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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground/xoa/ex_7_3.ma".
16 include "ground/xoa/ex_8_4.ma".
17 include "basic_2A/notation/relations/lrsubeqv_5.ma".
18 include "basic_2A/dynamic/shnv.ma".
20 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
22 (* Note: this is not transitive *)
23 inductive lsubsv (h) (g) (G): relation lenv ≝
24 | lsubsv_atom: lsubsv h g G (⋆) (⋆)
25 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g G L1 L2 →
26 lsubsv h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
27 | lsubsv_beta: ∀L1,L2,W,V,d1. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, g, d1] → ⦃G, L2⦄ ⊢ W ¡[h, g] →
28 ⦃G, L1⦄ ⊢ V ▪[h, g] d1+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d1 →
29 lsubsv h g G L1 L2 → lsubsv h g G (L1.ⓓⓝW.V) (L2.ⓛW)
33 "local environment refinement (stratified native validity)"
34 'LRSubEqV h g G L1 L2 = (lsubsv h g G L1 L2).
36 (* Basic inversion lemmas ***************************************************)
38 fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 = ⋆ → L2 = ⋆.
39 #h #g #G #L1 #L2 * -L1 -L2
41 | #I #L1 #L2 #V #_ #H destruct
42 | #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
46 lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃¡[h, g] L2 → L2 = ⋆.
47 /2 width=6 by lsubsv_inv_atom1_aux/ qed-.
49 fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
50 ∀I,K1,X. L1 = K1.ⓑ{I}X →
51 (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
52 ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
53 ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
55 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
56 #h #g #G #L1 #L2 * -L1 -L2
57 [ #J #K1 #X #H destruct
58 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
59 | #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K1 #X #H destruct /3 width=11 by or_intror, ex8_4_intro/
63 lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃¡[h, g] L2 →
64 (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
65 ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
66 ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
68 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
69 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
71 fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L2 = ⋆ → L1 = ⋆.
72 #h #g #G #L1 #L2 * -L1 -L2
74 | #I #L1 #L2 #V #_ #H destruct
75 | #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
79 lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃¡[h, g] ⋆ → L1 = ⋆.
80 /2 width=6 by lsubsv_inv_atom2_aux/ qed-.
82 fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
83 ∀I,K2,W. L2 = K2.ⓑ{I}W →
84 (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
85 ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
86 ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
87 G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
88 #h #g #G #L1 #L2 * -L1 -L2
89 [ #J #K2 #U #H destruct
90 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
91 | #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K2 #U #H destruct /3 width=8 by or_intror, ex7_3_intro/
95 lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃¡[h, g] K2.ⓑ{I}W →
96 (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
97 ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
98 ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
99 G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
100 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
102 (* Basic forward lemmas *****************************************************)
104 lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 ⫃ L2.
105 #h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
108 (* Basic properties *********************************************************)
110 lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ⫃¡[h, g] L.
111 #h #g #G #L elim L -L /2 width=1 by lsubsv_pair/
114 lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
115 ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
116 /3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
119 (* Note: the constant 0 cannot be generalized *)
120 lemma lsubsv_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
121 ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
122 ∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
123 #h #g #G #L1 #L2 #H elim H -L1 -L2
124 [ /2 width=3 by ex2_intro/
125 | #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
126 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
128 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
129 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
130 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
132 | #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #s #m #H
133 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
135 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
136 <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
137 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
142 (* Note: the constant 0 cannot be generalized *)
143 lemma lsubsv_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
144 ∀K2,s, m. ⬇[s, 0, m] L2 ≡ K2 →
145 ∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
146 #h #g #G #L1 #L2 #H elim H -L1 -L2
147 [ /2 width=3 by ex2_intro/
148 | #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
149 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
151 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
152 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
153 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
155 | #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #s #m #H
156 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
158 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
159 <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
160 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/