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

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  13 (**************************************************************************)
  14
  15 (* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
  16  * Initial invocation: - Patience on me to gain peace and perfection! -
  17  *)
  18
  19 include "ground_2/star.ma".
  20 include "basic_2/notation.ma".
  21
  22 (* ATOMIC ARITY *************************************************************)
  23
  24 inductive aarity: Type[0] ≝
  25   | AAtom: aarity                   (* atomic aarity construction *)
  26   | APair: aarity → aarity → aarity (* binary aarity construction *)
  27 .
  28
  29 interpretation "aarity construction (atomic)"
  30    'Item0 = AAtom.
  31
  32 interpretation "aarity construction (binary)"
  33    'SnItem2 A1 A2 = (APair A1 A2).
  34
  35 (* Basic inversion lemmas ***************************************************)
  36
  37 lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥.
  38 #A #B elim B -B
  39 [ #H destruct
  40 | #Y #X #IHY #_ #H destruct
  41   -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
  42   /2 width=1/
  43 ]
  44 qed-.
  45
  46 lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥.
  47 #B #A elim A -A
  48 [ #H destruct
  49 | #Y #X #_ #IHX #H destruct
  50   -H (**) (* destruct: the destucted equality is not erased *)
  51   /2 width=1/
  52 ]
  53 qed-.
  54
  55 (* Basic properties *********************************************************)
  56
  57 lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
  58 #A1 elim A1 -A1
  59 [ #A2 elim A2 -A2 /2 width=1/
  60   #B2 #A2 #_ #_ @or_intror #H destruct
  61 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
  62   [ -IHB1 -IHA1 @or_intror #H destruct
  63   | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
  64     [ #H destruct elim (IHA1 A2) -IHA1
  65       [ #H destruct /2 width=1/
  66       | #HA12 @or_intror #H destruct /2 width=1/
  67       ]
  68     | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
  69     ]
  70   ]
  71 ]
  72 qed-.