New version of the library.

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index 6ba305d98c05d2b8b50468f47dee83f5b8b41023..f59d7b369f8607454b24ebc4c3f74c617f2e65ed 100644 (file)
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -14,8 +14,9 @@
 
 set "baseuri" "cic:/matita/Z/".
 
-include "nat/nat.ma".
-include "datatypes/bool.ma".
+include "nat/compare.ma".
+include "nat/minus.ma".
+include "higher_order_defs/functions.ma".
 
 inductive Z : Set \def
   OZ : Z
@@ -34,23 +35,26 @@ definition neg_Z_of_nat \def
 [ O \Rightarrow  OZ 
 | (S n)\Rightarrow  neg n].
 
-definition absZ \def
+definition abs \def
 \lambda z.
  match z with 
 [ OZ \Rightarrow O
 | (pos n) \Rightarrow n
 | (neg n) \Rightarrow n].
 
-definition OZ_testb \def
+definition OZ_test \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.reflexivity.
+theorem OZ_test_to_Prop :\forall z:Z.
+match OZ_test z with
+[true \Rightarrow eq Z z OZ 
+|false \Rightarrow Not (eq Z z OZ)].
+intros.elim z.
+simplify.reflexivity.
 simplify.intros.
 cut match neg e1 with 
 [ OZ \Rightarrow True 
@@ -83,21 +87,22 @@ definition Zpred \def
          | (S p) \Rightarrow pos p]
 | (neg n) \Rightarrow neg (S n)].
 
-theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+theorem Zpred_Zsucc: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
 intros.elim z.reflexivity.
 elim e1.reflexivity.
 reflexivity.
 reflexivity.
 qed.
 
-theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+theorem Zsucc_Zpred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
 intros.elim z.reflexivity.
 reflexivity.
 elim e2.reflexivity.
 reflexivity.
 qed.
 
-let rec Zplus x y : Z \def
+definition Zplus :Z \to Z \to Z \def
+\lambda x,y.
   match x with
     [ OZ \Rightarrow y
     | (pos m) \Rightarrow
@@ -119,32 +124,34 @@ let rec Zplus x y : Z \def
                 | 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.
+theorem Zplus_z_OZ:  \forall z:Z. eq Z (Zplus z OZ) z.
 intro.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
+(* theorem symmetric_Zplus: symmetric Z Zplus. *)
+
 theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
-intros.elim x.simplify.rewrite > Zplus_z_O.reflexivity.
+intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
 elim y.simplify.reflexivity.
 simplify.
 rewrite < sym_plus.reflexivity.
 simplify.
-rewrite > nat_compare_invert.
+rewrite > nat_compare_n_m_m_n.
 simplify.elim nat_compare ? ?.simplify.reflexivity.
 simplify. reflexivity.
 simplify. reflexivity.
 elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_invert.
+simplify.rewrite > nat_compare_n_m_m_n.
 simplify.elim nat_compare ? ?.simplify.reflexivity.
 simplify. reflexivity.
 simplify. reflexivity.
-simplify.elim (sym_plus ? ?).reflexivity.
+simplify.rewrite < sym_plus.reflexivity.
 qed.
 
-theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+theorem Zpred_Zplus_neg_O : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
 intros.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
@@ -152,7 +159,7 @@ elim e2.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+theorem Zsucc_Zplus_pos_O : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
 intros.elim z.
 simplify.reflexivity.
 elim e1.simplify.reflexivity.
@@ -160,7 +167,7 @@ simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_pp :
+theorem Zplus_pos_pos:
 \forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
 intros.
 elim n.elim m.
@@ -173,12 +180,12 @@ simplify.
 rewrite < plus_n_Sm.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_pn :
+theorem Zplus_pos_neg:
 \forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
 intros.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_np :
+theorem Zplus_neg_pos :
 \forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
 intros.
 elim n.elim m.
@@ -189,7 +196,7 @@ simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_nn:
+theorem Zplus_neg_neg:
 \forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
 intros.
 elim n.elim m.
@@ -200,32 +207,34 @@ simplify.rewrite < plus_n_Sm.reflexivity.
 simplify.rewrite > plus_n_Sm.reflexivity.
 qed.
 
