X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Flambda-delta%2Fground.ma;h=54f9da02a162a1e307e99a4e72ea143c9dbfbe22;hb=dde568d876a5d2e1b6e554a526c98b09b145d25a;hp=f4279588b251aa360158fab7826fc6f3b4abbc19;hpb=95aa0f13b906e2b145c60bde078b752869976e7f;p=helm.git

diff --git a/matita/matita/lib/lambda-delta/ground.ma b/matita/matita/lib/lambda-delta/ground.ma
index f4279588b..54f9da02a 100644
--- a/matita/matita/lib/lambda-delta/ground.ma
+++ b/matita/matita/lib/lambda-delta/ground.ma
@@ -12,39 +12,94 @@
 include "basics/list.ma".
 include "lambda-delta/xoa_defs.ma".
 include "lambda-delta/xoa_notation.ma".
-include "lambda-delta/notation.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
 lemma plus_plus_comm_23: ∀m,n,p. m + n + p = m + p + n.
 // qed.
 
+lemma minus_plus_comm: ∀a,b,c. a - b - c = a - (c + b).
+// qed.
+
+lemma arith8: ∀a,b. a < a + b + 1.
+// qed.
+
+lemma arith9: ∀a,b,c. c < a + (b + c + 1) + 1.
+// qed.
+
 lemma minus_le: ∀m,n. m - n ≤ m.
 /2/ qed.
 
 lemma plus_plus_minus_m_m: ∀e1,e2,d. e1 ≤ e2 → d + e1 + (e2 - e1) = d + e2.
 /2/ qed.
 
+lemma le_O_to_eq_O: ∀n. n ≤ 0 → n = 0.
+/2/ qed.
+
+lemma plus_le_minus: ∀a,b,c. a + b ≤ c → a ≤ c - b.
+/2/ qed.
+
 lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → (m + n) - p = (m - p) + n.
 #n #m #p #lepm @plus_to_minus <associative_plus
 >(commutative_plus p) <plus_minus_m_m //
 qed.
 
-lemma lt_false: ∀n. n < n → False.
-#n #H lapply (lt_to_not_eq … H) -H #H elim H -H /2/
+lemma le_plus_minus: ∀a,b,c. c ≤ b → a + b - c = a + (b - c).
+/2/ qed.
+
+lemma minus_le_minus_minus_comm: ∀m,p,n. 
+                                 p ≤ m → m - p ≤ n → n + p - m = n - (m - p).
+#m elim m -m
+[ #p #n #H >(le_O_to_eq_O … H) -H //
+| #m #IHm #p elim p -p //
+  #p #_ #n #Hpm <plus_n_Sm @IHm /2/
+]
 qed.
 
-lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
-/2/ qed.
+lemma lt_refl_false: ∀n. n < n → False.
+#n #H elim (lt_to_not_eq … H) -H /2/
+qed.
 
-lemma arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
-/3/ qed.
+lemma lt_zero_false: ∀n. n < 0 → False.
+#n #H elim (lt_to_not_le … H) -H /2/
+qed.
 
-lemma arith3: ∀m,n,p. p ≤ m → m + n - (m - p + n) = p.
+lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
+#m #n elim (decidable_lt m n) /3/
+qed.
+
+lemma plus_lt_false: ∀m,n. m + n < m → False.
+#m #n #H1 lapply (le_plus_n_r n m) #H2
+lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H
+elim (lt_refl_false … H)
+qed.
+
+lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
+#n #m <plus_n_Sm #H destruct
+qed. 
+
+lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
 /3/ qed.
 
+lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
+/2/ qed.
+
+lemma arith6: ∀m,n. m < n → n - (n - m - 1) = m + 1.
+#m #n #H >minus_plus <minus_minus //
+qed.
+
 lemma arith4: ∀h,d,e1,e2. d ≤ e1 + e2 → d + h ≤ e1 + h + e2.
 /2/ qed.
 
 lemma arith5: ∀i,h,d. i + h ≤ d → d - i - h + (i + h) = d.
 /2/ qed.
+
+lemma arith7: ∀i,d. i ≤ d → d - i + i = d.
+/2/ qed.
+
+lemma arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
+/3/ qed.
+
+lemma arith3: ∀m,n,p. p ≤ m → m + n - (m - p + n) = p.
+/3/ qed.
+