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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "static_2/relocation/lex_lex.ma".
16 include "basic_2/rt_transition/cpm_lsubr.ma".
17 include "basic_2/rt_transition/cpr.ma".
18 include "basic_2/rt_transition/cpr_drops.ma".
19 include "basic_2/rt_transition/lpr_drops.ma".
21 (* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
23 definition IH_cpr_conf_lpr (h): relation3 genv lenv term ≝ λG,L,T.
24 ∀T1. ❨G,L❩ ⊢ T ➡[h,0] T1 → ∀T2. ❨G,L❩ ⊢ T ➡[h,0] T2 →
25 ∀L1. ❨G,L❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G,L❩ ⊢ ➡[h,0] L2 →
26 ∃∃T0. ❨G,L1❩ ⊢ T1 ➡[h,0] T0 & ❨G,L2❩ ⊢ T2 ➡[h,0] T0.
28 (* Main properties with context-sensitive parallel reduction for terms ******)
30 fact cpr_conf_lpr_atom_atom (h):
31 ∀I,G,L1,L2. ∃∃T. ❨G,L1❩ ⊢ ⓪[I] ➡[h,0] T & ❨G,L2❩ ⊢ ⓪[I] ➡[h,0] T.
32 /2 width=3 by cpr_refl, ex2_intro/ qed-.
34 fact cpr_conf_lpr_atom_delta (h):
36 ∀G,L,T. ❨G0,L0,#i❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
38 ∀K0,V0. ⇩[i] L0 ≘ K0.ⓓV0 →
39 ∀V2. ❨G0,K0❩ ⊢ V0 ➡[h,0] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
40 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
41 ∃∃T. ❨G0,L1❩ ⊢ #i ➡[h,0] T & ❨G0,L2❩ ⊢ T2 ➡[h,0] T.
42 #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
43 elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
44 elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
45 elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
46 elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
47 lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
48 lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
49 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
50 elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -V2 -HLK2 #T #HVT #HT2
51 /3 width=6 by cpm_delta_drops, ex2_intro/
54 (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
55 fact cpr_conf_lpr_delta_delta (h):
57 ∀G,L,T. ❨G0,L0,#i❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
59 ∀K0,V0. ⇩[i] L0 ≘ K0.ⓓV0 →
60 ∀V1. ❨G0,K0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ⇧[↑i] V1 ≘ T1 →
61 ∀KX,VX. ⇩[i] L0 ≘ KX.ⓓVX →
62 ∀V2. ❨G0,KX❩ ⊢ VX ➡[h,0] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
63 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
64 ∃∃T. ❨G0,L1❩ ⊢ T1 ➡[h,0] T & ❨G0,L2❩ ⊢ T2 ➡[h,0] T.
65 #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
66 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
67 lapply (drops_mono … H … HLK0) -H #H destruct
68 elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
69 elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
70 lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1
71 elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
72 elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
73 lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
74 lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
75 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
76 elim (cpm_lifts_sn … HV1 … HLK1 … HVT1) -V1 -HLK1 #T #HVT #HT1
77 /3 width=11 by cpm_lifts_bi, ex2_intro/
80 fact cpr_conf_lpr_bind_bind (h):
82 ∀G,L,T. ❨G0,L0,ⓑ[p,I]V0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
84 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ❨G0,L0.ⓑ[I]V0❩ ⊢ T0 ➡[h,0] T1 →
85 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀T2. ❨G0,L0.ⓑ[I]V0❩ ⊢ T0 ➡[h,0] T2 →
86 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
87 ∃∃T. ❨G0,L1❩ ⊢ ⓑ[p,I]V1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓑ[p,I]V2.T2 ➡[h,0] T.
88 #h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
89 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
90 elim (IH … HV01 … HV02 … HL01 … HL02) //
91 elim (IH … HT01 … HT02 (L1.ⓑ[I]V1) … (L2.ⓑ[I]V2)) -IH
92 /3 width=5 by lpr_pair, cpm_bind, ex2_intro/
95 fact cpr_conf_lpr_bind_zeta (h):
97 ∀G,L,T. ❨G0,L0,+ⓓV0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
99 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ❨G0,L0.ⓓV0❩ ⊢ T0 ➡[h,0] T1 →
100 ∀T2. ⇧[1]T2 ≘ T0 → ∀X2. ❨G0,L0❩ ⊢ T2 ➡[h,0] X2 →
101 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
102 ∃∃T. ❨G0,L1❩ ⊢ +ⓓV1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ X2 ➡[h,0] T.
