(*** plus *)
definition nplus: nat → nat → nat ≝
- λm,n. nsucc^n m.
+ λm,n. (nsucc^n) m.
interpretation
"plus (non-negative integers)"
// qed.
(*** plus_SO_dx *)
-lemma nplus_one_dx (n): ↑n = n + 𝟏.
+lemma nplus_unit_dx (n): ↑n = n + 𝟏.
// qed.
(*** plus_n_Sm *)
/3 width=5 by compose_repl_fwd_sn, compose_repl_fwd_dx/
qed.
-(* Advanved constructions (semigroup properties) ****************************)
+(* Advanced constructions (semigroup properties) ****************************)
(*** plus_S1 *)
lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
#m #n @(niter_appl … nsucc)
qed.
-(*** plus_O_n.con *)
+(*** plus_O_n *)
lemma nplus_zero_sn (m): m = 𝟎 + m.
#m @(nat_ind_succ … m) -m //
qed.
(*** commutative_plus *)
lemma nplus_comm: commutative … nplus.
#m @(nat_ind_succ … m) -m //
-qed-. (**) (* gets in the way with auto *)
+qed-. (* * gets in the way with auto *)
(*** associative_plus *)
lemma nplus_assoc: associative … nplus.
(* Helper constructions *****************************************************)
(*** plus_SO_sn *)
-lemma nplus_one_sn (n): ↑n = 𝟏 + n.
+lemma nplus_unit_sn (n): ↑n = 𝟏 + n.
#n <nplus_comm // qed.
lemma nplus_succ_shift (m) (n): ↑m + n = m + ↑n.
-// qed-.
+// qed.
(*** assoc_plus1 *)
lemma nplus_plus_comm_12 (o) (m) (n): m + n + o = n + (m + o).
(*** plus_plus_comm_23 *)
lemma nplus_plus_comm_23 (o) (m) (n): o + m + n = o + n + m.
#o #m #n >nplus_assoc >nplus_assoc <nplus_comm in ⊢ (??(??%)?); //
-qed-.
+qed.
(* Basic inversions *********************************************************)
-(*** plus_inv_O3 zero_eq_plus *)
+(*** zero_eq_plus *)
lemma eq_inv_zero_nplus (m) (n): 𝟎 = m + n → ∧∧ 𝟎 = m & 𝟎 = n.
#m #n @(nat_ind_succ … n) -n
[ /2 width=1 by conj/
]
qed-.
+(*** plus_inv_O3 *)
+lemma eq_inv_nplus_zero (m) (n):
+ m + n = 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n.
+/2 width=1 by eq_inv_zero_nplus/ qed-.
+
(*** injective_plus_l *)
lemma eq_inv_nplus_bi_dx (o) (m) (n): m + o = n + o → m = n.
#o @(nat_ind_succ … o) -o /3 width=1 by eq_inv_nsucc_bi/