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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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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 CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
21 (* Properties with degree-based equivalence for terms ***********************)
23 (* Basic_2A1: was just: cpx_lleq_conf *)
24 lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
25 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
26 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
27 elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
28 /4 width=3 by tdeq_sort, deg_next, ex2_intro/
29 | #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
30 >(tdeq_inv_lref1 … H0) -H0
31 elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
32 elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
33 elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
34 elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
35 | #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
36 >(tdeq_inv_lref1 … H0) -H0
37 elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
38 elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
39 elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
40 elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
41 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
42 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
43 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
44 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
45 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
46 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
47 elim (IHT … HT02 … H1 … H2) -L0 -T0
48 /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
49 | #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
50 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
51 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
52 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
53 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
54 elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
55 /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
56 | #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
57 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
58 elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
59 elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
60 lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
61 elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
62 elim (tdeq_inv_lifts … HT1 … HUT1) -T1
63 /3 width=5 by cpx_zeta, ex2_intro/
64 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
65 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
66 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
67 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
68 elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
69 /3 width=3 by cpx_eps, ex2_intro/
70 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
71 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
72 elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
73 elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
74 elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
75 /3 width=3 by cpx_ee, ex2_intro/
76 | #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
77 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
78 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
79 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
80 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
81 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
82 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
83 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
84 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
85 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
86 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
87 /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
88 | #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
89 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
90 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
91 elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
92 elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
93 elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
94 elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
95 lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
96 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
97 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
98 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
99 elim (tdeq_lifts … HV1 … HVU1) -V1
100 /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
104 lemma cpx_tdeq_conf: ∀h,o,G,L,T0,T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
106 ∃∃T. T1 ≡[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
107 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
108 elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
109 /2 width=3 by lfxs_refl, ex2_intro/
112 lemma tdeq_cpx_trans: ∀h,o,G,L,T2,T0. T2 ≡[h, o] T0 →
113 ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
114 ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≡[h, o] T1.
115 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
116 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
120 lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
121 ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2.
122 /3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
124 lemma cpx_lleq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h o G) (lleq 0).
125 /3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
127 lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
128 ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
129 /4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.