Notation for equality used everywhere.

[helm.git] / helm / matita / library / nat / minus.ma

diff --git a/helm/matita/library/nat/minus.ma b/helm/matita/library/nat/minus.ma
index b6e7fc5e2c5cb098bddcf1354e5483f4cb99b598..b4cff5d584ca3c16bf85ce285b3cf6d51f0e1307 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -27,28 +27,28 @@ let rec minus n m \def
         | (S q) \Rightarrow minus p q ]].
 
 
-theorem minus_n_O: \forall n:nat.eq nat n (minus n O).
+theorem minus_n_O: \forall n:nat.n=minus n O.
 intros.elim n.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem minus_n_n: \forall n:nat.eq nat O (minus n n).
+theorem minus_n_n: \forall n:nat.O=minus n n.
 intros.elim n.simplify.
 reflexivity.
 simplify.apply H.
 qed.
 
-theorem minus_Sn_n: \forall n:nat.eq nat (S O) (minus (S n) n).
+theorem minus_Sn_n: \forall n:nat. S O = minus (S n) n.
 intro.elim n.
 simplify.reflexivity.
 elim H.reflexivity.
 qed.
 
 theorem minus_Sn_m: \forall n,m:nat. 
-le m n \to eq nat (minus (S n) m) (S (minus n m)).
+le m n \to minus (S n) m = S (minus n m).
 intros 2.
 apply nat_elim2
-(\lambda n,m.le m n \to eq nat (minus (S n) m) (S (minus n m))).
+(\lambda n,m.le m n \to minus (S n) m = S (minus n m)).
 intros.apply le_n_O_elim n1 H.
 simplify.reflexivity.
 intros.simplify.reflexivity.
@@ -57,10 +57,10 @@ apply le_S_S_to_le. assumption.
 qed.
 
 theorem plus_minus:
-\forall n,m,p:nat. le m n \to eq nat (plus (minus n m) p) (minus (plus n p) m).
+\forall n,m,p:nat. le m n \to plus (minus n m) p = minus (plus n p) m.
 intros 2.
 apply nat_elim2
-(\lambda n,m.\forall p:nat.le m n \to eq nat (plus (minus n m) p) (minus (plus n p) m)).
+(\lambda n,m.\forall p:nat.le m n \to plus (minus n m) p = minus (plus n p) m).
 intros.apply le_n_O_elim ? H.
 simplify.rewrite < minus_n_O.reflexivity.
 intros.simplify.reflexivity.
@@ -68,9 +68,9 @@ intros.simplify.apply H.apply le_S_S_to_le.assumption.
 qed.
 
 theorem plus_minus_m_m: \forall n,m:nat.
-le m n \to eq nat n (plus (minus n m) m).
+le m n \to n = plus (minus n m) m.
 intros 2.
-apply nat_elim2 (\lambda n,m.le m n \to eq nat n (plus (minus n m) m)).
+apply nat_elim2 (\lambda n,m.le m n \to n = plus (minus n m) m).
 intros.apply le_n_O_elim n1 H.
 reflexivity.
 intros.simplify.rewrite < plus_n_O.reflexivity.
@@ -79,8 +79,8 @@ apply eq_f.rewrite < sym_plus.apply H.
 apply le_S_S_to_le.assumption.
 qed.
 
-theorem minus_to_plus :\forall n,m,p:nat.le m n \to eq nat (minus n m) p \to 
-eq nat n (plus m p).
+theorem minus_to_plus :\forall n,m,p:nat.le m n \to minus n m = p \to 
+n = plus m p.
 intros.apply trans_eq ? ? (plus (minus n m) m) ?.
 apply plus_minus_m_m.
 apply H.elim H1.
@@ -88,7 +88,7 @@ apply sym_plus.
 qed.
 
 theorem plus_to_minus :\forall n,m,p:nat.le m n \to
-eq nat n (plus m p) \to eq nat (minus n m) p.
+n = plus m p \to minus n m = p.
 intros.
 apply inj_plus_r m.
 rewrite < H1.
@@ -98,9 +98,9 @@ apply plus_minus_m_m.assumption.
 qed.
 
 theorem eq_minus_n_m_O: \forall n,m:nat.
-le n m \to eq nat (minus n m) O.
+le n m \to minus n m = O.
 intros 2.
-apply nat_elim2 (\lambda n,m.le n m \to eq nat (minus n m) O).
+apply nat_elim2 (\lambda n,m.le n m \to minus n m = O).
 intros.simplify.reflexivity.
 intros.apply False_ind.
 (* ancora problemi con il not *)
@@ -130,8 +130,8 @@ theorem distributive_times_minus: distributive nat times minus.
 simplify.
 intros.
 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)).
+cut 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 x y).
@@ -151,7 +151,7 @@ apply not_le_to_lt.assumption.
 qed.
 
 theorem distr_times_minus: \forall n,m,p:nat.
-eq nat (times n (minus m p)) (minus (times n m) (times n p))
+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.