X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FGround_2%2Farith.ma;h=37fac204b70452b7cb98881b30f63b303a844633;hb=6ed2537d49a307259db46481469fa44b2cfc56e6;hp=a18844e9ba805359bd53b7d772c597e099167e02;hpb=035e3f52f8da3cb3cdb493aa20568ad673cc2cf5;p=helm.git

diff --git a/matita/matita/contribs/lambda_delta/Ground_2/arith.ma b/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
index a18844e9b..37fac204b 100644
--- a/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
+++ b/matita/matita/contribs/lambda_delta/Ground_2/arith.ma
@@ -13,179 +13,89 @@
 (**************************************************************************)
 
 include "arithmetics/nat.ma".
-include "Ground_2/xoa_props.ma".
+include "Ground_2/star.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
-lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
-#n #m <plus_n_Sm #H destruct
-qed.
-
-lemma plus_S_le_to_pos: ∀n,m,p. n + S m ≤ p → 0 < p.
-#n #m #p <plus_n_Sm #H @(lt_to_le_to_lt … H) //
-qed.
-
-lemma minus_le: ∀m,n. m - n ≤ m.
-/2/ qed.
-
-lemma le_O_to_eq_O: ∀n. n ≤ 0 → n = 0.
-/2/ qed.
-
-lemma lt_to_le: ∀a,b. a < b → a ≤ b.
-/2/ qed.
-
-lemma lt_refl_false: ∀n. n < n → False.
-#n #H elim (lt_to_not_eq … H) -H /2/
-qed.
-
-lemma lt_zero_false: ∀n. n < 0 → False.
-#n #H elim (lt_to_not_le … H) -H /2/
-qed.
-
-lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
-#m #n elim (decidable_lt m n) /3/
-qed.
+(* equations ****************************************************************)
 
-lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
-#m elim m -m
-[ * /2/
-| #m #IHm * [ /2/ ]
-  #n elim (IHm n) -IHm #H 
-  [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *)
-  qed.
-
-lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n.
-/2/ qed. (**) (* REMOVE: this is le_to_or_lt_eq *)
-
-lemma plus_le_weak: ∀m,n,p. m + n ≤ p → n ≤ p.
-/2/ 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 le_fwd_plus_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
-#m1 #m2 #H elim H -H m1
-[ /2/
-| #m1 #_ #IHm1 #n1 #n2 #H @IHm1 /2/
-]
-qed.
-
-lemma monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
-#p #q #n #H1 #H2
-@lt_plus_to_minus_r <plus_minus_m_m //.
-qed.
-
-lemma plus_le_minus: ∀a,b,c. a + b ≤ c → a ≤ c - b.
-/2/ qed.
-
-lemma lt_plus_minus: ∀i,u,d. u ≤ i → i < d + u → i - u < d.
-/2/ qed.
-
-lemma plus_plus_comm_23: ∀m,n,p. m + n + p = m + p + n.
+lemma plus_plus_comm_23: ∀x,y,z. x + y + z = x + z + y.
 // 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 le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
+/2 by plus_minus/ 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_O_to_eq_O … 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/
-]
-qed.
-
-lemma minus_plus_comm: ∀a,b,c. a - b - c = a - (c + b).
-// 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/ qed.
-
-lemma le_plus_minus: ∀a,b,c. c ≤ b → a + b - c = a + (b - c).
-/2/ qed.
+/3 by monotonic_le_minus_l, le_to_le_to_eq/ qed.
 
-lemma plus_minus_m_m_comm: ∀n,m. m ≤ n → n = m + (n - m).
-/2/ qed.
-
-lemma minus_plus_m_m_comm: ∀n,m. n = (m + n) - m.
-/2/ qed.
-
-lemma arith_a2: ∀a,c1,c2. c1 + c2 ≤ a → a - c1 - c2 + (c1 + c2) = a.
-/2/ 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_plus @eq_f2 /2/
+#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.
-#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2/
+#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1/
 qed.
 
-lemma arith_c1: ∀a,b,c1. a + c1 - (b + c1) = a - b.
-// qed.
-
 lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
-#x #a #b #c1 >plus_plus_comm_23 //
-qed.
-
-lemma arith_d1: ∀a,b,c1. c1 ≤ b → a + c1 + (b - c1) = a + b.
-/2/ qed.
-
-lemma arith_e2: ∀a,c1,c2. a ≤ c1 → c1 + c2 - (c1 - a + c2) = a.
-/3/ qed.
-
-lemma arith_f1: ∀a,b,c1. a + b ≤ c1 → c1 - (c1 - a - b) = a + b.
-#a #b #c1 #H >minus_plus <minus_minus //
-qed.
-
-lemma arith_g1: ∀a,b,c1. c1 ≤ b → a - (b - c1) - c1 = a - b.
-/2/ qed.
+/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
 
 lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
                 a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
-#a1 #a2 #b #c1 #H1 #H2 <le_plus_minus_comm /2/
+#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/
 qed.
 
-lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a.
-/2/ qed.
+(* inversion & forward lemmas ***********************************************)
 
-lemma arith_z1: ∀a,b,c1. a + c1 - b - c1 = a - b.
-// qed.
+axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
 
-(* unstable *****************************************************************)
+axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
 
-lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
-/2/ qed.
+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 arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
-#j #i #e #d #H lapply (plus_le_minus … H) -H /2/
-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 arith3: ∀a1,a2,b,c1. a1 + a2 ≤ b → a1 + c1 + a2 ≤ b + c1.
-/2/ qed.
+lemma le_inv_plus_plus_r: ∀x,y,z. x + z ≤ y + z → x ≤ y.
+/2 by le_plus_to_le/ qed-.
 
-lemma arith4: ∀h,d,e1,e2. d ≤ e1 + e2 → d + h ≤ e1 + h + e2.
-/2/ qed.
+lemma le_inv_plus_l: ∀x,y,z. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
+/3 width=2/ qed-.
 
-lemma arith5: ∀a,b1,b2,c1. c1 ≤ b1 → c1 ≤ a → a < b1 + b2 → a - c1 < b1 - c1 + b2.
-#a #b1 #b2 #c1 #H1 #H2 #H3
-<le_plus_minus_comm // @monotonic_lt_minus_l //
-qed.
+lemma lt_inv_plus_l: ∀x,y,z. x + y < z → x < z ∧ y < z - x.
+/3 width=2/ qed-.
 
-lemma arith8: ∀a,b. a < a + b + 1.
-// qed.
+lemma lt_refl_false: ∀n. n < n → False.
+#n #H elim (lt_to_not_eq … H) -H /2 width=1/
+qed-.
 
-lemma arith9: ∀a,b,c. c < a + (b + c + 1) + 1.
-// qed.
+lemma lt_zero_false: ∀n. n < 0 → False.
+#n #H elim (lt_to_not_le … H) -H /2 width=1/
+qed-.
 
-lemma arith10: ∀a,b,c,d,e. a ≤ b → c + (a - d - e) ≤ c + (b - d - e).
-#a #b #c #d #e #H
->minus_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l //
-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 #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/
+*)