matita/matita/nlibrary/nat/plus.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "nat/big_ops.ma".
  16 include "algebra/unital_magmas.ma".
  17 include "algebra/abelian_magmas.ma".
  18
  19 nlet rec plus (n:nat) (m:nat) on n : nat ≝
  20  match n with
  21   [ O ⇒ m
  22   | S n' ⇒ S (plus n' m) ].
  23
  24 interpretation "natural plus" 'plus x y = (plus x y).
  25
  26 ndefinition plus_magma_type: magma_type.
  27  napply mk_magma_type
  28   [ napply NAT
  29   | napply mk_binary_morphism
  30      [ napply plus
  31      | #a; #a'; #b; #b'; #Ha; #Hb; nrewrite < Ha; nrewrite < Hb; napply refl ]##]
  32 nqed.
  33
  34 ndefinition plus_abelian_magma_type: abelian_magma_type.
  35  napply mk_abelian_magma_type
  36   [ napply plus_magma_type
  37   | nnormalize; #x;
  38      (* nelim non va *) napply (nat_rect_CProp0 ??? x);
  39      ##[ #y; napply (nat_rect_CProp0 ??? y) [ napply refl | #n; #H; nnormalize; nrewrite < H; napply refl]
  40      ##| #n; #H; #y; nnormalize;
  41          (* rewrite qui non va *)
  42          napply (eq_rect_CProp0_r ????? (H y));
  43          napply (nat_rect_CProp0 ??? y)
  44           [ napply refl
  45           | #n0; #K; nnormalize in K; nnormalize;
  46             napply (eq_rect_CProp0 ????? K); napply refl] ##]
  47 nqed.
  48
  49 ndefinition plus_unital_magma_type: unital_magma_type.
  50  napply mk_unital_magma_type
  51   [ napply plus_magma_type
  52   | napply O
  53   | #x; napply refl
  54   | #x; (* qua manca ancora l'hint *) napply (symm plus_abelian_magma_type) ]
  55 nqed.
  56
  57 ndefinition big_plus ≝ λn.λf.big_op plus_magma_type n f O.