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

diff --git a/matita/tests/coercions_propagation.ma b/matita/tests/coercions_propagation.ma
index af5f2435ca52b76525321bdb6ae881a2966cd239..63c48e66b9d42267b3d3455cc855565b80527095 100644 (file)
--- a/matita/tests/coercions_propagation.ma
+++ b/matita/tests/coercions_propagation.ma
@@ -16,7 +16,6 @@ set "baseuri" "cic:/matita/test/coercions_propagation/".
 
 include "logic/connectives.ma".
 include "nat/orders.ma".
-alias num (instance 0) = "natural number".
 
 inductive sigma (A:Type) (P:A → Prop) : Type ≝
  sigma_intro: ∀a:A. P a → sigma A P.
@@ -33,7 +32,6 @@ definition eject ≝ λP.λc: ∃n:nat.P n. match c with [ sigma_intro w _ ⇒ w
 coercion cic:/matita/test/coercions_propagation/eject.con.
 
 alias num (instance 0) = "natural number".
-
 theorem test: ∃n. 0 ≤ n.
  apply (S O : ∃n. 0 ≤ n).
  autobatch.
@@ -50,11 +48,10 @@ 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.
 
-(* guarded troppo debole 
 theorem test5: nat → ∃n. 1 ≤ n.
 apply (
  let rec aux n : nat ≝
@@ -65,95 +62,77 @@ apply (
  in
   aux
 : nat → ∃n. 1 ≤ n);
-cases name_con;
-simplify;
- [ autobatch
- | cases (aux n);
-   simplify;
-   apply lt_O_S
- ]
+[ cases (aux n1); simplify; ] autobatch;
 qed.
-*)
 
-(*
-theorem test5: nat → ∃n. 1 ≤ n.
+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 (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))) ? *)
+ 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
- : nat → ∃n. 1 ≤ n);
- [ autobatch
- | elim (aux m);
-   simplify;
-   autobatch
- ]
+: ∀n:nat. NN A n → ∃m:NN A (S n). PN ? ? m);
+[2: cases (aux x m); simplify; autobatch ] unfold PN; 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
+theorem f : nat -> nat -> ∃n:nat.O <= n.
+apply (λx,y:nat.y : nat -> nat -> ∃n:nat. O <= n).
+apply le_O_n; 
+qed.
 
-nat => nat
-nat => X
+axiom F : nat -> nat -> nat.
 
-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) ?
+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 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));
- ].
+  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.
 
-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)).
-
-   
+(* guarded troppo debole *)
+theorem test522: nat → ∃n. 1 ≤ n.
 apply (
  let rec aux n : nat ≝
   match n with
-   [ O ⇒ O
+   [ O ⇒ 1
    | S m ⇒ S (aux m)
    ]
  in
   aux
-: (∃n. 2 ≤ n) → ∃n. 1 ≤ n);
-
+: nat → ∃n. 1 ≤ n);
+[ cases (aux n1); simplify;
+  autobatch
+| autobatch].
 qed.
 
-(*
-theorem test5: nat → ∃n. 0 ≤ n.
- apply (λn:nat.?);
- apply
-  (match n return λ_.∃n.0 ≤ n with [O ⇒ (0 : ∃n.0 ≤ n) | S n' ⇒ ex_intro ? ? n' ?]
-  : ∃n. 0 ≤ n);
- autobatch.
-qed.
-*)
-*)
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