packaging cleanup: get rid of ancient debhelpers, use dh_install

[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..1b790031401c2aeb8dc04a3ad769fd9f43fa86c6 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -15,7 +15,7 @@
 
 set "baseuri" "cic:/matita/nat/minus".
 
-include "nat/orders_op.ma".
+include "nat/le_arith.ma".
 include "nat/compare.ma".
 
 let rec minus n m \def 
@@ -26,29 +26,30 @@ let rec minus n m \def
        [O \Rightarrow (S p)
         | (S q) \Rightarrow minus p q ]].
 
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural minus" 'minus x y = (cic:/matita/nat/minus/minus.con x y).
 
-theorem minus_n_O: \forall n:nat.eq nat n (minus n O).
+theorem minus_n_O: \forall n:nat.n=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=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 = (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)).
+theorem minus_Sn_m: \forall n,m:nat. m \leq n \to (S n)-m = S (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.m \leq n \to (S n)-m = S (n-m)).
 intros.apply le_n_O_elim n1 H.
 simplify.reflexivity.
 intros.simplify.reflexivity.
@@ -57,10 +58,10 @@ 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. m \leq n \to (n-m)+p = (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.m \leq n \to (n-m)+p = (n+p)-m).
 intros.apply le_n_O_elim ? H.
 simplify.rewrite < minus_n_O.reflexivity.
 intros.simplify.reflexivity.
@@ -68,9 +69,9 @@ 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).
+m \leq n \to n = (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.m \leq n \to n = (n-m)+m).
 intros.apply le_n_O_elim n1 H.
 reflexivity.
 intros.simplify.rewrite < plus_n_O.reflexivity.
@@ -79,16 +80,16 @@ 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).
-intros.apply trans_eq ? ? (plus (minus n m) m) ?.
+theorem minus_to_plus :\forall n,m,p:nat.m \leq n \to n-m = p \to 
+n = m+p.
+intros.apply trans_eq ? ? ((n-m)+m) ?.
 apply plus_minus_m_m.
 apply H.elim H1.
 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.
+theorem plus_to_minus :\forall n,m,p:nat.m \leq n \to
+n = m+p \to n-m = p.
 intros.
 apply inj_plus_r m.
 rewrite < H1.
@@ -97,10 +98,24 @@ symmetry.
 apply plus_minus_m_m.assumption.
 qed.
 
+theorem minus_S_S : \forall n,m:nat.
+eq nat (minus (S n) (S m)) (minus n m).
+intros.
+reflexivity.
+qed.
+
+theorem minus_pred_pred : \forall n,m:nat. lt O n \to lt O m \to 
+eq nat (minus (pred n) (pred m)) (minus n m).
+intros.
+apply lt_O_n_elim n H.intro.
+apply lt_O_n_elim m H1.intro.
+simplify.reflexivity.
+qed.
+
 theorem eq_minus_n_m_O: \forall n,m:nat.
-le n m \to eq nat (minus n m) O.
+n \leq m \to 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.n \leq m \to n-m = O).
 intros.simplify.reflexivity.
 intros.apply False_ind.
 (* ancora problemi con il not *)
@@ -109,7 +124,7 @@ intros.
 simplify.apply H.apply le_S_S_to_le. apply H1.
 qed.
 
-theorem le_SO_minus: \forall n,m:nat.le (S n) m \to le (S O) (minus m n).
+theorem le_SO_minus: \forall n,m:nat.S n \leq m \to S O \leq m-n.
 intros.elim H.elim minus_Sn_n n.apply le_n.
 rewrite > minus_Sn_m.
 apply le_S.assumption.
@@ -117,9 +132,9 @@ apply lt_to_le.assumption.
 qed.
 
 (*
-theorem le_plus_minus: \forall n,m,p. (le (plus n m) p) \to (le n (minus p m)).
+theorem le_plus_minus: \forall n,m,p. n+m \leq p \to n \leq p-m.
 intros 3.
-elim p.simplify.apply trans_le ? (plus n m) ?.
+elim p.simplify.apply trans_le ? (n+m) ?.
 elim sym_plus ? ?.
 apply plus_le.assumption.
 apply le_n_Sm_elim ? ? H1.
@@ -130,11 +145,10 @@ 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)).
-apply inj_plus_l (times x z).
+cut x*(y-z)+x*z = (x*y-x*z)+x*z.
+apply inj_plus_l (x*z).
 assumption.
-apply trans_eq nat ? (times x y).
+apply trans_eq nat ? (x*y).
 rewrite < times_plus_distr. 
 rewrite < plus_minus_m_m ? ? H.reflexivity.
 rewrite < plus_minus_m_m ? ? ?.reflexivity.
@@ -142,7 +156,7 @@ apply le_times_r.
 assumption.
 intro.
 rewrite > eq_minus_n_m_O.
-rewrite > eq_minus_n_m_O (times x y).
+rewrite > eq_minus_n_m_O (x*y).
 rewrite < sym_times.simplify.reflexivity.
 apply lt_to_le.
 apply not_le_to_lt.assumption.
@@ -150,11 +164,10 @@ apply le_times_r.apply lt_to_le.
 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))
+theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
 \def distributive_times_minus.
 
-theorem le_minus_m: \forall n,m:nat. le (minus n m) n.
+theorem le_minus_m: \forall n,m:nat. n-m \leq n.
 intro.elim n.simplify.apply le_n.
 elim m.simplify.apply le_n.
 simplify.apply le_S.apply H.