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

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