ooops, missing )

[helm.git] / matita / tests / coercions_propagation.ma

diff --git a/matita/tests/coercions_propagation.ma b/matita/tests/coercions_propagation.ma
index c8529631b7a57515a4a2c59714ed634e30050982..8ff8f9c759ba41d26541571040df79e6b11586af 100644 (file)
--- a/matita/tests/coercions_propagation.ma
+++ b/matita/tests/coercions_propagation.ma
@@ -50,10 +50,171 @@ qed.
 
 theorem test4: (∃n. 1 ≤ n) → ∃n. 0 < n.
  apply ((λn:nat.n) : (∃n. 1 ≤ n) → ∃n. 0 < n);
- cases name_con;
+ cases s;
  assumption.
 qed.
 
+theorem test5: nat → ∃n. 1 ≤ n.
+apply (
+ let rec aux n : nat ≝
+  match n with
+   [ O ⇒ 1
+   | S m ⇒ S (aux m)
+   ]
+ in
+  aux
+: nat → ∃n. 1 ≤ n);
+[ cases (aux n1); simplify; ] autobatch;
+qed.
+
+inductive NN (A:Type) : nat -> Type ≝
+ | NO : NN A O
+ | NS : ∀n:nat. NN A n → NN A (S n).
+ 
+definition injectN ≝ λA,k.λP.λa:NN A k.λp:P a. sigma_intro ? P ? p.
+
+coercion cic:/matita/test/coercions_propagation/injectN.con 0 1.
+
+definition ejectN ≝ λA,k.λP.λc: ∃n:NN A k.P n. match c with [ sigma_intro w _ ⇒ w].
+
+coercion cic:/matita/test/coercions_propagation/ejectN.con.
+
+definition PN :=
+ λA,k.λx:NN A k. 1 <= k.
+
+theorem test51_byhand: ∀A,k. NN A k → ∃n:NN A (S k). PN ? ? n.
+intros 1;
+apply (
+ let rec aux (w : nat) (n : NN A w) on n : ∃p:NN A (S w).PN ? ? p ≝
+  match n in NN return λx.λm:NN A x.∃p:NN A (S x).PN ? ? p with
+   [ NO ⇒ injectN ? ? ? (NS A ? (NO A)) ?
+   | NS x m ⇒ injectN ? ? ? (NS A (S x) (ejectN ? ? ? (aux ? m))) ?  
+   ]
+ in
+  aux
+: ∀n:nat. NN A n → ∃m:NN A (S n). PN ? ? m);
+[2: cases (aux x m); simplify; autobatch ] unfold PN; autobatch;
+qed.
+
+theorem f : nat -> nat -> ∃n:nat.O <= n.
+apply (λx,y:nat.y : nat -> nat -> ∃n:nat. O <= n).
+apply le_O_n; 
+qed.
+
+axiom F : nat -> nat -> nat.
+
+theorem f1 : nat -> nat -> ∃n:nat.O <= n.
+apply (F : nat -> nat -> ∃n:nat. O <= n).
+apply le_O_n; 
+qed.
+
+theorem test51: ∀A,k. NN A k → ∃n:NN A (S k). PN ? ? n.
+intros 1;
+letin xxx ≝ (
+ let rec aux (w : nat) (n : NN A w) on n : NN A (S w) ≝
+  match n in NN return λx.λm:NN A x.NN A (S x) with
+   [ NO ⇒ NS A ? (NO A)
+   | NS x m ⇒ NS A (S x) (aux ? m)  
+   ]
+ in
+  aux
+: ∀n:nat. NN A n → ∃m:NN A (S n). PN ? ? m); [3: apply xxx];
+unfold PN in aux ⊢ %; [cases (aux n2 n3)] autobatch
+qed.
+
+(* guarded troppo debole 
+theorem test5: nat → ∃n. 1 ≤ n.
+apply (
+ let rec aux n : nat ≝
+  match n with
+   [ O ⇒ 1
+   | S m ⇒ S (aux m)
+   ]
+ in
+  aux
+: nat → ∃n. 1 ≤ n);
+cases name_con;
+simplify;
+ [ autobatch
+ | cases (aux n);
+   simplify;
+   apply lt_O_S
+ ]
+qed.
+*)
+
+(*
+theorem test5: nat → ∃n. 1 ≤ n.
+apply (
+ let rec aux (n : nat) : ∃n. 1 ≤ n ≝ 
+   match n with
+   [ O => (S O : ∃n. 1 ≤ n)
+   | S m => (S (aux m) : ∃n. 1 ≤ n)
+(*      
+   inject ? (S (eject ? (aux m))) ? *)
+   ]
+ in
+  aux
+ : nat → ∃n. 1 ≤ n);
+ [ autobatch
+ | elim (aux m);
+   simplify;
+   autobatch
+ ]
+qed.
+
+Re1: nat => nat |- body[Rel1] : nat => nat
+Re1: nat => X |- body[Rel1] : nat => nat : nat => X
+Re1: Y => X |- body[Rel1] : nat => nat : Y => X
+
+nat => nat
+nat => X
+
+theorem test5: (∃n. 2 ≤ n) → ∃n. 1 ≤ n.
+apply (
+ λk: ∃n. 2 ≤ n.
+ let rec aux n : eject ? n = eject ? k → ∃n. 1 ≤ n ≝
+  match eject ? n return λx:nat. x = eject ? k → ∃n. 1 ≤ n with
+   [ O ⇒ λH: 0 = eject ? k.
+          sigma_intro ? ? 0 ?
+   | S m ⇒ λH: S m = eject ? k.
+          sigma_intro ? ? (S m) ?
+   ]
+ in
+  aux k (refl_eq ? (eject ? k)));
+
+
+intro;
+cases s; clear s;
+generalize in match H; clear H;
+elim a;
+ [ apply (sigma_intro ? ? 0);
+ | apply (sigma_intro ? ? (S n));
+ ].
+
+apply (
+ λk.
+ let rec aux n : ∃n. 1 ≤ n ≝
+  inject ?
+   (match n with
+     [ O ⇒ O
+     | S m ⇒ S (eject ? (aux m))
+     ]) ?
+ in aux (eject ? k)).
+
+   
+apply (
+ let rec aux n : nat ≝
+  match n with
+   [ O ⇒ O
+   | S m ⇒ S (aux m)
+   ]
+ in
+  aux
+: (∃n. 2 ≤ n) → ∃n. 1 ≤ n);
+
+qed.
+
 (*
 theorem test5: nat → ∃n. 0 ≤ n.
  apply (λn:nat.?);
@@ -63,3 +224,4 @@ theorem test5: nat → ∃n. 0 ≤ n.
  autobatch.
 qed.
 *)
+*)
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