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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/equivalence/lsubse.ma".
16 include "basic_2/dynamic/snv.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
20 (* Note: this is not transitive *)
21 inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
22 | lsubsv_atom: lsubsv h g (⋆) (⋆)
23 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
24 lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
25 | lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → ⦃h, L1⦄ ⊢ V1 •[g, l+1] W1 →
26 L1 ⊢ W1 ⬌* W2 → ⦃h, L2⦄ ⊩ W2 :[g] → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
27 lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2)
31 "local environment refinement (stratified native validity)"
32 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2).
34 (* Basic inversion lemmas ***************************************************)
36 fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
37 #h #g #L1 #L2 * -L1 -L2
39 | #I #L1 #L2 #V #_ #H destruct
40 | #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
44 lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆.
45 /2 width=5 by lsubsv_inv_atom1_aux/ qed-.
47 fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
48 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
49 (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
50 ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
51 K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
52 h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
53 #h #g #L1 #L2 * -L1 -L2
54 [ #J #K1 #U1 #H destruct
55 | #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
56 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=11/
60 lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 →
61 (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
62 ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
63 K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
64 h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
65 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
67 fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
68 #h #g #L1 #L2 * -L1 -L2
70 | #I #L1 #L2 #V #_ #H destruct
71 | #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
75 lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆.
76 /2 width=5 by lsubsv_inv_atom2_aux/ qed-.
78 fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
79 ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
80 (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
81 ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
82 K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
83 h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
84 #h #g #L1 #L2 * -L1 -L2
85 [ #J #K2 #U2 #H destruct
86 | #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
87 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
91 lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 →
92 (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
93 ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
94 K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
95 h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
96 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
98 (* Basic_forward lemmas *****************************************************)
100 lemma lsubsv_fwd_lsubse: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → h ⊢ L1 ⊢•⊑[g] L2.
101 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ /2 width=6/
104 lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2.
105 /3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs1/
108 lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2.
109 /3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs2/
112 (* Basic properties *********************************************************)
114 lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L.
115 #h #g #L elim L -L // /2 width=1/
118 lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
119 ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
120 /3 width=5 by lsubsv_fwd_lsubse, lsubse_cprs_trans/