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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/notation/relations/lsubeqx_6.ma".
16 include "basic_2/rt_computation/lfsx.ma".
18 (* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNCOUNTED RT-TRANSITION ********)
20 (* Note: this should be an instance of a more general lexs *)
21 (* Basic_2A1: uses: lcosx *)
22 inductive lsubsx (h) (o) (G): rtmap → relation lenv ≝
23 | lsubsx_atom: ∀f. lsubsx h o G f (⋆) (⋆)
24 | lsubsx_push: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
25 lsubsx h o G (↑f) (K1.ⓘ{I}) (K2.ⓘ{I})
26 | lsubsx_unit: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
27 lsubsx h o G (⫯f) (K1.ⓤ{I}) (K2.ⓧ)
28 | lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ →
29 lsubsx h o G f K1 K2 → lsubsx h o G (⫯f) (K1.ⓑ{I}V) (K2.ⓧ)
33 "local environment refinement (clear)"
34 'LSubEqX h o f G L1 L2 = (lsubsx h o G f L1 L2).
36 (* Basic inversion lemmas ***************************************************)
38 fact lsubsx_inv_atom_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
40 #h #o #g #G #L1 #L2 * -g -L1 -L2 //
41 [ #f #I #K1 #K2 #_ #H destruct
42 | #f #I #K1 #K2 #_ #H destruct
43 | #f #I #K1 #K2 #V #_ #_ #H destruct
47 lemma lsubsx_inv_atom_sn: ∀h,o,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, o, g] L2 → L2 = ⋆.
48 /2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
50 fact lsubsx_inv_push_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
51 ∀f,I,K1. g = ↑f → L1 = K1.ⓘ{I} →
52 ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
53 #h #o #g #G #L1 #L2 * -g -L1 -L2
54 [ #f #g #J #L1 #_ #H destruct
55 | #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
56 <(injective_push … H1) -g /2 width=3 by ex2_intro/
57 | #f #I #K1 #K2 #_ #g #J #L1 #H
58 elim (discr_next_push … H)
59 | #f #I #K1 #K2 #V #_ #_ #g #J #L1 #H
60 elim (discr_next_push … H)
64 lemma lsubsx_inv_push_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, o, ↑f] L2 →
65 ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
66 /2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
68 fact lsubsx_inv_unit_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
69 ∀f,I,K1. g = ⫯f → L1 = K1.ⓤ{I} →
70 ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
71 #h #o #g #G #L1 #L2 * -g -L1 -L2
72 [ #f #g #J #L1 #_ #H destruct
73 | #f #I #K1 #K2 #_ #g #J #L1 #H
74 elim (discr_push_next … H)
75 | #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
76 <(injective_next … H1) -g /2 width=3 by ex2_intro/
77 | #f #I #K1 #K2 #V #_ #_ #g #J #L1 #_ #H destruct
81 lemma lsubsx_inv_unit_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, o, ⫯f] L2 →
82 ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
83 /2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
85 fact lsubsx_inv_pair_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
86 ∀f,I,K1,V. g = ⫯f → L1 = K1.ⓑ{I}V →
87 ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
88 G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
89 #h #o #g #G #L1 #L2 * -g -L1 -L2
90 [ #f #g #J #L1 #W #_ #H destruct
91 | #f #I #K1 #K2 #_ #g #J #L1 #W #H
92 elim (discr_push_next … H)
93 | #f #I #K1 #K2 #_ #g #J #L1 #W #_ #H destruct
94 | #f #I #K1 #K2 #V #HV #HK12 #g #J #L1 #W #H1 #H2 destruct
95 <(injective_next … H1) -g /2 width=4 by ex3_intro/
99 (* Basic_2A1: uses: lcosx_inv_pair *)
100 lemma lsubsx_inv_pair_sn: ∀h,o,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, ⫯f] L2 →
101 ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
102 G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
103 /2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
105 (* Advanced inversion lemmas ************************************************)
107 lemma lsubsx_inv_pair_sn_gen: ∀h,o,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, g] L2 →
108 ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ↑f & L2 = K2.ⓑ{I}V
109 | ∃∃f,K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
110 G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ⫯f & L2 = K2.ⓧ.
111 #h #o #g #I #G #K1 #L2 #V #H
112 elim (pn_split g) * #f #Hf destruct
113 [ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
114 | elim (lsubsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
118 (* Advanced forward lemmas **************************************************)
120 lemma lsubsx_fwd_bind_sn: ∀h,o,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, o, g] L2 →
121 ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, o, ⫱g] K2 & L2 = K2.ⓘ{I2}.
122 #h #o #g #I1 #G #K1 #L2
123 elim (pn_split g) * #f #Hf destruct
124 [ #H elim (lsubsx_inv_push_sn … H) -H
126 [ #H elim (lsubsx_inv_unit_sn … H) -H
127 | #V #H elim (lsubsx_inv_pair_sn … H) -H
130 /2 width=4 by ex2_2_intro/
133 (* Basic properties *********************************************************)
135 lemma lsubsx_eq_repl_back: ∀h,o,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
136 #h #o #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
137 [ #f #I #L1 #L2 #_ #IH #x #H
138 elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
139 | #f #I #L1 #L2 #_ #IH #x #H
140 elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_unit/
141 | #f #I #L1 #L2 #V #HV #_ #IH #x #H
142 elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_pair/
146 lemma lsubsx_eq_repl_fwd: ∀h,o,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
147 #h #o #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
150 (* Advanced properties ******************************************************)
152 (* Basic_2A1: uses: lcosx_O *)
153 lemma lsubsx_refl: ∀h,o,f,G. 𝐈⦃f⦄ → reflexive … (lsubsx h o G f).
154 #h #o #f #G #Hf #L elim L -L
155 /3 width=3 by lsubsx_eq_repl_back, lsubsx_push, eq_push_inv_isid/
158 (* Basic_2A1: removed theorems 2:
159 lcosx_drop_trans_lt lcosx_inv_succ