X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Flib%2Farith.ma;h=9fedf60c0c180fd43f5e2ce90139254ba59dbfca;hb=f6b452b9c9be141740d4058dfbcf81a4b75fd00b;hp=1afc42ebcbccd6d0365a2869eb9035071a03916f;hpb=98c91e19a9cc31c77a0151f5df7f7690813cbd07;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 1afc42ebc..9fedf60c0 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -12,13 +12,51 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/notation/functions/successor_1.ma".
+include "ground_2/notation/functions/predecessor_1.ma".
 include "arithmetics/nat.ma".
 include "ground_2/lib/star.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
+interpretation "nat successor" 'Successor m = (S m).
+
+interpretation "nat predecessor" 'Predecessor m = (pred m).
+
+interpretation "nat min" 'and x y = (min x y).
+
+interpretation "nat max" 'or x y = (max x y).
+
+(* Iota equations ***********************************************************)
+
+lemma pred_O: pred 0 = 0.
+normalize // qed.
+
+lemma pred_S: ∀m. pred (S m) = m.
+// qed.
+
+lemma plus_S1: ∀x,y. ⫯(x+y) = (⫯x) + y.
+// qed.
+
+lemma max_O1: ∀n. n = (0 ∨ n).
+// qed.
+
+lemma max_O2: ∀n. n = (n ∨ 0).
+// qed.
+
+lemma max_SS: ∀n1,n2. ⫯(n1∨n2) = (⫯n1 ∨ ⫯n2).
+#n1 #n2 elim (decidable_le n1 n2) #H normalize
+[ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
+qed.
+
 (* Equations ****************************************************************)
 
+lemma plus_SO: ∀n. n + 1 = ⫯n.
+// qed.
+
+lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
+// qed-.
+
 (* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
 lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
 #x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
@@ -60,9 +98,13 @@ lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
 
 (* Properties ***************************************************************)
 
-axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
-
-axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
+lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
+#n1 elim n1 -n1 [| #n1 #IHn1 ] * [2,4: #n2 ]
+[1,4: @or_intror #H destruct
+| elim (IHn1 n2) -IHn1 /3 width=1 by or_intror, or_introl/
+| /2 width=1 by or_introl/
+]
+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 by or3_intro0/
@@ -85,6 +127,18 @@ lemma monotonic_lt_pred: ∀m,n. m < n → O < m → pred m < pred n.
 @le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
 qed.
 
+lemma lt_S_S: ∀x,y. x < y → ⫯x < ⫯y.
+/2 width=1 by le_S_S/ qed.
+
+lemma lt_S: ∀n,m. n < m → n < ⫯m.
+/2 width=1 by le_S/ qed.
+
+lemma max_S1_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (⫯n1 ∨ n2) ≤ ⫯n.
+/4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
+
+lemma max_S2_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (n1 ∨ ⫯n2) ≤ ⫯n.
+/2 width=1 by max_S1_le_S/ qed-.
+
 lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
 /3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
 
@@ -104,6 +158,15 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
+/2 width=1 by plus_le_0/ qed-.
+
+lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
+// qed-.
+
+lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
+/2 width=2 by le_plus_minus_comm/ qed-.
+
 lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
 /2 width=1 by monotonic_pred/ qed-.
 
@@ -115,9 +178,26 @@ lemma lt_zero_false: ∀n. n < 0 → ⊥.
 #n #H elim (lt_to_not_le … H) -H /2 width=1 by/
 qed-.
 
-lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x.
-#x #y #H elim (decidable_lt x y) /2 width=1 by not_lt_to_le/
-#Hxy elim (H Hxy)
+lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
+/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
+
+lemma succ_inv_refl_sn: ∀x. ⫯x = x → ⊥.
+#x #H @(lt_le_false x (⫯x)) //
+qed-.
+
+lemma lt_inv_O1: ∀n. 0 < n → ∃m. ⫯m = n.
+* /2 width=2 by ex_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_S1: ∀m,n. ⫯m < n → ∃∃p. m < p & ⫯p = n.
+#m * /3 width=3 by lt_S_S_to_lt, ex2_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_gen: ∀y,x. x < y → ∃∃z. x ≤ z & ⫯z = y.
+* /3 width=3 by le_S_S_to_le, ex2_intro/
+#x #H elim (lt_le_false … H) -H //
 qed-.
 
 lemma pred_inv_refl: ∀m. pred m = m → m = 0.
@@ -139,10 +219,45 @@ lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
 lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
 /2 width=4 by plus_xySz_x_false/ qed-.
 
+(* Note this should go in nat.ma *)
+lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
+#x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
+#x #_ #y @(nat_ind_plus … y) -y /2 width=1 by or_intror/
+#y #_ >minus_plus_plus_l
+#H lapply (discr_plus_xy_minus_xz … H) -H
+#H destruct
+qed-.
+
+lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
+* /2 width=1 by conj/ #x #y normalize #H destruct
+qed-.
+
+lemma lt_S_S_to_lt: ∀x,y. ⫯x < ⫯y → x < y.
+/2 width=1 by le_S_S_to_le/ qed-.
+
+lemma lt_elim: ∀R:relation nat.
+               (∀n2. R O (⫯n2)) →
+               (∀n1,n2. R n1 n2 → R (⫯n1) (⫯n2)) →
+               ∀n2,n1. n1 < n2 → R n1 n2.
+#R #IH1 #IH2 #n2 elim n2 -n2
+[ #n1 #H elim (lt_le_false … H) -H //
+| #n2 #IH * /4 width=1 by lt_S_S_to_lt/
+]
+qed-.
+
+lemma le_elim: ∀R:relation nat.
+               (∀n2. R O (n2)) →
+               (∀n1,n2. R n1 n2 → R (⫯n1) (⫯n2)) →
+               ∀n1,n2. n1 ≤ n2 → R n1 n2.
+#R #IH1 #IH2 #n1 #n2 @(nat_elim2 … n1 n2) -n1 -n2
+/4 width=1 by monotonic_pred/ -IH1 -IH2
+#n1 #H elim (lt_le_false … H) -H //
+qed-.
+
 (* Iterators ****************************************************************)
 
-(* Note: see also: lib/arithemetcs/bigops.ma *)
-let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
+(* Note: see also: lib/arithemetics/bigops.ma *)
+rec definition iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
   match n with
    [ O   ⇒ nil
    | S k ⇒ op (iter k B op nil)
@@ -150,6 +265,12 @@ let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
 
 interpretation "iterated function" 'exp op n = (iter n ? op).
 
+lemma iter_O: ∀B:Type[0]. ∀f:B→B.∀b. f^0 b = b.
+// qed.
+
+lemma iter_S: ∀B:Type[0]. ∀f:B→B.∀b,l. f^(S l) b = f (f^l b).
+// qed.
+
 lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
 #B #f #b #l >commutative_plus //
 qed.
@@ -165,7 +286,7 @@ qed.
 (* Trichotomy operator ******************************************************)
 
 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
-let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
+rec definition tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
   match n1 with
   [ O    ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
   | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]