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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "Basic_2/static/aaa.ma".
16 include "Basic_2/computation/acp_cr.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
20 inductive lsubc (RP:lenv→predicate term): relation lenv ≝
21 | lsubc_atom: lsubc RP (⋆) (⋆)
22 | lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. 𝕓{I} V) (L2. 𝕓{I} V)
23 | lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ÷ A →
24 lsubc RP L1 L2 → lsubc RP (L1. 𝕓{Abbr} V) (L2. 𝕓{Abst} W)
28 "local environment refinement (abstract candidates of reducibility)"
29 'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
31 (* Basic inversion lemmas ***************************************************)
33 fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. 𝕓{I} W →
34 (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. 𝕓{I} W) ∨
35 ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
37 L1 = K1. 𝕓{Abbr} V & I = Abst.
39 [ #I #K2 #W #H destruct
40 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
41 | #L1 #L2 #V1 #W2 #A #H #HV1 #HW2 #I #K2 #W #H destruct /3 width=7/
45 lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. 𝕓{I} W →
46 (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. 𝕓{I} W) ∨
47 ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
49 L1 = K1. 𝕓{Abbr} V & I = Abst.
52 (* Basic properties *********************************************************)
54 lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
55 #RP #L elim L -L // /2 width=1/