update in ground

[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 9455ca7447ec71159d8b204b184ebeaa8d209282..bb2300f021a2e549d6f25bbb13dd49ace62a07cc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
@@ -16,11 +16,19 @@ 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 nsucc_pos (m): pnat ≝
+match m with
+[ nzero  ⇒ 𝟏
+| ninj p ⇒ ↑p
 ].
 
+interpretation
+  "positive successor (non-negative integers)"
+  'UpArrow m = (nsucc_pos m).
+
+definition nsucc (m): nat ≝
+           ninj (↑m).
+
 interpretation
   "successor (non-negative integers)"
   'UpArrow m = (nsucc m).
@@ -43,7 +51,7 @@ lemma nat_ind_succ (Q:predicate …):
 qed-.
 
 (*** nat_elim2 *)
-lemma nat_ind_succ_2 (Q:relation2 …):
+lemma nat_ind_2_succ (Q:relation2 …):
       (∀n. Q (𝟎) n) →
       (∀m. Q (↑m) (𝟎)) →
       (∀m,n. Q m n → Q (↑m) (↑n)) →
@@ -52,7 +60,7 @@ lemma nat_ind_succ_2 (Q:relation2 …):
 #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.
@@ -67,6 +75,14 @@ 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-.