ocaml 3.09 transition

[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 7945a76e01c3c36ebe2d68ad3eda850af1bf4186..710418d72644022cb4d799fb95f67354da4af89b 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -48,9 +48,9 @@ qed.
 
 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.m \leq n \to (S n)-m = S (n-m)).
-intros.apply le_n_O_elim n1 H.
+apply (nat_elim2
+(\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.
 intros.rewrite < H.reflexivity.
@@ -60,19 +60,31 @@ qed.
 theorem plus_minus:
 \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.m \leq n \to (n-m)+p = (n+p)-m).
-intros.apply le_n_O_elim ? H.
+apply (nat_elim2
+(\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.
 intros.simplify.apply H.apply le_S_S_to_le.assumption.
 qed.
 
+theorem minus_plus_m_m: \forall n,m:nat.n = (n+m)-m.
+intros 2.
+generalize in match n.
+elim m.
+rewrite < minus_n_O.apply plus_n_O.
+elim n2.simplify.
+apply minus_n_n.
+rewrite < plus_n_Sm.
+change with (S n3 = (S n3 + n1)-n1).
+apply H.
+qed.
+
 theorem plus_minus_m_m: \forall n,m:nat.
 m \leq n \to n = (n-m)+m.
 intros 2.
-apply nat_elim2 (\lambda n,m.m \leq n \to n = (n-m)+m).
-intros.apply le_n_O_elim n1 H.
+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.
 intros.simplify.rewrite < sym_plus.simplify.
@@ -82,20 +94,22 @@ qed.
 
 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) ?.
+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.m \leq n \to
+theorem plus_to_minus :\forall n,m,p:nat.
 n = m+p \to n-m = p.
 intros.
-apply inj_plus_r m.
-rewrite < H1.
+apply (inj_plus_r m).
+rewrite < H.
 rewrite < sym_plus.
 symmetry.
-apply plus_minus_m_m.assumption.
+apply plus_minus_m_m.rewrite > H.
+rewrite > sym_plus.
+apply le_plus_n.
 qed.
 
 theorem minus_S_S : \forall n,m:nat.
@@ -107,32 +121,32 @@ 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.
+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.
 n \leq m \to n-m = O.
 intros 2.
-apply nat_elim2 (\lambda n,m.n \leq m \to 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 *)
-apply not_le_Sn_O n1 H.
+apply not_le_Sn_O.
+goal 13.apply H.
 intros.
 simplify.apply H.apply le_S_S_to_le. apply H1.
 qed.
 
 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.
+intros.elim H.elim (minus_Sn_n n).apply le_n.
 rewrite > minus_Sn_m.
 apply le_S.assumption.
 apply lt_to_le.assumption.
 qed.
 
 theorem minus_le_S_minus_S: \forall n,m:nat. m-n \leq S (m-(S n)).
-intros.apply nat_elim2 (\lambda n,m.m-n \leq S (m-(S n))).
+intros.apply (nat_elim2 (\lambda n,m.m-n \leq S (m-(S n)))).
 intro.elim n1.simplify.apply le_n_Sn.
 simplify.rewrite < minus_n_O.apply le_n.
 intros.simplify.apply le_n_Sn.
@@ -141,21 +155,27 @@ qed.
 
 theorem lt_minus_S_n_to_le_minus_n : \forall n,m,p:nat. m-(S n) < p \to m-n \leq p. 
 intros 3.simplify.intro.
-apply trans_le (m-n) (S (m-(S n))) p.
+apply (trans_le (m-n) (S (m-(S n))) p).
 apply minus_le_S_minus_S.
 assumption.
 qed.
 
 theorem le_minus_m: \forall n,m:nat. n-m \leq n.
-intros.apply nat_elim2 (\lambda m,n. n-m \leq n).
+intros.apply (nat_elim2 (\lambda m,n. n-m \leq n)).
 intros.rewrite < minus_n_O.apply le_n.
 intros.simplify.apply le_n.
 intros.simplify.apply le_S.assumption.
 qed.
 
+theorem lt_minus_m: \forall n,m:nat. O < n \to O < m \to n-m \lt n.
+intros.apply (lt_O_n_elim n H).intro.
+apply (lt_O_n_elim m H1).intro.
+simplify.unfold lt.apply le_S_S.apply le_minus_m.
+qed.
+
 theorem minus_le_O_to_le: \forall n,m:nat. n-m \leq O \to n \leq m.
 intros 2.
-apply nat_elim2 (\lambda n,m:nat.n-m \leq O \to n \leq m).
+apply (nat_elim2 (\lambda n,m:nat.n-m \leq O \to n \leq m)).
 intros.apply le_O_n.
 simplify.intros. assumption.
 simplify.intros.apply le_S_S.apply H.assumption.
@@ -164,9 +184,9 @@ qed.
 (* galois *)
 theorem monotonic_le_minus_r: 
 \forall p,q,n:nat. q \leq p \to n-p \le n-q.
-simplify.intros 2.apply nat_elim2 
-(\lambda p,q.\forall a.q \leq p \to a-p \leq a-q).
-intros.apply le_n_O_elim n H.apply le_n.
+simplify.intros 2.apply (nat_elim2 
+(\lambda p,q.\forall a.q \leq p \to a-p \leq a-q)).
+intros.apply (le_n_O_elim n H).apply le_n.
 intros.rewrite < minus_n_O.
 apply le_minus_m.
 intros.elim a.simplify.apply le_n.
@@ -174,7 +194,7 @@ simplify.apply H.apply le_S_S_to_le.assumption.
 qed.
 