-theorem Zplus_succ_pred:
+theorem Zplus_Zsucc_Zpred:
 \forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
 intros.
 elim x. elim y.
 simplify.reflexivity.
 simplify.reflexivity.
-rewrite < Zsucc_pos.rewrite > Zsucc_pred.reflexivity.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zsucc_Zpred.reflexivity.
 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_neg.rewrite > Zpred_succ.
+rewrite < Zpred_Zplus_neg_O.
+rewrite > Zpred_Zsucc.
 simplify.reflexivity.
-rewrite < Zplus_succ_pred_nn.reflexivity.
-apply Zplus_succ_pred_np.
+rewrite < Zplus_neg_neg.reflexivity.
+apply Zplus_neg_pos.
 elim y.simplify.reflexivity.
-apply Zplus_succ_pred_pn.
-apply Zplus_succ_pred_pp.
+apply Zplus_pos_neg.
+apply Zplus_pos_pos.
 qed.
 
-theorem Zsucc_plus_pp : 
+theorem Zplus_Zsucc_pos_pos : 
 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
 intros.reflexivity.
 qed.
 
-theorem Zsucc_plus_pn : 
+theorem Zplus_Zsucc_pos_neg: 
 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
 intros.elim n1.
 simplify. reflexivity.
@@ -235,14 +244,14 @@ intros. elim n1.
 simplify. reflexivity.
 simplify.reflexivity.
 intros.
-rewrite < (Zplus_succ_pred_pn ? m1).
+rewrite < (Zplus_pos_neg ? m1).
 elim H.reflexivity.
 qed.
 
-theorem Zsucc_plus_nn : 
+theorem Zplus_Zsucc_neg_neg : 
 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
 intros.elim n1.
 simplify. reflexivity.
@@ -252,14 +261,14 @@ intros. elim n1.
 simplify. reflexivity.
 simplify.reflexivity.
 intros.
-rewrite < (Zplus_succ_pred_nn ? m1).
+rewrite < (Zplus_neg_neg ? m1).
 reflexivity.
 qed.
 
-theorem Zsucc_plus_np : 
+theorem Zplus_Zsucc_neg_pos: 
 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
 intros.elim n1.
 simplify. reflexivity.
@@ -270,53 +279,76 @@ simplify. reflexivity.
 simplify.reflexivity.
 intros.
 rewrite < H.
-rewrite < (Zplus_succ_pred_np ? (S m1)).
+rewrite < (Zplus_neg_pos ? (S m1)).
 reflexivity.
 qed.
 
-
-theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+theorem Zplus_Zsucc : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
 intros.elim x.elim y.
 simplify. reflexivity.
-rewrite < Zsucc_pos.reflexivity.
+rewrite < Zsucc_Zplus_pos_O.reflexivity.
 simplify.reflexivity.
 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zsucc_plus_nn.
-apply Zsucc_plus_np.
+apply Zplus_Zsucc_neg_neg.
+apply Zplus_Zsucc_neg_pos.
 elim y.
 rewrite < sym_Zplus OZ.reflexivity.
-apply Zsucc_plus_pn.
-apply Zsucc_plus_pp.
+apply Zplus_Zsucc_pos_neg.
+apply Zplus_Zsucc_pos_pos.
 qed.
 
-theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+theorem Zplus_Zpred: \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)).
 rewrite > Hcut.
-rewrite > Zsucc_plus.
-rewrite > Zpred_succ.
+rewrite > Zplus_Zsucc.
+rewrite > Zpred_Zsucc.
+reflexivity.
+rewrite > Zsucc_Zpred.
+reflexivity.
+qed.
+
+
+theorem associative_Zplus: associative Z Zplus.
+(* change with (\forall x,y,z. eq ?(Zplus (Zplus x y) z) (Zplus x (Zplus y z))).*)
+simplify.
+intros.elim x.simplify.reflexivity.
+elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
+drop.
+rewrite < (Zpred_Zplus_neg_O y).
+rewrite < Zplus_Zpred.
 reflexivity.
-rewrite > Zsucc_pred.
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (Zplus (neg e) y).
+apply eq_f.assumption.
+elim e2.rewrite < Zsucc_Zplus_pos_O.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zplus_Zsucc.
 reflexivity.
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (Zplus (pos e1) y).
+apply eq_f.assumption.
 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.reflexivity.
-elim e1.rewrite < (Zpred_neg (Zplus y z)).
-rewrite < (Zpred_neg y).
-rewrite < Zpred_plus.
+elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
+rewrite < (Zpred_Zplus_neg_O y).
+rewrite < Zplus_Zpred.
 reflexivity.
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (Zplus (neg e) y).
-apply f_equal.assumption.
-elim e2.rewrite < Zsucc_pos.
-rewrite < Zsucc_pos.
-rewrite > Zsucc_plus.
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (Zplus (neg e) y).
+apply eq_f.assumption.
+elim e2.rewrite < Zsucc_Zplus_pos_O.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zplus_Zsucc.
 reflexivity.
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (Zplus (pos e1) y).
-apply f_equal.assumption.
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (Zplus (pos e1) y).
+apply eq_f.assumption.
 qed.

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index d148dfd310b2b8be46fdb8a5e4b854e71d6ac34f..fec46ae2fe1f65f8106020bc4a038553beb1667b 100644 (file)
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -16,6 +16,7 @@ set "baseuri" "cic:/matita/nat/compare".
 
 include "nat/orders.ma".
 include "datatypes/bool.ma".
+include "datatypes/compare.ma".
 
 let rec leb n m \def 
 match n with 
@@ -41,7 +42,7 @@ simplify.apply le_S_S.apply H.
 simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
-theorem le_elim: \forall n,m:nat. \forall P:bool \to Prop. 
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
 ((le n m) \to (P true)) \to ((Not (le n m)) \to (P false)) \to
 P (leb n m).
 intros.
@@ -55,5 +56,70 @@ apply (H H2).
 apply (H1 H2).
 qed.
 
+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]].
 
+theorem nat_compare_n_n: \forall n:nat.(eq compare (nat_compare n n) EQ).
+intro.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem nat_compare_S_S: \forall n,m:nat. 
+eq compare (nat_compare n m) (nat_compare (S n) (S m)).
+intros.simplify.reflexivity.
+qed.
 
+theorem nat_compare_to_Prop: \forall n,m:nat. 
+match (nat_compare n m) with
+  [ LT \Rightarrow (lt n m)
+  | EQ \Rightarrow (eq nat n m)
+  | GT \Rightarrow (lt m n) ].
+intros.
+apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
+  [ LT \Rightarrow (lt n m)
+  | EQ \Rightarrow (eq nat n m)
+  | GT \Rightarrow (lt m n) ]).
+intro.elim n1.simplify.reflexivity.
+simplify.apply le_S_S.apply le_O_n.
+intro.simplify.apply le_S_S. apply le_O_n.
+intros 2.simplify.elim (nat_compare n1 m1).
+simplify. apply le_S_S.apply H.
+simplify. apply le_S_S.apply H.
+simplify. apply eq_f. apply H.
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat. 
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros. 
+apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intro.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intros.simplify.elim H.reflexivity.
+qed.
+     
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to 
+(P (nat_compare n m)).
+intros.
+cut match (nat_compare n m) with
+[ LT \Rightarrow (lt n m)
+| EQ \Rightarrow (eq nat n m)
+| GT \Rightarrow (lt m n)] \to
+(P (nat_compare n m)).
+apply Hcut.apply nat_compare_to_Prop.
+elim (nat_compare n m).
+apply (H H3).
+apply (H2 H3).
+apply (H1 H3).
+qed.

