matita/matita/contribs/lambda_delta/basic_2/grammar/aarity.ma

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  14
  15 (* THE FORMAL SYSTEM λδ - MATITA SOURCE FILES
  16  * Support for abstract candidates of reducibility closed: 2012 January 27
  17  * Confluence of context-sensitive parallel reduction closed: 2011 September 21
  18  * Confluence of context-free parallel reduction closed: 2011 September 6
  19  * Specification started: 2011 April 17
  20  * - Patience on me to gain peace and perfection! -
  21  * [ suggested invocation to start formal specifications with ]
  22  *)
  23
  24 include "ground_2/star.ma".
  25 include "basic_2/notation.ma".
  26
  27 (* ATOMIC ARITY *************************************************************)
  28
  29 inductive aarity: Type[0] ≝
  30   | AAtom: aarity                   (* atomic aarity construction *)
  31   | APair: aarity → aarity → aarity (* binary aarity construction *)
  32 .
  33
  34 interpretation "aarity construction (atomic)"
  35    'Item0 = AAtom.
  36
  37 interpretation "aarity construction (binary)"
  38    'SnItem2 A1 A2 = (APair A1 A2).
  39
  40 (* Basic inversion lemmas ***************************************************)
  41
  42 lemma discr_apair_xy_x: ∀A,B. ②B. A = B → False.
  43 #A #B elim B -B
  44 [ #H destruct
  45 | #Y #X #IHY #_ #H destruct
  46   -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
  47   /2 width=1/
  48 ]
  49 qed-.
  50
  51 lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → False.
  52 #B #A elim A -A
  53 [ #H destruct
  54 | #Y #X #_ #IHX #H destruct
  55   -H (**) (* destruct: the destucted equality is not erased *)
  56   /2 width=1/
  57 ]
  58 qed-.
  59
  60 (* Basic properties *********************************************************)
  61
  62 lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
  63 #A1 elim A1 -A1
  64 [ #A2 elim A2 -A2 /2 width=1/
  65   #B2 #A2 #_ #_ @or_intror #H destruct
  66 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
  67   [ -IHB1 -IHA1 @or_intror #H destruct
  68   | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
  69     [ #H destruct elim (IHA1 A2) -IHA1
  70       [ #H destruct /2 width=1/
  71       | #HA12 @or_intror #H destruct /2 width=1/
  72       ]
  73     | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
  74     ]
  75   ]
  76 ]
  77 qed-.