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15 include "basic_2/notation/relations/lrsubeqa_3.ma".
16 include "basic_2/substitution/lsubr.ma".
17 include "basic_2/static/aaa.ma".
19 (* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
21 inductive lsuba (G:genv): relation lenv ≝
22 | lsuba_atom: lsuba G (⋆) (⋆)
23 | lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
24 | lsuba_abbr: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
25 lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
29 "local environment refinement (atomic arity assigment)"
30 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
32 (* Basic inversion lemmas ***************************************************)
34 fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
37 | #I #L1 #L2 #V #_ #H destruct
38 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
42 lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⁝⊑ L2 → L2 = ⋆.
43 /2 width=4 by lsuba_inv_atom1_aux/ qed-.
45 fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
46 (∃∃K2. G ⊢ K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
47 ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
48 G ⊢ K1 ⁝⊑ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
50 [ #J #K1 #X #H destruct
51 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
52 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9/
56 lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⁝⊑ L2 →
57 (∃∃K2. G ⊢ K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
58 ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⁝⊑ K2 &
59 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
60 /2 width=3 by lsuba_inv_pair1_aux/ qed-.
62 fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆.
65 | #I #L1 #L2 #V #_ #H destruct
66 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
70 lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⁝⊑ ⋆ → L1 = ⋆.
71 /2 width=4 by lsuba_inv_atom2_aux/ qed-.
73 fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
74 (∃∃K1. G ⊢ K1 ⁝⊑ K2 & L1 = K1.ⓑ{I}W) ∨
75 ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
76 G ⊢ K1 ⁝⊑ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
78 [ #J #K2 #U #H destruct
79 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
80 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7/
84 lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⁝⊑ K2.ⓑ{I}W →
85 (∃∃K1. G ⊢ K1 ⁝⊑ K2 & L1 = K1.ⓑ{I}W) ∨
86 ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⁝⊑ K2 &
87 I = Abst & L1 = K1.ⓓⓝW.V.
88 /2 width=3 by lsuba_inv_pair2_aux/ qed-.
90 (* Basic forward lemmas *****************************************************)
92 lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L1 ⊑ L2.
93 #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
96 (* Basic properties *********************************************************)
98 lemma lsuba_refl: ∀G,L. G ⊢ L ⁝⊑ L.
99 #G #L elim L -L // /2 width=1/