- transcript: bugfix

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

diff --git a/helm/software/matita/library/nat/minus.ma b/helm/software/matita/library/nat/minus.ma
index 710418d72644022cb4d799fb95f67354da4af89b..20785c9e88e87c49b8aa386dd59d8947ef3fbe4e 100644 (file)
--- a/helm/software/matita/library/nat/minus.ma
+++ b/helm/software/matita/library/nat/minus.ma
@@ -13,8 +13,6 @@
 (**************************************************************************)
 
 
-set "baseuri" "cic:/matita/nat/minus".
-
 include "nat/le_arith.ma".
 include "nat/compare.ma".
 
@@ -26,8 +24,7 @@ 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).
+interpretation "natural minus" 'minus x y = (minus x y).
 
 theorem minus_n_O: \forall n:nat.n=n-O.
 intros.elim n.simplify.reflexivity.
@@ -57,6 +54,14 @@ intros.rewrite < H.reflexivity.
 apply le_S_S_to_le. assumption.
 qed.
 
+theorem eq_minus_S_pred: \forall n,m. n - (S m) = pred(n -m).
+apply nat_elim2
+  [intro.reflexivity
+  |intro.simplify.autobatch
+  |intros.simplify.assumption
+  ]
+qed.
+
 theorem plus_minus:
 \forall n,m,p:nat. m \leq n \to (n-m)+p = (n+p)-m.
 intros 2.
@@ -70,13 +75,12 @@ qed.
 
 theorem minus_plus_m_m: \forall n,m:nat.n = (n+m)-m.
 intros 2.
-generalize in match n.
-elim m.
+elim m in n ⊢ %.
 rewrite < minus_n_O.apply plus_n_O.
-elim n2.simplify.
+elim n1.simplify.
 apply minus_n_n.
 rewrite < plus_n_Sm.
-change with (S n3 = (S n3 + n1)-n1).
+change with (S n2 = (S n2 + n)-n).
 apply H.
 qed.
 
@@ -132,8 +136,8 @@ intros 2.
 apply (nat_elim2 (\lambda n,m.n \leq m \to n-m = O)).
 intros.simplify.reflexivity.
 intros.apply False_ind.
-apply not_le_Sn_O.
-goal 13.apply H.
+apply not_le_Sn_O;
+[2: apply H | skip].
 intros.
 simplify.apply H.apply le_S_S_to_le. apply H1.
 qed.
@@ -226,6 +230,49 @@ rewrite > plus_n_Sm.assumption.
 qed.
 
 (* minus and lt - to be completed *)
+theorem lt_minus_l: \forall m,l,n:nat. 
+  l < m \to m \le n \to n - m < n - l.
+apply nat_elim2
+  [intros.apply False_ind.apply (not_le_Sn_O ? H)
+  |intros.rewrite < minus_n_O.
+   autobatch
+  |intros.
+   generalize in match H2.
+   apply (nat_case n1)
+    [intros.apply False_ind.apply (not_le_Sn_O ? H3)
+    |intros.simplify.
+     apply H
+      [
+       apply lt_S_S_to_lt.
+       assumption
+      |apply le_S_S_to_le.assumption
+      ]
+    ]
+  ]
+qed.
+
+theorem lt_minus_r: \forall n,m,l:nat. 
+  n \le l \to l < m \to l -n < m -n.
+intro.elim n
+  [applyS H1
+  |rewrite > eq_minus_S_pred.
+   rewrite > eq_minus_S_pred.
+   apply lt_pred
+    [unfold lt.apply le_plus_to_minus_r.applyS H1
+    |apply H[autobatch|assumption]
+    ]
+  ]
+qed.
+
+lemma lt_to_lt_O_minus : \forall m,n.
+  n < m \to O < m - n.
+intros.  
+unfold. apply le_plus_to_minus_r. unfold in H. rewrite > sym_plus. 
+rewrite < plus_n_Sm. 
+rewrite < plus_n_O. 
+assumption.
+qed.  
+
 theorem lt_minus_to_plus: \forall n,m,p. (lt n (p-m)) \to (lt (n+m) p).
 intros 3.apply (nat_elim2 (\lambda m,p.(lt n (p-m)) \to (lt (n+m) p))).
 intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
@@ -236,6 +283,56 @@ rewrite < plus_n_Sm.
 apply H.apply H1.
 qed.
 
+theorem lt_O_minus_to_lt: \forall a,b:nat.
+O \lt b-a \to a \lt b.
+intros.
+rewrite > (plus_n_O a).
+rewrite > (sym_plus a O).
+apply (lt_minus_to_plus O  a b).
+assumption.
+qed.
+
+theorem lt_minus_to_lt_plus:
+\forall n,m,p. n - m < p \to n < m + p.
+intros 2.
+apply (nat_elim2 ? ? ? ? n m)
+  [simplify.intros.autobatch.
+  |intros 2.rewrite < minus_n_O.
+   intro.assumption
+  |intros.
+   simplify.
+   cut (n1 < m1+p)
+    [autobatch
+    |apply H.
+     apply H1
+    ]
+  ]
+qed.
+
+theorem lt_plus_to_lt_minus:
+\forall n,m,p. m \le n \to n < m + p \to n - m < p.
+intros 2.
+apply (nat_elim2 ? ? ? ? n m)
+  [simplify.intros 3.
+   apply (le_n_O_elim ? H).
+   simplify.intros.assumption
+  |simplify.intros.assumption.
+  |intros.
+   simplify.
+   apply H
+    [apply le_S_S_to_le.assumption
+    |apply le_S_S_to_le.apply H2
+    ]
+  ]
+qed. 
+
+theorem minus_m_minus_mn: \forall n,m. n\le m \to n=m-(m-n).
+intros.
+apply sym_eq.
+apply plus_to_minus.
+autobatch.
+qed.
+
 theorem distributive_times_minus: distributive nat times minus.
 unfold distributive.
 intros.