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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground/arith/nat_succ_iter.ma".
17 (* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************)
20 definition nplus: nat → nat → nat ≝
24 "plus (positive integers)"
25 'plus m n = (nplus m n).
27 (* Basic constructions ******************************************************)
30 lemma nplus_zero_dx (m): m = m + 𝟎.
34 lemma nplus_one_dx (n): ↑n = n + 𝟏.
38 lemma nplus_succ_dx (m) (n): ↑(m+n) = m + ↑n.
39 #m #n @(niter_succ … nsucc)
42 (* Constructions with niter *************************************************)
45 lemma niter_plus (A) (f) (a) (n1) (n2):
46 f^n1 (f^n2 a) = f^{A}(n1+n2) a.
47 #A #f #a #n1 #n2 @(nat_ind_succ … n2) -n2 //
48 #n2 #IH <nplus_succ_dx <niter_succ <niter_succ <niter_appl //
51 (* Advanved constructions (semigroup properties) ****************************)
54 lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
55 #m #n @(niter_appl … nsucc)
59 lemma nplus_zero_sn (m): m = 𝟎 + m.
60 #m @(nat_ind_succ … m) -m //
63 (*** commutative_plus *)
64 lemma nplus_comm: commutative … nplus.
65 #m @(nat_ind_succ … m) -m //
68 (*** associative_plus *)
69 lemma nplus_assoc: associative … nplus.
70 #m #n #o @(nat_ind_succ … o) -o //
71 #o #IH <nplus_succ_dx <nplus_succ_dx <nplus_succ_dx <IH -IH //
74 (* Helper constructions *****************************************************)
77 lemma nplus_one_sn (n): ↑n = 𝟏 + n.
78 #n <nplus_comm // qed.
80 lemma nplus_succ_shift (m) (n): ↑m + n = m + ↑n.
84 lemma nplus_plus_comm_12 (o) (m) (n): m + n + o = n + (m + o).
85 #o #m #n <nplus_comm in ⊢ (??(?%?)?); // qed.
87 (*** plus_plus_comm_23 *)
88 lemma nplus_plus_comm_23 (o) (m) (n): o + m + n = o + n + m.
89 #o #m #n >nplus_assoc >nplus_assoc <nplus_comm in ⊢ (??(??%)?); //
92 (* Basic inversions *********************************************************)
94 lemma eq_inv_nzero_plus (m) (n): 𝟎 = m + n → ∧∧ 𝟎 = m & 𝟎 = n.
95 #m #n @(nat_ind_succ … n) -n
97 | #n #_ <nplus_succ_dx #H
98 elim (eq_inv_nzero_succ … H)
102 (*** injective_plus_l *)
103 lemma eq_inv_nplus_bi_dx (o) (m) (n): m + o = n + o → m = n.
104 #o @(nat_ind_succ … o) -o /3 width=1 by eq_inv_nsucc_bi/
107 (*** injective_plus_r *)
108 lemma eq_inv_nplus_bi_sn (o) (m) (n): o + m = o + n → m = n.
109 #o #m #n <nplus_comm <nplus_comm in ⊢ (???%→?);
110 /2 width=2 by eq_inv_nplus_bi_dx/
113 (* Advanced eliminations ****************************************************)
116 lemma nat_ind_plus (Q:predicate …):
117 Q (𝟎) → (∀n. Q n → Q (𝟏+n)) → ∀n. Q n.
118 #Q #IH1 #IH2 #n @(nat_ind_succ … n) -n /2 width=1 by/