Added Z, Zplus and its properties.

diff --git a/helm/matita/tests/match.ma b/helm/matita/tests/match.ma
index bc8caa22332b5c4a18bb65e7a941def31f2e47c8..036cc393b45214e60a96442f8bd1a4477b01017d 100644 (file)
--- a/helm/matita/tests/match.ma
+++ b/helm/matita/tests/match.ma
@@ -295,15 +295,338 @@ simplify.apply le_n_S.apply H.
 simplify.intros.apply H.apply le_S_n.assumption.
 qed.
 
-(*CSC: this requires too much time
-theorem prova :
-let three \def (S (S (S O))) in
-let nine \def (times three three) in
-let eightyone \def (times nine nine) in 
-let square \def (times eightyone eightyone) in
-(eq nat square O).
-intro.
-intro.
-intro.intro.
-normalize goal at (? ? % ?).
-*)
\ No newline at end of file
+inductive Z : Set \def
+  OZ : Z
+| pos : nat \to Z
+| neg : nat \to Z.
+
+definition Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow  OZ 
+| (S n)\Rightarrow  pos n].
+
+coercion Z_of_nat.
+
+definition neg_Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow  OZ 
+| (S n)\Rightarrow  neg n].
+
+definition absZ \def
+\lambda z.
+ match z with 
+[ OZ \Rightarrow O
+| (pos n) \Rightarrow n
+| (neg n) \Rightarrow n].
+
+definition OZ_testb \def
+\lambda z.
+match z with 
+[ OZ \Rightarrow true
+| (pos n) \Rightarrow false
+| (neg n) \Rightarrow false].
+
+theorem OZ_discr :
+\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)). 
+intros.elim z.simplify.apply refl_equal.
+simplify.intros.
+cut match neg e with 
+[ OZ \Rightarrow True 
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+simplify.intros.
+cut match pos e with 
+[ OZ \Rightarrow True 
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+qed.
+
+definition Zsucc \def
+\lambda z. match z with
+[ OZ \Rightarrow pos O
+| (pos n) \Rightarrow pos (S n)
+| (neg n) \Rightarrow 
+         match n with
+         [ O \Rightarrow OZ
+         | (S p) \Rightarrow neg p]].
+
+definition Zpred \def
+\lambda z. match z with
+[ OZ \Rightarrow neg O
+| (pos n) \Rightarrow 
+         match n with
+         [ O \Rightarrow OZ
+         | (S p) \Rightarrow pos p]
+| (neg n) \Rightarrow neg (S n)].
+
+theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+intros.elim z.apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.          
+apply refl_equal.
+qed.
+
+theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+intros.elim z.apply refl_equal.
+apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+qed.
+
+inductive compare :Set \def
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
+
+definition compare_invert: compare \to compare \def
+  \lambda c.
+    match c with
+      [ LT \Rightarrow GT
+      | EQ \Rightarrow EQ
+      | GT \Rightarrow LT ].
+
+theorem nat_compare_invert: \forall n,m:nat. 
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros. 
+apply nat_double_ind (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intro.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intros.simplify.elim H.apply refl_equal.
+qed.
+
+let rec Zplus x y : Z \def
+  match x with
+    [ OZ \Rightarrow y
+    | (pos m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow x
+         | (pos n) \Rightarrow (pos (S (plus m n)))
+         | (neg n) \Rightarrow 
+              match nat_compare m n with
+                [ LT \Rightarrow (neg (pred (minus n m)))
+                | EQ \Rightarrow OZ
+                | GT \Rightarrow (pos (pred (minus m n)))]]
+    | (neg m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow x
+         | (pos n) \Rightarrow 
+              match nat_compare m n with
+                [ LT \Rightarrow (pos (pred (minus n m)))
+                | EQ \Rightarrow OZ
+                | GT \Rightarrow (neg (pred (minus m n)))]     
+         | (neg n) \Rightarrow (neg (S (plus m n)))]].
+         
+theorem Zplus_z_O:  \forall z:Z. eq Z (Zplus z OZ) z.
+intro.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
+intros.elim x.simplify.elim (sym_eq ? ? ? (Zplus_z_O y)).apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_plus e e1).apply refl_equal.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+simplify.elim (sym_plus e1 e).apply refl_equal.
+qed.
+
+theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+intros.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+intros.elim z.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pp :
+\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_O ?).apply refl_equal.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pn :
+\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_np :
+\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_nn:
+\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+simplify.elim (sym_eq ? ? ? (plus_n_Sm ? ?)).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred:
+\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
+intros.
+elim x. elim y.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim (Zsucc_pos ?).elim (sym_eq ? ? ? (Zsucc_pred ?)).apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus (Zpred OZ) ?.
+elim (Zpred_neg ?).elim (sym_eq ? ? ? (Zpred_succ ?)).
+simplify.apply refl_equal.
+apply Zplus_succ_pred_nn.
+apply Zplus_succ_pred_np.
+elim y.simplify.apply refl_equal.
+apply Zplus_succ_pred_pn.
+apply Zplus_succ_pred_pp.
+qed.
+
+theorem Zsucc_plus_pp : 
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zsucc_plus_pn : 
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_pn ? m1).
+elim H.apply refl_equal.
+qed.
+
+theorem Zsucc_plus_nn : 
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_nn ? m1).
+apply refl_equal.
+qed.
+
+theorem Zsucc_plus_np : 
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim H.
+elim (Zplus_succ_pred_np ? (S m1)).
+apply refl_equal.
+qed.
+
+
+theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+intros.elim x.elim y.
+simplify. apply refl_equal.
+elim (Zsucc_pos ?).apply refl_equal.
+simplify.apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus OZ ?.simplify.apply refl_equal.
+apply Zsucc_plus_nn.
+apply Zsucc_plus_np.
+elim y.
+elim (sym_Zplus OZ ?).apply refl_equal.
+apply Zsucc_plus_pn.
+apply Zsucc_plus_pp.
+qed.
+
+theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+intros.
+cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
+elim (sym_eq ? ? ? Hcut).
+elim (sym_eq ? ? ? (Zsucc_plus ? ?)).
+elim (sym_eq ? ? ? (Zpred_succ ?)).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zsucc_pred ?)).
+apply refl_equal.
+qed.
+
+theorem assoc_Zplus : 
+\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
+intros.elim x.simplify.apply refl_equal.
+elim e.elim (Zpred_neg (Zplus y z)).
+elim (Zpred_neg y).
+elim (Zpred_plus ? ?).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (Zplus (neg e1) y) ?)).
+apply f_equal.assumption.
+elim e.elim (Zsucc_pos ?).
+elim (Zsucc_pos ?).
+apply (sym_eq ? ? ? (Zsucc_plus ? ?)) .
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (Zplus (pos e1) y) ?)).
+apply f_equal.assumption.
+qed.
+
+
+