initial properies of the "same top term constructor" predicate

[helm.git] / matita / matita / contribs / lambda_delta / Ground_2 / arith.ma

diff --git a/matita/matita/contribs/lambda_delta/Ground_2/arith.ma b/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
index 37fac204b70452b7cb98881b30f63b303a844633..39d28c959182d17c31fa4a851751928c3fc12b8d 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
+++ b/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
@@ -17,10 +17,7 @@ include "Ground_2/star.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
-(* equations ****************************************************************)
-
-lemma plus_plus_comm_23: ∀x,y,z. x + y + z = x + z + y.
-// qed.
+(* Equations ****************************************************************)
 
 lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
 /2 by plus_minus/ qed.
@@ -28,13 +25,6 @@ lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
 lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
 /2 by plus_minus/ qed.
 
-lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
-/3 by monotonic_le_minus_l, le_to_le_to_eq/ qed.
-
-lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
-#b #c #a #H >(plus_minus_m_m b c) in ⊢ (? ? ?%); //
-qed.
-
 lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
 #a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
 qed.
@@ -57,25 +47,12 @@ axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
 
 axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
 
-lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
-#m #n elim (decidable_lt m n) /2 width=1/ /3 width=1/
-qed-.
-
 lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
 #m #n elim (lt_or_ge m n) /2 width=1/
 #H elim H -m /2 width=1/
 #m #Hm * #H /2 width=1/ /3 width=1/
 qed-.
 
-lemma le_inv_plus_plus_r: ∀x,y,z. x + z ≤ y + z → x ≤ y.
-/2 by le_plus_to_le/ qed-.
-
-lemma le_inv_plus_l: ∀x,y,z. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
-/3 width=2/ qed-.
-
-lemma lt_inv_plus_l: ∀x,y,z. x + y < z → x < z ∧ y < z - x.
-/3 width=2/ qed-.
-
 lemma lt_refl_false: ∀n. n < n → False.
 #n #H elim (lt_to_not_eq … H) -H /2 width=1/
 qed-.
@@ -89,13 +66,11 @@ lemma false_lt_to_le: ∀x,y. (x < y → False) → y ≤ x.
 #Hxy elim (H Hxy)
 qed-.
 
-lemma le_fwd_plus_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
-#m1 #m2 #H #n1 #n2 >commutative_plus
-#H elim (le_inv_plus_l … H) -H >commutative_plus <minus_le_minus_minus_comm //
-#H #_ @(transitive_le … H) /2 width=1/
-qed-. 
-
 (*
 lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
 /3 width=2/
+
+lemma le_or_ge: ∀m,n. m ≤ n ∨ n ≤ m.
+#m #n elim (lt_or_ge m n) /2 width=1/ /3 width=2/
+qed-.
 *)