(* 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
- "successor (non-negative integers"
+ "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).
-(* Basic rewrites ***********************************************************)
+(* Basic constructions ******************************************************)
lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
// 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,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.
* [ <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-.