A little bit more of notation here and there.

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

diff --git a/helm/matita/library/nat/div_and_mod.ma b/helm/matita/library/nat/div_and_mod.ma
index 4c43d33bd9d1d57ba4b283016c850f0cb0bf7a29..ca8b6c3faaaec0cd8c89c5bce373ba0efd9a588a 100644 (file)
--- a/helm/matita/library/nat/div_and_mod.ma
+++ b/helm/matita/library/nat/div_and_mod.ma
@@ -24,7 +24,7 @@ match (leb m n) with
 | false \Rightarrow
   match p with
   [O \Rightarrow m
-  |(S q) \Rightarrow mod_aux q (minus m (S n)) n]].
+  |(S q) \Rightarrow mod_aux q (m-(S n)) n]].
 
 definition mod : nat \to nat \to nat \def
 \lambda n,m.
@@ -38,7 +38,7 @@ match (leb m n) with
 | false \Rightarrow
   match p with
   [O \Rightarrow O
-  |(S q) \Rightarrow S (div_aux q (minus m (S n)) n)]].
+  |(S q) \Rightarrow S (div_aux q (m-(S n)) n)]].
 
 definition div : nat \to nat \to nat \def
 \lambda n,m.
@@ -47,15 +47,15 @@ match m with
 | (S p) \Rightarrow div_aux n n p]. 
 
 theorem le_mod_aux_m_m: 
-\forall p,n,m. (le n p) \to (le (mod_aux p n m) m).
+\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
 intro.elim p.
-apply le_n_O_elim n H (\lambda n.(le (mod_aux O n m) m)).
+apply le_n_O_elim n H (\lambda n.(mod_aux O n m) \leq m).
 simplify.apply le_O_n.
 simplify.
 apply leb_elim n1 m.
 simplify.intro.assumption.
 simplify.intro.apply H.
-cut (le n1 (S n)) \to (le (minus n1 (S m)) n).
+cut n1 \leq (S n) \to n1-(S m) \leq n.
 apply Hcut.assumption.
 elim n1.
 simplify.apply le_O_n.
@@ -63,7 +63,7 @@ simplify.apply trans_le ? n2 n.
 apply le_minus_m.apply le_S_S_to_le.assumption.
 qed.
 
-theorem lt_mod_m_m: \forall n,m. (lt O m) \to (lt (mod n m) m).
+theorem lt_mod_m_m: \forall n,m. O < m \to (mod n m) < m.
 intros 2.elim m.apply False_ind.
 apply not_le_Sn_O O H.
 simplify.apply le_S_S.apply le_mod_aux_m_m.
@@ -71,7 +71,7 @@ apply le_n.
 qed.
 
 theorem div_aux_mod_aux: \forall p,n,m:nat. 
-(n=plus (times (div_aux p n m) (S m)) (mod_aux p n m)).
+(n=(div_aux p n m)*(S m) + (mod_aux p n m)).
 intro.elim p.
 simplify.elim leb n m.
 simplify.apply refl_eq.
@@ -81,38 +81,36 @@ apply leb_elim n1 m.
 simplify.intro.apply refl_eq.
 simplify.intro.
 rewrite > assoc_plus. 
-elim (H (minus n1 (S m)) m).
-change with (n1=plus (S m) (minus n1 (S m))).
+elim (H (n1-(S m)) m).
+change with (n1=(S m)+(n1-(S m))).
 rewrite < sym_plus.
 apply plus_minus_m_m.
-change with lt m n1.
+change with m < n1.
 apply not_le_to_lt.exact H1.
 qed.
 
-theorem div_mod: \forall n,m:nat. 
-(lt O m) \to n=plus (times (div n m) m) (mod n m).
+theorem div_mod: \forall n,m:nat. O < m \to n=(div n m)*m+(mod n m).
 intros 2.elim m.elim (not_le_Sn_O O H).
 simplify.
 apply div_aux_mod_aux.
 qed.
 
 inductive div_mod_spec (n,m,q,r:nat) : Prop \def
-div_mod_spec_intro: 
-(lt r m) \to n=plus (times q m) r \to (div_mod_spec n m q r).
+div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
 
 (* 
 definition div_mod_spec : nat \to nat \to nat \to nat \to Prop \def
-\lambda n,m,q,r:nat.(And (lt r m) n=plus (times q m) r).
+\lambda n,m,q,r:nat.r < m \land n=q*m+r).
 *)
 
-theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to Not (m=O).
+theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to \lnot m=O.
 intros 4.simplify.intros.elim H.absurd le (S r) O.
 rewrite < H1.assumption.
 exact not_le_Sn_O r.
 qed.
 
 theorem div_mod_spec_div_mod: 
-\forall n,m. (lt O m) \to (div_mod_spec n m (div n m) (mod n m)).
+\forall n,m. O < m \to (div_mod_spec n m (div n m) (mod n m)).
 intros.
 apply div_mod_spec_intro.
 apply lt_mod_m_m.assumption.
@@ -125,12 +123,12 @@ theorem div_mod_spec_to_eq :\forall a,b,q,r,q1,r1.
 intros.elim H.elim H1.
 apply nat_compare_elim q q1.intro.
 apply False_ind.
-cut eq nat (plus (times (minus q1 q) b) r1) r.
-cut le b (plus (times (minus q1 q) b) r1).
-cut le b r.
+cut eq nat ((q1-q)*b+r1) r.
+cut b \leq (q1-q)*b+r1.
+cut b \leq r.
 apply lt_to_not_le r b H2 Hcut2.
 elim Hcut.assumption.
-apply trans_le ? (times (minus q1 q) b) ?.
+apply trans_le ? ((q1-q)*b) ?.
 apply le_times_n.
 apply le_SO_minus.exact H6.
 rewrite < sym_plus.
@@ -153,12 +151,12 @@ intros.assumption.
 (* the following case is symmetric *)
 intro.
 apply False_ind.
-cut eq nat (plus (times (minus q q1) b) r) r1.
-cut le b (plus (times (minus q q1) b) r).
-cut le b r1.
+cut eq nat ((q-q1)*b+r) r1.
+cut b \leq (q-q1)*b+r.
+cut b \leq r1.
 apply lt_to_not_le r1 b H4 Hcut2.
 elim Hcut.assumption.
-apply trans_le ? (times (minus q q1) b) ?.
+apply trans_le ? ((q-q1)*b) ?.
 apply le_times_n.
 apply le_SO_minus.exact H6.
 rewrite < sym_plus.