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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/relocation/lifts_tdeq.ma".
16 include "basic_2/static/lfdeq_fqup.ma".
17 include "basic_2/rt_transition/lfpx.ma".
19 (* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
21 (* Properties with degree-based equivalence for local environments **********)
23 lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
24 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
25 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
26 elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
27 /4 width=3 by tdeq_sort, deg_next, ex2_intro/
28 | #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
29 >(tdeq_inv_lref1 … H0) -H0
30 elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
31 elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
32 elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
33 elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
34 | #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
35 >(tdeq_inv_lref1 … H0) -H0
36 elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
37 elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
38 elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
39 elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
40 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
41 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
42 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
43 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
44 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
45 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
46 elim (IHT … HT02 … H1 … H2) -L0 -T0
47 /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
48 | #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
49 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
50 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
51 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
52 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
53 elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
54 /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
55 | #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
56 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
57 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
58 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
59 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
60 elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
61 elim (tdeq_inv_lifts … HT1 … HUT1) -T1
62 /3 width=5 by cpx_zeta, ex2_intro/
63 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
64 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
65 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
66 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
67 elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
68 /3 width=3 by cpx_eps, ex2_intro/
69 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
70 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
71 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
72 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
73 elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
74 /3 width=3 by cpx_ee, ex2_intro/
75 | #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
76 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
77 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
78 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
79 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
80 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
81 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
82 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
83 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
84 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
85 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
86 /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
87 | #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
88 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
89 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
90 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
91 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
92 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
93 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
94 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
95 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
96 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
97 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
98 elim (tdeq_lifts … HV1 … HVU1) -V1
99 /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
103 lemma cpx_tdeq_conf: ∀h,o,G,L,T0,T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
105 ∃∃T. T1 ≡[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
106 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
107 elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
108 /2 width=3 by lfxs_refl, ex2_intro/
111 lemma tdeq_cpx_trans: ∀h,o,G,L,T2,T0. T2 ≡[h, o] T0 →
112 ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
113 ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≡[h, o] T1.
114 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
115 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
118 (* Basic_2A1: was just: cpx_lleq_conf *)
119 lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
120 ∀L2. L0 ≡[h, o, T0] L2 →
121 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≡[h, o] T.
122 #h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
123 elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
124 /2 width=3 by lfxs_refl, ex2_intro/
127 (* Basic_2A1: was just: lleq_cpx_trans *)
128 lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≡[h, o, T0] L0 →
129 ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
130 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≡[h, o] T1.
131 #h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
132 elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
133 /3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
136 include "basic_2/static/lfxs_lfxs.ma".
138 axiom lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
140 #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
143 (* Basic_2A1: was just: lleq_lpx_trans *)
144 lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
145 ∀L1. L1 ≡[h, o, T] L2 →
146 ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≡[h, o, T] K2.
147 #h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
148 elim (lfpx_lfdeq_conf … o … HLK2 L1)
149 /3 width=3 by lfdeq_sym, ex2_intro/
152 (* Properties with supclosure ***********************************************)
154 lemma lpx_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
155 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
156 ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
157 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
158 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
159 #K0 #V0 #H1KL1 #_ #H destruct
160 elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
161 #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
162 /2 width=4 by fqu_lref_O, ex3_intro/
163 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
164 [ elim (lleq_inv_bind … H)
165 | elim (lleq_inv_flat … H)
166 ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
167 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
168 /3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
169 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
170 /2 width=4 by fqu_flat_dx, ex3_intro/
171 | #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
172 elim (drop_O1_le (Ⓕ) (k+1) K1)
173 [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
174 #H2KL elim (lpx_drop_trans_O1 … H1KL1 … HL1) -L1
175 #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
176 /3 width=4 by fqu_drop, ex3_intro/
177 | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
178 lapply (lleq_fwd_length … H2KL1) //
183 lemma lpx_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
184 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
185 ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
186 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
187 elim (fquq_inv_gen … H) -H
188 [ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
189 /3 width=4 by fqu_fquq, ex3_intro/
190 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
194 lemma lpx_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
195 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
196 ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
197 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
198 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
199 /3 width=4 by fqu_fqup, ex3_intro/
200 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
201 #K #HT1 #H1KL #H2KL elim (lpx_lleq_fqu_trans … HT2 … H1KL H2KL) -L
202 /3 width=5 by fqup_strap1, ex3_intro/
206 lemma lpx_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
207 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
208 ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
209 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
210 elim (fqus_inv_gen … H) -H
211 [ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
212 /3 width=4 by fqup_fqus, ex3_intro/
213 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/