propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_succ.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
index 923b235e6f4dc5cf9ffbbf16b4ea9e9dc120cfa0..742c6b25776f397bfddd57541f8a007ad5a0b4cc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
@@ -16,16 +16,30 @@ include "ground/arith/nat.ma".
 
 (* SUCCESSOR FOR NON-NEGATIVE INTEGERS **************************************)
 
-definition nsucc: nat → nat ≝ λm. match m with
-[ nzero  ⇒ ninj (𝟏)
-| ninj p ⇒ ninj (↑p)
+definition npsucc (m): pnat ≝
+match m with
+[ nzero  ⇒ 𝟏
+| ninj p ⇒ ↑p
 ].
 
+interpretation
+  "positive successor (non-negative integers)"
+  'UpArrow m = (npsucc m).
+
+definition nsucc (m): nat ≝
+           ninj (↑m).
+
 interpretation
   "successor (non-negative integers)"
   'UpArrow m = (nsucc m).
 
-(* Basic rewrites ***********************************************************)
+(* Basic constructions ******************************************************)
+
+lemma npsucc_zero: (𝟏) = ↑𝟎.
+// qed.
+
+lemma npsucc_inj (p): (↑p) = ↑(ninj p).
+// qed.
 
 lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
 // qed.
@@ -33,40 +47,56 @@ lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
 lemma nsucc_inj (p): ninj (↑p) = ↑(ninj p).
 // qed.
 
+lemma npsucc_succ (n): psucc (npsucc n) = npsucc (nsucc n).
+// qed.
+
 (* Basic eliminations *******************************************************)
 
 (*** nat_ind *)
-lemma nat_ind (Q:predicate …):
+lemma nat_ind_succ (Q:predicate …):
       Q (𝟎) → (∀n. Q n → Q (↑n)) → ∀n. Q n.
 #Q #IH1 #IH2 * //
 #p elim p -p /2 width=1 by/
 qed-.
 
 (*** nat_elim2 *)
-lemma nat_ind_2 (Q:relation2 …):
+lemma nat_ind_2_succ (Q:relation2 …):
       (∀n. Q (𝟎) n) →
-      (∀m. Q (↑m) (𝟎)) →
+      (∀m. Q m (𝟎) → Q (↑m) (𝟎)) →
       (∀m,n. Q m n → Q (↑m) (↑n)) →
       ∀m,n. Q m n.
-#Q #IH1 #IH2 #IH3 #m elim m -m [ // ]
-#m #IH #n elim n -n /2 width=1 by/
+#Q #IH1 #IH2 #IH3 #m @(nat_ind_succ … m) -m [ // ]
+#m #IH #n @(nat_ind_succ … n) -n /2 width=1 by/
 qed-.
 
-(* Basic inversions ***************************************************************)
+(* Basic inversions *********************************************************)
 
-(*** injective_S *)
-lemma eq_inv_nsucc_bi: injective … nsucc.
+lemma eq_inv_npsucc_bi: injective … npsucc.
 * [| #p1 ] * [2,4: #p2 ]
-[1,4: <nsucc_zero <nsucc_inj #H destruct
-| <nsucc_inj <nsucc_inj #H destruct //
+[ 1,4: <npsucc_zero <npsucc_inj #H destruct
+| <npsucc_inj <npsucc_inj #H destruct //
 | //
 ]
 qed-.
 
+(*** injective_S *)
+lemma eq_inv_nsucc_bi: injective … nsucc.
+#n1 #n2 #H
+@eq_inv_npsucc_bi @eq_inv_ninj_bi @H
+qed-.
+
 lemma eq_inv_nsucc_zero (m): ↑m = 𝟎 → ⊥.
 * [ <nsucc_zero | #p <nsucc_inj ] #H destruct
 qed-.
 
-lemma eq_inv_nzero_succ (m): 𝟎 = ↑m → ⊥.
+lemma eq_inv_zero_nsucc (m): 𝟎 = ↑m → ⊥.
 * [ <nsucc_zero | #p <nsucc_inj ] #H destruct
 qed-.
+
+(*** succ_inv_refl_sn *)
+lemma nsucc_inv_refl (n): n = ↑n → ⊥.
+#n @(nat_ind_succ … n) -n
+[ /2 width=2 by eq_inv_zero_nsucc/
+| #n #IH #H /3 width=1 by eq_inv_nsucc_bi/
+]
+qed-.