matita/matita/contribs/lambda_delta/Basic_2/grammar/aarity.ma

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  13 (**************************************************************************)
  14
  15 include "Ground_2/list.ma".
  16 include "Basic_2/notation.ma".
  17
  18 (* ATOMIC ARITY *************************************************************)
  19
  20 inductive aarity: Type[0] ≝
  21   | AAtom: aarity                   (* atomic aarity construction *)
  22   | APair: aarity → aarity → aarity (* binary aarity construction *)
  23 .
  24
  25 interpretation "aarity construction (atomic)" 'SItem = AAtom.
  26
  27 interpretation "aarity construction (binary)" 'SItem A1 A2 = (APair A1 A2).
  28
  29 (* Basic inversion lemmas ***************************************************)
  30
  31 lemma discr_apair_xy_x: ∀A,B. 𝕔 B. A = B → False.
  32 #A #B elim B -B
  33 [ #H destruct
  34 | #Y #X #IHY #_ #H destruct
  35   -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
  36   /2 width=1/
  37 ]
  38 qed-.
  39
  40 lemma discr_tpair_xy_y: ∀B,A. 𝕔 B. A = A → False.
  41 #B #A elim A -A
  42 [ #H destruct
  43 | #Y #X #_ #IHX #H destruct
  44   -H (**) (* destruct: the destucted equality is not erased *)
  45   /2 width=1/
  46 ]
  47 qed-.
  48
  49 (* Basic properties *********************************************************)
  50
  51 lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
  52 #A1 elim A1 -A1
  53 [ #A2 elim A2 -A2 /2 width=1/
  54   #B2 #A2 #_ #_ @or_intror #H destruct
  55 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
  56   [ -IHB1 -IHA1 @or_intror #H destruct
  57   | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
  58     [ #H destruct elim (IHA1 A2) -IHA1
  59       [ #H destruct /2 width=1/
  60       | #HA12 @or_intror #H destruct /2 width=1/
  61       ]
  62     | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
  63     ]
  64   ]
  65 ]
  66 qed-.