercion tele_app as return - )- -  java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

(** Inversion lemma for [tele_arg] *)
Lemma tele_arg_inv {:tele  ) :
  match TT as TT return TT -  end.
  |TeleOfun a = a = TargO
  | TeleS
  end a.
Proof. induction a; eauto. Qed
Lemma tele_arg_O_inv (a Lemmatele_app_bindU {T :telef   ->
Proof. exact (tele_arg_inv]; simpl in *.
Lemma tele_arg_S_inv {X} {f : X -> tele} (a :
  exists x a',   destruct ( x)  ['[' -].simpl
Proof ( a)..

(** Operate below [tele_fun]s with argument telescope [TT]. *)
java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  match TT  return -U)>TT  with
  | TeleO =>    binder so that, after simplifying, this matches   notations
eleS)(:X) >(* b x -t> U *)
                  tele_bind (fun a => F (TargS x a))
  end.
Arguments tele_bind {_ !_} _ /.

(* Show that tele_app ∘ tele_bind is the identity. *)
Lemma tele_app_bind {U} {TT : tele} (f : TT -> U) x :
  (tele_app (tele_bind f)) x = f x.
Proof.
  induction TT as [|X b IH]; simpl in *.
  - rewrite (tele_arg_O_inv x). auto.
  - destruct (tele_arg_S_inv x) as [x' [a' ->]]. simpl.
    rewrite IH. auto.
Qed.

(** Notation-compatible telescope mapping *)
(* 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 "'λ..' x .. y , e" :=
  (tele_app (tele_bind (fun x => .. (tele_app (tele_bind (fun y => e))) .. )))
  (at level 200, x binder, y binder, right associativity,
   format "'[ ' 'λ..' x .. y ']' , e").

(* The testcase *)
Lemma test {TA TB : tele} {X} (α' β' γ' : X -> Prop) (Φ : TA -> TB -> Prop) x' :
  (forall P Q, ((P /\ Q) = Q) * ((P -> Q) = Q)) ->
  forall a b, Φ a b = tele_bind (fun x : TA => tele_bind (fun y : TB => β' x' /\ (γ' x' -> Φ x y))) a b.
Proof.
intros cheat a b.
rewrite !tele_app_bind.
by rewrite !cheat.
Qed.

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