matita/matita/contribs/lambda_delta/Basic_2/computation/lsubc.ma

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  14
  15 include "Basic_2/computation/acp_cr.ma".
  16
  17 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
  18
  19 inductive lsubc (RP:lenv→predicate term): relation lenv ≝
  20 | lsubc_atom: lsubc RP (⋆) (⋆)
  21 | lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. 𝕓{I} V) (L2. 𝕓{I} V)
  22 | lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → ⦃L2, W⦄ [RP] ϵ 〚A〛 →
  23               lsubc RP L1 L2 → lsubc RP (L1. 𝕓{Abbr} V) (L2. 𝕓{Abst} W)
  24 .
  25
  26 interpretation
  27   "local environment refinement (abstract candidates of reducibility)"
  28   'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
  29
  30 (* Basic inversion lemmas ***************************************************)
  31
  32 fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. 𝕓{I} W →
  33                           (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. 𝕓{I} W) ∨
  34                           ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & ⦃K2, W⦄ [RP] ϵ 〚A〛 &
  35                                     K1 [RP] ⊑ K2 & L1 = K1. 𝕓{Abbr} V &
  36                                     I = Abst.
  37 #RP #L1 #L2 * -L1 -L2
  38 [ #I #K2 #W #H destruct
  39 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
  40 | #L1 #L2 #V1 #W2 #A #H #HV1 #HW2 #I #K2 #W #H destruct /3 width=7/
  41 ]
  42 qed.
  43
  44 lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. 𝕓{I} W →
  45                        (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. 𝕓{I} W) ∨
  46                        ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & ⦃K2, W⦄ [RP] ϵ 〚A〛 &
  47                                  K1 [RP] ⊑ K2 & L1 = K1. 𝕓{Abbr} V &
  48                                  I = Abst.
  49 /2 width=3/ qed-.
  50
  51 (* Basic properties *********************************************************)
  52
  53 lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
  54 #RP #L elim L -L // /2 width=1/
  55 qed.