--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+
+include "coq.ma".
+
+alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
+alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
+alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
+alias id "le" = "cic:/matita/tests/fguidi/le.ind#xpointer(1/1)".
+alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con".
+alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
+alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
+alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
+alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
+
+alias symbol "and" (instance 0) = "Coq's logical and".
+alias symbol "eq" (instance 0) = "Coq's leibnitz's equality".
+alias symbol "exists" (instance 0) = "Coq's exists".
+
+definition is_S: nat \to Prop \def
+ \lambda n. match n with
+ [ O \Rightarrow False
+ | (S n) \Rightarrow True
+ ].
+
+definition pred: nat \to nat \def
+ \lambda n. match n with
+ [ O \Rightarrow O
+ | (S n) \Rightarrow n
+ ].
+
+theorem eq_gen_S_O: \forall x. (S x = O) \to \forall P:Prop. P.
+intros. apply False_ind. cut (is_S O). elim Hcut. rewrite < H. apply I.
+qed.
+
+theorem eq_gen_S_O_cc: (\forall P:Prop. P) \to \forall x. (S x = O).
+intros. apply H.
+qed.
+
+theorem eq_gen_S_S: \forall m,n. (S m) = (S n) \to m = n.
+intros. cut ((pred (S m)) = (pred (S n))).
+assumption. elim H. autobatch.
+qed.
+
+theorem eq_gen_S_S_cc: \forall m,n. m = n \to (S m) = (S n).
+intros. elim H. autobatch.
+qed.
+
+inductive le: nat \to nat \to Prop \def
+ le_zero: \forall n. (le O n)
+ | le_succ: \forall m, n. (le m n) \to (le (S m) (S n)).
+
+theorem le_refl: \forall x. (le x x).
+intros. elim x; autobatch.
+qed.
+
+theorem le_gen_x_O_aux: \forall x, y. (le x y) \to (y =O) \to
+ (x = O).
+intros 3. elim H. autobatch. apply eq_gen_S_O. exact n1. autobatch.
+qed.
+
+theorem le_gen_x_O: \forall x. (le x O) \to (x = O).
+intros. apply le_gen_x_O_aux. exact O. autobatch. autobatch.
+qed.
+
+theorem le_gen_x_O_cc: \forall x. (x = O) \to (le x O).
+intros. elim H. autobatch.
+qed.
+
+theorem le_gen_S_x_aux: \forall m,x,y. (le y x) \to (y = S m) \to
+ (\exists n. x = (S n) \land (le m n)).
+intros 4. elim H; clear H x y.
+apply eq_gen_S_O. exact m. elim H1. autobatch.
+clear H2. cut (n = m).
+elim Hcut. apply ex_intro. exact n1. split; autobatch.
+apply eq_gen_S_S. autobatch.
+qed.
+
+theorem le_gen_S_x: \forall m,x. (le (S m) x) \to
+ (\exists n. x = (S n) \land (le m n)).
+intros. apply le_gen_S_x_aux. exact (S m). autobatch. autobatch.
+qed.
+
+theorem le_gen_S_x_cc: \forall m,x. (\exists n. x = (S n) \land (le m n)) \to
+ (le (S m) x).
+intros. elim H. elim H1. cut ((S x1) = x). elim Hcut. autobatch.
+elim H2. autobatch.
+qed.
+
+theorem le_gen_S_S: \forall m,n. (le (S m) (S n)) \to (le m n).
+intros.
+lapply le_gen_S_x to H as H0. elim H0. elim H1.
+lapply eq_gen_S_S to H2 as H4. rewrite > H4. assumption.
+qed.
+
+theorem le_gen_S_S_cc: \forall m,n. (le m n) \to (le (S m) (S n)).
+intros. autobatch.
+qed.
+
+(*
+theorem le_trans: \forall x,y. (le x y) \to \forall z. (le y z) \to (le x z).
+intros 1. elim x; clear H. clear x.
+autobatch.
+fwd H1 [H]. decompose.
+*)
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