diff --git a/helm/matita/library/nat/minus.ma b/helm/matita/library/nat/minus.ma
index 0091cb9b09b93ace2834862b1b84d64d7bdd219d..b6e7fc5e2c5cb098bddcf1354e5483f4cb99b598 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -12,10 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
+
 set "baseuri" "cic:/matita/nat/minus".
 
 include "nat/orders_op.ma".
-include "nat/times.ma".
+include "nat/compare.ma".
 
 let rec minus n m \def 
  match n with 
@@ -96,7 +97,7 @@ symmetry.
 apply plus_minus_m_m.assumption.
 qed.
 
-theorem minus_ge_O: \forall n,m:nat.
+theorem eq_minus_n_m_O: \forall n,m:nat.
 le n m \to eq nat (minus n m) O.
 intros 2.
 apply nat_elim2 (\lambda n,m.le n m \to eq nat (minus n m) O).
@@ -124,36 +125,36 @@ apply plus_le.assumption.
 apply le_n_Sm_elim ? ? H1.
 intros.
 *)
-check distributive.
 
-theorem times_minus_distr: \forall n,m,p:nat.
-eq nat (times n (minus m p)) (minus (times n m) (times n p)).
+theorem distributive_times_minus: distributive nat times minus.
+simplify.
 intros.
-apply (leb_ind p m).intro.
-cut eq nat (plus (times n (minus m p)) (times n p)) (plus (minus (times n m) (times n p)) (times n p)).
-apply plus_injective_right ? ? (times n p).
+apply (leb_elim z y).intro.
+cut eq nat (plus (times x (minus y z)) (times x z)) 
+           (plus (minus (times x y) (times x z)) (times x z)).
+apply inj_plus_l (times x z).
 assumption.
-apply trans_eq nat ? (times n m).
-elim (times_plus_distr ? ? ?).
-elim (minus_plus ? ? H).apply refl_equal.
-elim (minus_plus ? ? ?).apply refl_equal.
-apply times_le_monotony_left.
+apply trans_eq nat ? (times x y).
+rewrite < times_plus_distr. 
+rewrite < plus_minus_m_m ? ? H.reflexivity.
+rewrite < plus_minus_m_m ? ? ?.reflexivity.
+apply le_times_r.
 assumption.
 intro.
-elim sym_eq ? ? ? (minus_ge_O ? ? ?).
-elim sym_eq ? ? ? (minus_ge_O ? ? ?).
-elim (sym_times ? ?).simplify.apply refl_equal.
-simplify.
-apply times_le_monotony_left.
-cut (lt m p) \to (le m p).
-apply Hcut.simplify.apply not_le_lt ? ? H.
-intro.apply lt_le.apply H1.
-cut (lt m p) \to (le m p).
-apply Hcut.simplify.apply not_le_lt ? ? H.
-intro.apply lt_le.apply H1.
+rewrite > eq_minus_n_m_O.
+rewrite > eq_minus_n_m_O (times x y).
+rewrite < sym_times.simplify.reflexivity.
+apply lt_to_le.
+apply not_le_to_lt.assumption.
+apply le_times_r.apply lt_to_le.
+apply not_le_to_lt.assumption.
 qed.
 
-theorem minus_le: \forall n,m:nat. le (minus n m) n.
+theorem distr_times_minus: \forall n,m,p:nat.
+eq nat (times n (minus m p)) (minus (times n m) (times n p))
+\def distributive_times_minus.
+
+theorem le_minus_m: \forall n,m:nat. le (minus n m) n.
 intro.elim n.simplify.apply le_n.
 elim m.simplify.apply le_n.
 simplify.apply le_S.apply H.