rc-1

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

diff --git a/matita/tests/coercions_propagation.ma b/matita/tests/coercions_propagation.ma
index c8529631b7a57515a4a2c59714ed634e30050982..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.
@@ -50,16 +48,91 @@ 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. 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.
+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 test522: 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
+| autobatch].
+qed.
+