 theorem le_minus_to_plus: \forall n,m,p. (le (n-m) p) \to (le n (p+m)).
-intros 2.apply nat_elim2 (\lambda n,m.\forall p.(le (n-m) p) \to (le n (p+m))).
+intros 2.apply (nat_elim2 (\lambda n,m.\forall p.(le (n-m) p) \to (le n (p+m)))).
 intros.apply le_O_n.
 simplify.intros.rewrite < plus_n_O.assumption.
 intros.
@@ -184,7 +204,7 @@ exact H1.
 qed.
 
 theorem le_plus_to_minus: \forall n,m,p. (le n (p+m)) \to (le (n-m) p).
-intros 2.apply nat_elim2 (\lambda n,m.\forall p.(le n (p+m)) \to (le (n-m) p)).
+intros 2.apply (nat_elim2 (\lambda n,m.\forall p.(le n (p+m)) \to (le (n-m) p))).
 intros.simplify.apply le_O_n.
 intros 2.rewrite < plus_n_O.intro.simplify.assumption.
 intros.simplify.apply H.
@@ -193,11 +213,11 @@ qed.
 
 (* the converse of le_plus_to_minus does not hold *)
 theorem le_plus_to_minus_r: \forall n,m,p. (le (n+m) p) \to (le n (p-m)).
-intros 3.apply nat_elim2 (\lambda m,p.(le (n+m) p) \to (le n (p-m))).
+intros 3.apply (nat_elim2 (\lambda m,p.(le (n+m) p) \to (le n (p-m)))).
 intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
-intro.intro.cut n=O.rewrite > Hcut.apply le_O_n.
+intro.intro.cut (n=O).rewrite > Hcut.apply le_O_n.
 apply sym_eq. apply le_n_O_to_eq.
-apply trans_le ? (n+(S n1)).
+apply (trans_le ? (n+(S n1))).
 rewrite < sym_plus.
 apply le_plus_n.assumption.
 intros.simplify.
@@ -205,30 +225,76 @@ apply H.apply le_S_S_to_le.
 rewrite > plus_n_Sm.assumption.
 qed.
 
+(* minus and lt - to be completed *)
+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.
+simplify.intros.apply False_ind.apply (not_le_Sn_O n H).
+simplify.intros.unfold lt.
+apply le_S_S.
+rewrite < plus_n_Sm.
+apply H.apply H1.
+qed.
 
 theorem distributive_times_minus: distributive nat times minus.
-simplify.
+unfold distributive.
 intros.
-apply (leb_elim z y).intro.
-cut x*(y-z)+x*z = (x*y-x*z)+x*z.
-apply inj_plus_l (x*z).
-assumption.
-apply trans_eq nat ? (x*y).
-rewrite < distr_times_plus. 
-rewrite < plus_minus_m_m ? ? H.reflexivity.
-rewrite < plus_minus_m_m ? ? ?.reflexivity.
-apply le_times_r.
-assumption.
-intro.
-rewrite > eq_minus_n_m_O.
-rewrite > eq_minus_n_m_O (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.
+apply ((leb_elim z y)).
+  intro.cut (x*(y-z)+x*z = (x*y-x*z)+x*z).
+    apply (inj_plus_l (x*z)).assumption.
+    apply (trans_eq nat ? (x*y)).
+      rewrite < distr_times_plus.rewrite < (plus_minus_m_m ? ? H).reflexivity.
+      rewrite < plus_minus_m_m.
+        reflexivity.
+        apply le_times_r.assumption.
+  intro.rewrite > eq_minus_n_m_O.
+    rewrite > (eq_minus_n_m_O (x*y)).
+      rewrite < sym_times.simplify.reflexivity.
+        apply le_times_r.apply lt_to_le.apply not_le_to_lt.assumption.
+        apply lt_to_le.apply not_le_to_lt.assumption.
 qed.
 
 theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
 \def distributive_times_minus.
 
+theorem eq_minus_plus_plus_minus: \forall n,m,p:nat. p \le m \to (n+m)-p = n+(m-p).
+intros.
+apply plus_to_minus.
+rewrite > sym_plus in \vdash (? ? ? %).
+rewrite > assoc_plus.
+rewrite < plus_minus_m_m.
+reflexivity.assumption.
+qed.
+
+theorem eq_minus_minus_minus_plus: \forall n,m,p:nat. (n-m)-p = n-(m+p).
+intros.
+cut (m+p \le n \or m+p \nleq n).
+  elim Hcut.
+    symmetry.apply plus_to_minus.
+    rewrite > assoc_plus.rewrite > (sym_plus p).rewrite < plus_minus_m_m.
+      rewrite > sym_plus.rewrite < plus_minus_m_m.
+        reflexivity.
+        apply (trans_le ? (m+p)).
+          rewrite < sym_plus.apply le_plus_n.
+          assumption.
+      apply le_plus_to_minus_r.rewrite > sym_plus.assumption.   
+    rewrite > (eq_minus_n_m_O n (m+p)).
+      rewrite > (eq_minus_n_m_O (n-m) p).
+        reflexivity.
+      apply le_plus_to_minus.apply lt_to_le. rewrite < sym_plus.
+       apply not_le_to_lt. assumption.
+    apply lt_to_le.apply not_le_to_lt.assumption.          
+  apply (decidable_le (m+p) n).
+qed.
+
+theorem eq_plus_minus_minus_minus: \forall n,m,p:nat. p \le m \to m \le n \to
+p+(n-m) = n-(m-p).
+intros.
+apply sym_eq.
+apply plus_to_minus.
+rewrite < assoc_plus.
+rewrite < plus_minus_m_m.
+rewrite < sym_plus.
+rewrite < plus_minus_m_m.reflexivity.
+assumption.assumption.
+qed.