- the definition of the framework for strong normalization continues ...

[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 80e23e6956a2ea78bccda3a080348a853774207b..418a176d5024ab7e8190c2c3220d7e45c1b3d0b2 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
+++ b/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
@@ -19,30 +19,14 @@ include "Ground_2/star.ma".
 
 (* equations ****************************************************************)
 
-lemma plus_plus_comm_23: ∀x,y,z. x + y + z = x + z + y.
-// qed.
-
 lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
 /2 by plus_minus/ qed.
 
 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 elim b -b
-[ #c #a #H >(le_n_O_to_eq … H) -H //
-| #b #IHb #c elim c -c //
-  #c #_ #a #Hcb
-  lapply (le_S_S_to_le … Hcb) -Hcb #Hcb
-  <plus_n_Sm normalize /2 width=1/
-]
-qed.
-
 lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
-#a #b #c1 #H >minus_plus /3 width=1/
+#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
 qed.
 
 lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
@@ -63,24 +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 elim m -m
-[ * /2 width=1/
-| #m #IHm * /2 width=1/
-  #n elim (IHm n) -IHm #H 
-  [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] // /2 width=1/ (**) (* /3 width=1/ is slow *)
+#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_refl_false: ∀n. n < n → False.
 #n #H elim (lt_to_not_eq … H) -H /2 width=1/
 qed-.
@@ -89,24 +61,12 @@ lemma lt_zero_false: ∀n. n < 0 → False.
 #n #H elim (lt_to_not_le … H) -H /2 width=1/
 qed-.
 
-lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
-#n #m <plus_n_Sm #H destruct
-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 false_lt_to_le: ∀x,y. (x < y → False) → y ≤ x.
 #x #y #H elim (decidable_lt x y) /2 width=1/
 #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 elim H -m1
-[ /2 width=2/
-| #m1 #_ #IHm1 #n1 #n2 #H @IHm1 /2 width=3/
-]
-qed-.
+(*
+lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
+/3 width=2/
+*)