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15 include "basic_2/static/lfdeq_fqup.ma".
16 include "basic_2/static/lfdeq_lfdeq.ma".
17 include "basic_2/rt_transition/lfpx_fsle.ma".
19 (* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
21 (* Properties with degree-based equivalence for local environments **********)
23 lemma lfpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
24 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
25 /3 width=5 by lfpx_fsge_comp, lfxs_pair_sn_split/ qed-.
27 lemma lfpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
28 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
29 /3 width=5 by lfpx_fsge_comp, lfxs_flat_dx_split/ qed-.
31 lemma lfpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
32 ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
33 /3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split/ qed-.
35 lemma lfpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
36 ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
37 /3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split_void/ qed-.
39 lemma lfpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (lfpx h G).
40 /2 width=5 by tdeq_lfxs_conf/ qed-.
42 lemma lfpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
43 ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
44 /2 width=5 by tdeq_lfxs_div/ qed-.
46 lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
47 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
48 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
49 elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
50 /4 width=3 by tdeq_sort, deg_next, ex2_intro/
51 | #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
52 >(tdeq_inv_lref1 … H0) -H0
53 elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
54 elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
55 elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
56 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
57 | #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
58 >(tdeq_inv_lref1 … H0) -H0
59 elim (lfpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
60 elim (lfdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
61 elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
62 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
63 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
64 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
65 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
66 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
67 lapply (lfdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
68 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
69 elim (IHT … HT02 … H1 … H2) -L0 -T0
70 /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
71 | #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
72 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
73 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
74 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
75 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
76 elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
77 /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
78 | #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
79 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
80 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
81 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
82 lapply (lfdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
83 elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
84 elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
85 /3 width=5 by cpx_zeta, ex2_intro/
86 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
87 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
88 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
89 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
90 elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
91 /3 width=3 by cpx_eps, ex2_intro/
92 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
93 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
94 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
95 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
96 elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
97 /3 width=3 by cpx_ee, ex2_intro/
98 | #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
99 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
100 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
101 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
102 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
103 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
104 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
105 lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
106 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
107 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
108 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
109 /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
110 | #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
111 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
112 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
113 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
114 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
115 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
116 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
117 lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
118 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
119 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
120 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
121 elim (tdeq_lifts_sn … HV1 … HVU1) -V1
122 /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
126 lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
128 ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
129 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
130 elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
131 /2 width=3 by lfxs_refl, ex2_intro/
134 lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
135 ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
136 ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
137 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
138 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
141 (* Basic_2A1: uses: cpx_lleq_conf *)
142 lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
143 ∀L2. L0 ≛[h, o, T0] L2 →
144 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
145 #h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
146 elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
147 /2 width=3 by lfxs_refl, ex2_intro/
150 (* Basic_2A1: uses: lleq_cpx_trans *)
151 lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
152 ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
153 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
154 #h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
155 elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
156 /3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
159 lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
160 /3 width=6 by lfpx_fsge_comp, lfdeq_fsge_comp, cpx_tdeq_conf_lexs, lfxs_conf/ qed-.
162 lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
163 ∀L1. L1 ≛[h, o, T] L2 →
164 ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
165 #h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
166 elim (lfpx_lfdeq_conf … o … HLK2 L1)
167 /3 width=3 by lfdeq_sym, ex2_intro/