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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 "static_2/static/rdeq_drops.ma".
16 include "static_2/static/rdeq_fqup.ma".
17 include "static_2/static/rdeq_rdeq.ma".
18 include "basic_2/rt_transition/rpx_fsle.ma".
20 (* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
22 (* Properties with degree-based equivalence for local environments **********)
24 lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
25 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
26 /3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
28 lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
29 ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
30 /3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
32 lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
33 ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
34 /3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
36 lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
37 ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
38 /3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
40 lemma rpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (rpx h G).
41 /2 width=5 by tdeq_rex_conf/ qed-.
43 lemma rpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
44 ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
45 /2 width=5 by tdeq_rex_div/ qed-.
47 lemma cpx_tdeq_conf_sex: ∀h,o,G. R_confluent2_rex … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
48 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
49 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
50 elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
51 /4 width=3 by tdeq_sort, deg_next, ex2_intro/
52 | #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
53 >(tdeq_inv_lref1 … H0) -H0
54 elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
55 elim (rdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
56 elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
57 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
58 | #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
59 >(tdeq_inv_lref1 … H0) -H0
60 elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
61 elim (rdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
62 elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
63 elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
64 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
65 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
66 elim (rpx_inv_bind … H1) -H1 #HL01 #H1
67 elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
68 lapply (rdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
69 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
70 elim (IHT … HT02 … H1 … H2) -L0 -T0
71 /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
72 | #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
73 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
74 elim (rpx_inv_flat … H1) -H1 #HL01 #H1
75 elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
76 elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
77 elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
78 /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
79 | #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
80 elim (tdeq_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
81 elim (rpx_inv_bind … H1) -H1 #HL01 #H1
82 elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
83 lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
84 lapply (rdeq_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
85 elim (tdeq_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
86 elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
87 /3 width=5 by cpx_zeta, ex2_intro/
88 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
89 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
90 elim (rpx_inv_flat … H1) -H1 #HL01 #H1
91 elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
92 elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
93 /3 width=3 by cpx_eps, ex2_intro/
94 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
95 elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
96 elim (rpx_inv_flat … H1) -H1 #HL01 #H1
97 elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
98 elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
99 /3 width=3 by cpx_ee, ex2_intro/
100 | #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
101 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
102 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
103 elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
104 elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
105 elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
106 elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
107 lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
108 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
109 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
110 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
111 /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
112 | #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
113 elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
114 elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
115 elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
116 elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
117 elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
118 elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
119 lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
120 elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
121 elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
122 elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
123 elim (tdeq_lifts_sn … HV1 … HVU1) -V1
124 /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
128 lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
130 ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
131 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
132 elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02
133 /2 width=3 by rex_refl, ex2_intro/
136 lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
137 ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
138 ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
139 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
140 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
143 (* Basic_2A1: uses: cpx_lleq_conf *)
144 lemma cpx_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
145 ∀L2. L0 ≛[h, o, T0] L2 →
146 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
147 #h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
148 elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02
149 /2 width=3 by rex_refl, ex2_intro/
152 (* Basic_2A1: uses: lleq_cpx_trans *)
153 lemma rdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
154 ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
155 ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
156 #h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
157 elim (cpx_rdeq_conf … o … HT01 L2) -HT01
158 /3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
161 lemma rpx_rdeq_conf: ∀h,o,G,T. confluent2 … (rpx h G T) (rdeq h o T).
162 /3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-.
164 lemma rdeq_rpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
165 ∀L1. L1 ≛[h, o, T] L2 →
166 ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
167 #h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
168 elim (rpx_rdeq_conf … o … HLK2 L1)
169 /3 width=3 by rdeq_sym, ex2_intro/