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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Petit bench vite fait, mal fait *)
Require Refine.
(************************************************************************)
Lemma essai : (x:nat)x=x.
Refine (([x0:nat]Cases x0 of
O => ?
| (S p) => ?
end) :: (x:nat)x=x). (* x0=x0 et x0=x0 *)
Restart.
Refine [x0:nat]<[n:nat]n=n>Case x0 of ? [p:nat]? end. (* OK *)
Restart.
Refine [x0:nat]<[n:nat]n=n>Cases x0 of O => ? | (S p) => ? end. (* OK *)
Restart.
(**
Refine [x0:nat]Cases x0 of O => ? | (S p) => ? end. (* cannot be executed *)
**)
Abort.
(************************************************************************)
Lemma T : nat.
Refine (S ?).
Abort.
(************************************************************************)
Lemma essai2 : (x:nat)x=x.
Refine Fix f{f/1 : (x:nat)x=x := [x:nat]? }.
Restart.
Refine Fix f{f/1 : (x:nat)x=x :=
[x:nat]<[n:nat](eq nat n n)>Case x of ? [p:nat]? end}.
Restart.
Refine Fix f{f/1 : (x:nat)x=x :=
[x:nat]<[n:nat]n=n>Cases x of O => ? | (S p) => ? end}.
Restart.
Refine Fix f{f/1 : (x:nat)x=x :=
[x:nat]<[n:nat](eq nat n n)>Case x of
?
[p:nat](f_equal nat nat S p p ?) end}.
Restart.
Refine Fix f{f/1 : (x:nat)x=x :=
[x:nat]<[n:nat](eq nat n n)>Cases x of
O => ?
| (S p) =>(f_equal nat nat S p p ?) end}.
Abort.
(************************************************************************)
Lemma essai : nat.
Parameter f : nat*nat -> nat -> nat.
Refine (f ? ([x:nat](? :: nat) O)).
Restart.
Refine (f ? O).
Abort.
(************************************************************************)
Parameter P : nat -> Prop.
Lemma essai : { x:nat | x=(S O) }.
Refine (exist nat ? (S O) ?). (* ECHEC *)
Restart.
(* mais si on contraint par le but alors ca marche : *)
(* Remarque : on peut toujours faire ça *)
Refine ((exist nat ? (S O) ?) :: { x:nat | x=(S O) }).
Restart.
Refine (exist nat [x:nat](x=(S O)) (S O) ?).
Abort.
(************************************************************************)
Lemma essai : (n:nat){ x:nat | x=(S n) }.
Refine [n:nat]<[n:nat]{x:nat|x=(S n)}>Case n of ? [p:nat]? end.
Restart.
Refine (([n:nat]Case n of ? [p:nat]? end) :: (n:nat){ x:nat | x=(S n) }).
Restart.
Refine [n:nat]<[n:nat]{x:nat|x=(S n)}>Cases n of O => ? | (S p) => ? end.
Restart.
Refine Fix f{f/1 :(n:nat){x:nat|x=(S n)} :=
[n:nat]<[n:nat]{x:nat|x=(S n)}>Case n of ? [p:nat]? end}.
Restart.
Refine Fix f{f/1 :(n:nat){x:nat|x=(S n)} :=
[n:nat]<[n:nat]{x:nat|x=(S n)}>Cases n of O => ? | (S p) => ? end}.
Exists (S O). Trivial.
Elim (f0 p).
Refine [x:nat][h:x=(S p)](exist nat [x:nat]x=(S (S p)) (S x) ?).
Rewrite h. Auto.
Save.
(* Quelques essais de recurrence bien fondée *)
Require Wf.
Require Wf_nat.
Lemma essai_wf : nat->nat.
Refine [x:nat](well_founded_induction
nat
lt ?
[_:nat]nat->nat
[phi0:nat][w:(phi:nat)(lt phi phi0)->nat->nat](w x ?)
x x).
Exact lt_wf.
Abort.
Require Compare_dec.
Require Lt.
Lemma fibo : nat -> nat.
Refine (well_founded_induction
nat
lt ?
[_:nat]nat
[x0:nat][fib:(x:nat)(lt x x0)->nat]
Cases (zerop x0) of
(left _) => (S O)
| (right h1) => Cases (zerop (pred x0)) of
(left _) => (S O)
| (right h2) => (plus (fib (pred x0) ?)
(fib (pred (pred x0)) ?))
end
end).
(*********
Refine (well_founded_induction
nat
lt ?
[_:nat]nat
[x0:nat][fib:(x:nat)(lt x x0)->nat]
Cases x0 of
O => (S O)
| (S O) => (S O)
| (S (S p)) => (plus (fib (pred x0) ?)
(fib (pred (pred x0)) ?))
end).
***********)
Exact lt_wf.
Auto.
Apply lt_trans with m:=(pred x0); Auto.
Save.
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