+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
set "baseuri" "cic:/matita/tests/fguidi/".
+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 "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) = "logical and".
-alias symbol "eq" (instance 0) = "leibnitz's equality".
-alias symbol "exists" (instance 0) = "exists".
+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
].
theorem eq_gen_S_O: \forall x. (S x = O) \to \forall P:Prop. P.
-intros. apply False_ind. cut (is_S O). auto. elim H. exact I.
+intros. apply False_ind. cut (is_S O). auto paramodulation. elim H. exact I.
qed.
theorem eq_gen_S_O_cc: (\forall P:Prop. P) \to \forall x. (S x = O).
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. auto.
+intros. cut ((pred (S m)) = (pred (S n))).
+assumption. elim H. auto paramodulation.
qed.
theorem eq_gen_S_S_cc: \forall m,n. m = n \to (S m) = (S n).
-intros. elim H. auto.
+intros. elim H. auto paramodulation.
qed.
inductive le: nat \to nat \to Prop \def
| 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. auto. auto.
+intros. elim x. auto paramodulation. auto paramodulation.
qed.
theorem le_gen_x_O_aux: \forall x, y. (le x y) \to (y =O) \to
(x = O).
-intros 3. elim H. auto. apply eq_gen_S_O. exact e3. auto.
+intros 3. elim H. auto paramodulation. apply eq_gen_S_O. exact n1. auto paramodulation.
qed.
theorem le_gen_x_O: \forall x. (le x O) \to (x = O).
-intros. apply le_gen_x_O_aux. exact O. auto. auto.
+intros. apply le_gen_x_O_aux. exact O. auto paramodulation. auto paramodulation.
qed.
theorem le_gen_x_O_cc: \forall x. (x = O) \to (le x O).
-intros. elim H. auto.
+intros. elim H. auto paramodulation.
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.
-apply eq_gen_S_O. exact m. elim H1. auto.
-cut e4 = m. elim Hcut. apply ex_intro. exact e3. auto. auto.
+apply eq_gen_S_O. exact m. elim H1. auto paramodulation.
+cut (n = m). elim Hcut. apply ex_intro. exact n1. auto paramodulation. auto. (* paramodulation non trova la prova *)
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). auto. auto.
+intros. apply le_gen_S_x_aux. exact (S m). auto paramodulation. auto paramodulation.
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. auto. elim H2. auto.
+intros. elim H. elim H1. cut ((S x1) = x). elim Hcut. auto paramodulation. elim H2. auto paramodulation.
qed.
theorem le_gen_S_S: \forall m,n. (le (S m) (S n)) \to (le m n).
qed.
theorem le_gen_S_S_cc: \forall m,n. (le m n) \to (le (S m) (S n)).
-intros. auto.
+intros. auto paramodulation.
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 *)
-auto.
+intros 1. elim x; clear H. clear x.
+auto paramodulation.
fwd H1 [H]. decompose H.
*)