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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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15 include "basic_2/relocation/lifts_tdeq.ma".
16 include "basic_2/static/lfxs_lfxs.ma".
17 include "basic_2/static/lfdeq_fqup.ma".
18 include "basic_2/rt_transition/lfpx_frees.ma".
19 include "basic_2/rt_transition/lfpx.ma". (**) (* should be in lfpx_frees.ma *)
21 (* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
23 (* Properties with degree-based equivalence for local environments **********)
25 lemma lfpx_pair_sn_split: ∀h,o,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
26 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≡[h, o, V] L2.
27 /3 width=5 by lfpx_frees_conf, lfxs_pair_sn_split/ qed-.
29 lemma lfpx_flat_dx_split: ∀h,o,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
30 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≡[h, o, T] L2.
31 /3 width=5 by lfpx_frees_conf, lfxs_flat_dx_split/ qed-.
33 lemma lfpx_bind_dx_split: ∀h,o,p,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 →
34 ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≡[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
35 /3 width=5 by lfpx_frees_conf, lfxs_bind_dx_split/ qed-.
37 lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
38 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
39 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
40 elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
41 /4 width=3 by tdeq_sort, deg_next, ex2_intro/
42 | #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
43 >(tdeq_inv_lref1 … H0) -H0
44 elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
45 elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
46 elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
47 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
48 | #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
49 >(tdeq_inv_lref1 … H0) -H0
50 elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
51 elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
52 elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
53 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
54 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
55 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
56 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
57 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
58 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
59 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
60 elim (IHT … HT02 … H1 … H2) -L0 -T0
61 /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
62 | #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
63 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
64 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
65 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
66 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
67 elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
68 /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
69 | #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
70 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
71 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
72 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
73 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
74 elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
75 elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
76 /3 width=5 by cpx_zeta, ex2_intro/
77 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
78 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
79 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
80 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
81 elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
82 /3 width=3 by cpx_eps, ex2_intro/
83 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
84 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
85 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
86 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
87 elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
88 /3 width=3 by cpx_ee, ex2_intro/
89 | #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
90 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
91 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
92 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
93 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
94 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
95 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
96 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
97 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
98 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
99 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
100 /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
101 | #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
102 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
103 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
104 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
105 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
106 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
107 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
108 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
109 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
110 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
111 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
112 elim (tdeq_lifts_sn … HV1 … HVU1) -V1
113 /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
117 lemma cpx_tdeq_conf: ∀h,o,G,L,T0,T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
119 ∃∃T. T1 ≡[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
120 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
121 elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
122 /2 width=3 by lfxs_refl, ex2_intro/
125 lemma tdeq_cpx_trans: ∀h,o,G,L,T2,T0. T2 ≡[h, o] T0 →
126 ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
127 ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≡[h, o] T1.
128 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
129 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
132 (* Basic_2A1: uses: cpx_lleq_conf *)
133 lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
134 ∀L2. L0 ≡[h, o, T0] L2 →
135 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≡[h, o] T.
136 #h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
137 elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
138 /2 width=3 by lfxs_refl, ex2_intro/
141 (* Basic_2A1: uses: lleq_cpx_trans *)
142 lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≡[h, o, T0] L0 →
143 ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
144 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≡[h, o] T1.
145 #h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
146 elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
147 /3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
150 lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
151 /3 width=6 by lfpx_frees_conf, cpx_tdeq_conf_lexs, frees_lfdeq_conf_lexs, lfxs_conf/ qed-.
153 (* Basic_2A1: uses: lleq_lpx_trans *)
154 lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
155 ∀L1. L1 ≡[h, o, T] L2 →
156 ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≡[h, o, T] K2.
157 #h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
158 elim (lfpx_lfdeq_conf … o … HLK2 L1)
159 /3 width=3 by lfdeq_sym, ex2_intro/