matitaB/matita/contribs/ng_assembly2/common/sigma.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* ********************************************************************** *)
  16 (*                          Progetto FreeScale                            *)
  17 (*                                                                        *)
  18 (*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
  19 (*   Sviluppo: 2008-2010                                                  *)
  20 (*                                                                        *)
  21 (* ********************************************************************** *)
  22
  23 include "common/theory.ma".
  24
  25 (* coppia dipendente *)
  26
  27 ninductive sigma (A:Type) (P:A → Type) : Type ≝
  28     sigma_intro: ∀x:A.P x → sigma A P.
  29
  30 notation < "hvbox(\Sigma ident i opt (: tx) break . p)"
  31   right associative with precedence 20
  32 for @{ 'Sigma ${default
  33   @{\lambda ${ident i} : $tx. $p}
  34   @{\lambda ${ident i} . $p}}}.
  35
  36 notation > "\Sigma list1 ident x sep , opt (: T). term 19 Px"
  37   with precedence 20
  38   for ${ default
  39           @{ ${ fold right @{$Px} rec acc @{'Sigma (λ${ident x}:$T.$acc)} } }
  40           @{ ${ fold right @{$Px} rec acc @{'Sigma (λ${ident x}.$acc)} } }
  41        }.
  42
  43 notation "\ll term 19 a, break term 19 b \gg"
  44 with precedence 90 for @{'dependent_pair (λx:?.? x) $a $b}.
  45 interpretation "dependent pair" 'dependent_pair \eta.c a b = (sigma_intro ? c a b).
  46
  47 interpretation "sigma" 'Sigma \eta.x = (sigma ? x).
  48
  49 ndefinition sigmaFst ≝
  50 λT:Type.λf:T → Type.λs:sigma T f.match s with [ sigma_intro x _ ⇒ x ].
  51 ndefinition sigmaSnd ≝
  52 λT:Type.λf:T → Type.λs:sigma T f.match s return λs.f (sigmaFst ?? s) with [ sigma_intro _ x ⇒ x ].