- ng_kernel: catched Invalid_argument "List.nth" in typeof of Rel i (raied when i...

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / arith.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 557957bea4ba1254ac01f3f96cf4c3d719c88b77..d181e88cf3f385f1e8ae993f172461810ca31036 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -12,13 +12,30 @@
 (*                                                                        *)
 (**************************************************************************)
 
+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).
+
+(* Iota equations ***********************************************************)
+
+lemma pred_O: pred 0 = 0.
+normalize // qed.
+
+lemma pred_S: ∀m. pred (S m) = m.
+// qed.
+
 (* Equations ****************************************************************)
 
+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 //
@@ -89,6 +106,9 @@ 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 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.
 
@@ -108,6 +128,9 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
+// qed-.
+
 lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
 /2 width=1 by monotonic_pred/ qed-.
 
@@ -119,6 +142,9 @@ lemma lt_zero_false: ∀n. n < 0 → ⊥.
 #n #H elim (lt_to_not_le … H) -H /2 width=1 by/
 qed-.
 
+lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
+/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
+
 lemma pred_inv_refl: ∀m. pred m = m → m = 0.
 * // normalize #m #H elim (lt_refl_false m) //
 qed-.
@@ -138,9 +164,35 @@ 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-.
+
 (* Iterators ****************************************************************)
 
-(* Note: see also: lib/arithemetcs/bigops.ma *)
+(* Note: see also: lib/arithemetics/bigops.ma *)
 let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
   match n with
    [ O   ⇒ nil
@@ -149,6 +201,12 @@ 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.