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15 include "basic_2/notation/relations/lrsubeqd_5.ma".
16 include "basic_2/static/lsubr.ma".
17 include "basic_2/static/da.ma".
19 (* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
21 inductive lsubd (h) (o) (G): relation lenv ≝
22 | lsubd_atom: lsubd h o G (⋆) (⋆)
23 | lsubd_pair: ∀I,L1,L2,V. lsubd h o G L1 L2 →
24 lsubd h o G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
25 | lsubd_beta: ∀L1,L2,W,V,d. ⦃G, L1⦄ ⊢ V ▪[h, o] d+1 → ⦃G, L2⦄ ⊢ W ▪[h, o] d →
26 lsubd h o G L1 L2 → lsubd h o G (L1.ⓓⓝW.V) (L2.ⓛW)
30 "local environment refinement (degree assignment)"
31 'LRSubEqD h o G L1 L2 = (lsubd h o G L1 L2).
33 (* Basic forward lemmas *****************************************************)
35 lemma lsubd_fwd_lsubr: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L1 ⫃ L2.
36 #h #o #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
39 (* Basic inversion lemmas ***************************************************)
41 fact lsubd_inv_atom1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L1 = ⋆ → L2 = ⋆.
42 #h #o #G #L1 #L2 * -L1 -L2
44 | #I #L1 #L2 #V #_ #H destruct
45 | #L1 #L2 #W #V #d #_ #_ #_ #H destruct
49 lemma lsubd_inv_atom1: ∀h,o,G,L2. G ⊢ ⋆ ⫃▪[h, o] L2 → L2 = ⋆.
50 /2 width=6 by lsubd_inv_atom1_aux/ qed-.
52 fact lsubd_inv_pair1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
53 ∀I,K1,X. L1 = K1.ⓑ{I}X →
54 (∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
55 ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
57 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
58 #h #o #G #L1 #L2 * -L1 -L2
59 [ #J #K1 #X #H destruct
60 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
61 | #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
65 lemma lsubd_inv_pair1: ∀h,o,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃▪[h, o] L2 →
66 (∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
67 ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
69 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
70 /2 width=3 by lsubd_inv_pair1_aux/ qed-.
72 fact lsubd_inv_atom2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L2 = ⋆ → L1 = ⋆.
73 #h #o #G #L1 #L2 * -L1 -L2
75 | #I #L1 #L2 #V #_ #H destruct
76 | #L1 #L2 #W #V #d #_ #_ #_ #H destruct
80 lemma lsubd_inv_atom2: ∀h,o,G,L1. G ⊢ L1 ⫃▪[h, o] ⋆ → L1 = ⋆.
81 /2 width=6 by lsubd_inv_atom2_aux/ qed-.
83 fact lsubd_inv_pair2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
84 ∀I,K2,W. L2 = K2.ⓑ{I}W →
85 (∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
86 ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
87 G ⊢ K1 ⫃▪[h, o] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
88 #h #o #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 #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
95 lemma lsubd_inv_pair2: ∀h,o,I,G,L1,K2,W. G ⊢ L1 ⫃▪[h, o] K2.ⓑ{I}W →
96 (∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
97 ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
98 G ⊢ K1 ⫃▪[h, o] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
99 /2 width=3 by lsubd_inv_pair2_aux/ qed-.
101 (* Basic properties *********************************************************)
103 lemma lsubd_refl: ∀h,o,G,L. G ⊢ L ⫃▪[h, o] L.
104 #h #o #G #L elim L -L /2 width=1 by lsubd_pair/
107 (* Note: the constant 0 cannot be generalized *)
108 lemma lsubd_drop_O1_conf: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
109 ∀K1,c,k. ⬇[c, 0, k] L1 ≡ K1 →
110 ∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & ⬇[c, 0, k] L2 ≡ K2.
111 #h #o #G #L1 #L2 #H elim H -L1 -L2
112 [ /2 width=3 by ex2_intro/
113 | #I #L1 #L2 #V #_ #IHL12 #K1 #c #k #H
114 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
116 elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
117 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
118 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
120 | #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #c #k #H
121 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
123 elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
124 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
125 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
130 (* Note: the constant 0 cannot be generalized *)
131 lemma lsubd_drop_O1_trans: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
132 ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
133 ∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & ⬇[c, 0, k] L1 ≡ K1.
134 #h #o #G #L1 #L2 #H elim H -L1 -L2
135 [ /2 width=3 by ex2_intro/
136 | #I #L1 #L2 #V #_ #IHL12 #K2 #c #k #H
137 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
139 elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
140 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
141 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
143 | #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #c #k #H
144 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
146 elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
147 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
148 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/