arithmetics for λδ

[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 c93a75d42fc4b9a35b4a93a812633a6291f6f23e..916e6be1e5785f651c9208fc4c18c0a7ac236709 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma
@@ -14,7 +14,7 @@
 
 include "ground/arith/nat.ma".
 
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* SUCCESSOR FOR NON-NEGATIVE INTEGERS **************************************)
 
 definition nsucc: nat → nat ≝ λm. match m with
 [ nzero  ⇒ ninj (𝟏)
@@ -22,10 +22,10 @@ definition nsucc: nat → nat ≝ λm. match m with
 ].
 
 interpretation
-  "successor (non-negative integers"
+  "successor (non-negative integers)"
   'UpArrow m = (nsucc m).
 
-(* Basic rewrites ***********************************************************)
+(* Basic constructions ******************************************************)
 
 lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
 // qed.
@@ -36,23 +36,23 @@ lemma nsucc_inj (p): ninj (↑p) = ↑(ninj p).
 (* 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,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.
@@ -67,6 +67,14 @@ 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-.