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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/dynamic/snv.ma".
17 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
19 (* Note: this is not transitive *)
20 inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
21 | lsubsv_atom: lsubsv h g (⋆) (⋆)
22 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
23 lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
24 | lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → L1 ⊢ W2 ⬌* W1 →
25 ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
26 lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2)
30 "local environment refinement (stratified native validity)"
31 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2).
33 (* Basic inversion lemmas ***************************************************)
35 fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
36 #h #g #L1 #L2 * -L1 -L2
38 | #I #L1 #L2 #V #_ #H destruct
39 | #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
43 lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆.
44 /2 width=5 by lsubsv_inv_atom1_aux/ qed-.
46 fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
47 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
48 (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
49 ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
50 K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
51 #h #g #L1 #L2 * -L1 -L2
52 [ #J #K1 #U1 #H destruct
53 | #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
54 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/
58 lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 →
59 (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
60 ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
61 K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
62 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
64 fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
65 #h #g #L1 #L2 * -L1 -L2
67 | #I #L1 #L2 #V #_ #H destruct
68 | #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
72 lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆.
73 /2 width=5 by lsubsv_inv_atom2_aux/ qed-.
75 fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
76 ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
77 (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
78 ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
79 K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
80 #h #g #L1 #L2 * -L1 -L2
81 [ #J #K2 #U2 #H destruct
82 | #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
83 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
87 lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 →
88 (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
89 ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
90 K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
91 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
93 (* Basic_forward lemmas *****************************************************)
95 lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2.
96 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
99 lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2.
100 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
103 (* Basic properties *********************************************************)
105 lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L.
106 #h #g #L elim L -L // /2 width=1/
109 lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
110 ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
111 /3 width=5 by lsubsv_fwd_lsubs2, cprs_lsubs_trans/