+
+theorem prova :
+ \forall n,m:nat.
+ \forall P:nat \to Prop.
+ \forall H:P (S (S O)).
+ \forall H:P (S (S (S O))).
+ \forall H1: \forall x.P x \to O = x.
+ O = S (S (S (S (S O)))).
+ intros.
+ auto paramodulation.
+ qed.
+
+theorem example2:
+\forall x: nat. (x+S O)*(x-S O) = x*x - S O.
+intro;
+apply (nat_case x);
+ [ auto paramodulation.|intro.auto paramodulation.]
+qed.
+
+theorem prova3:
+ \forall A:Set.
+ \forall m:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to (\exists z.m x z = y)).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);
+ [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
+ |elim (H6 ? ? Hcut).
+ cut (divides two b);
+ [ apply (H10 ? Hcut Hcut1).
+ | elim (H8 b b);[assumption.|assumption|
+ apply (H7 ? ? (m a1 a1));
+ apply (H5 two ? ?);rewrite < H9.
+ rewrite < H11.rewrite < H2.
+ apply eq_f.rewrite > H2.rewrite > H4.reflexivity.
+ ]
+ ]
+ ]
+ qed.
+
+theorem prova31:
+ \forall A:Set.
+ \forall m,f:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to m x (f x y) = y).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);
+ [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
+ |(*elim (H6 ? ? Hcut). *)
+ cut (divides two b);
+ [ apply (H10 ? Hcut Hcut1).
+ | elim (H8 b b);[assumption.|assumption|
+
+ apply (H7 ? ? (m (f two a) (f two a)));
+ apply (H5 two ? ?);
+ rewrite < H9.
+ rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
+ rewrite < H2.apply eq_f.
+ rewrite < H4 in \vdash (? ? ? %).
+ rewrite > H2.reflexivity.
+ ]
+ ]
+ ]
+ qed.
+
+theorem prova32:
+ \forall A:Set.
+ \forall m,f:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to m x (f x y) = y).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x.divides two (m x x) \to divides two x.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);[|apply H8;rewrite > H9.auto].
+ apply H10;
+ [ assumption.
+ | apply (H8 b);
+ apply (H7 ? ? (m (f two a) (f two a)));
+ apply (H5 two ? ?);
+ auto paramodulation.
+ (*
+ rewrite < H9.
+ rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
+ rewrite < H2.apply eq_f.
+ rewrite < H4 in \vdash (? ? ? %).
+ rewrite > H2.reflexivity.
+ *)
+ ]
+qed.
+