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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_plus.ma".
16 include "ground/arith/nat_minus.ma".
18 (* SUBTRACTION FOR NON-NEGATIVE INTEGERS ************************************)
20 (* Rewrites with nplus ******************************************************)
22 (*** minus_plus_m_m *)
23 lemma nminus_plus_sn_refl_sn (m) (n): m = m + n - n.
25 #n #IH <nplus_succ_dx <nminus_succ_bi //
28 lemma nminus_plus_sn_refl_dx (m) (n): m = n + m - n.
33 theorem nminus_plus_assoc (o) (m) (n): o-m-n = o-(m+n).
35 #n #IH <nplus_succ_dx <nminus_succ_dx <nminus_succ_dx //
38 (*** minus_plus_plus_l *)
39 lemma nminus_plus_dx_bi (m) (n) (o): m - n = (m + o) - (n + o).
40 #m #n #o <nminus_plus_assoc <nminus_minus_comm //
43 (*** plus_minus_plus_plus_l *) (**)
44 lemma plus_minus_plus_plus_l: ∀z,x,y,h. z + (x + h) - (y + h) = z + x - y.
47 (* Helper constructions with nplus ******************************************)
50 lemma nminus_plus_dx (o) (m) (n): o = m+n → n = o-m.
51 #o #m #n #H destruct //
54 lemma nminus_plus_sn (o) (m) (n): o = m+n → m = o-n.
55 #o #m #n #H destruct //
58 (* Inversions with nplus ****************************************************)
60 (*** discr_plus_xy_minus_xz *)
61 lemma eq_inv_plus_nminus_refl_sn (m) (n) (o):
66 [ /3 width=1 by or_introl, conj/
67 | #m #IH #n @(nat_ind … n) -n
69 lapply (eq_inv_nplus_bi_sn … (𝟎) Ho) -Ho
70 /3 width=1 by or_intror, conj/
72 <nminus_succ_bi >nplus_succ_shift #Ho
73 elim (IH … Ho) -IH -Ho * #_ #H
74 elim (eq_inv_nzero_succ … H)
79 (*** discr_minus_x_xy *)
80 lemma eq_inv_nminus_refl_sn (m) (n): m = m - n → ∨∨ 𝟎 = m | 𝟎 = n.
82 elim (eq_inv_plus_nminus_refl_sn … (𝟎) Hmn) -Hmn * #H1 #H2
83 /2 width=1 by or_introl, or_intror/