103 #h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
104 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
105 elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01 [| /3 width=1 by drops_refl, drops_drop/ ] #T #HT1 #HT2
106 elim (IH … HT2 … HTX2 … HL01 … HL02) [| /2 width=1 by fqup_zeta/ ] -L0 -V0 -T0 -T2 #T2 #HT2 #HXT2
107 /3 width=3 by cpm_zeta, ex2_intro/
110 fact cpr_conf_lpr_zeta_zeta (h):
112 ∀G,L,T. ❨G0,L0,+ⓓV0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
114 ∀T1. ⇧[1] T1 ≘ T0 → ∀X1. ❨G0,L0❩ ⊢ T1 ➡[h,0] X1 →
115 ∀T2. ⇧[1] T2 ≘ T0 → ∀X2. ❨G0,L0❩ ⊢ T2 ➡[h,0] X2 →
116 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
117 ∃∃T. ❨G0,L1❩ ⊢ X1 ➡[h,0] T & ❨G0,L2❩ ⊢ X2 ➡[h,0] T.
118 #h #G0 #L0 #V0 #T0 #IH #T1 #HT10 #X1 #HTX1
119 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
120 lapply (lifts_inj … HT20 … HT10) -HT20 #H destruct
121 elim (IH … HTX1 … HTX2 … HL01 … HL02) [| /2 width=1 by fqup_zeta/ ] -L0 -V0 -T0 -T1 #X #HX1 #HX2
122 /2 width=3 by ex2_intro/
125 fact cpr_conf_lpr_flat_flat (h):
127 ∀G,L,T. ❨G0,L0,ⓕ[I]V0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
129 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ❨G0,L0❩ ⊢ T0 ➡[h,0] T1 →
130 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀T2. ❨G0,L0❩ ⊢ T0 ➡[h,0] T2 →
131 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
132 ∃∃T. ❨G0,L1❩ ⊢ ⓕ[I]V1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓕ[I]V2.T2 ➡[h,0] T.
133 #h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
134 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
135 elim (IH … HV01 … HV02 … HL01 … HL02) //
136 elim (IH … HT01 … HT02 … HL01 … HL02) //
137 /3 width=5 by cpr_flat, ex2_intro/
140 fact cpr_conf_lpr_flat_eps (h):
142 ∀G,L,T. ❨G0,L0,ⓝV0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
144 ∀V1,T1. ❨G0,L0❩ ⊢ T0 ➡[h,0] T1 → ∀T2. ❨G0,L0❩ ⊢ T0 ➡[h,0] T2 →
145 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
146 ∃∃T. ❨G0,L1❩ ⊢ ⓝV1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ T2 ➡[h,0] T.
147 #h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
148 #T2 #HT02 #L1 #HL01 #L2 #HL02
149 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
150 /3 width=3 by cpm_eps, ex2_intro/
153 fact cpr_conf_lpr_eps_eps (h):
155 ∀G,L,T. ❨G0,L0,ⓝV0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
157 ∀T1. ❨G0,L0❩ ⊢ T0 ➡[h,0] T1 → ∀T2. ❨G0,L0❩ ⊢ T0 ➡[h,0] T2 →
158 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
159 ∃∃T. ❨G0,L1❩ ⊢ T1 ➡[h,0] T & ❨G0,L2❩ ⊢ T2 ➡[h,0] T.
160 #h #G0 #L0 #V0 #T0 #IH #T1 #HT01
161 #T2 #HT02 #L1 #HL01 #L2 #HL02
162 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
163 /2 width=3 by ex2_intro/
166 fact cpr_conf_lpr_flat_beta (h):
168 ∀G,L,T. ❨G0,L0,ⓐV0.ⓛ[p]W0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
170 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ❨G0,L0❩ ⊢ ⓛ[p]W0.T0 ➡[h,0] T1 →
171 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀W2. ❨G0,L0❩ ⊢ W0 ➡[h,0] W2 → ∀T2. ❨G0,L0.ⓛW0❩ ⊢ T0 ➡[h,0] T2 →
172 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
173 ∃∃T. ❨G0,L1❩ ⊢ ⓐV1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h,0] T.
174 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
175 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
176 elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
177 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
178 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
179 elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
180 lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
181 /4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/
184 (* Basic-1: includes:
185 pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
186 pr0_cong_upsilon_cong pr0_cong_upsilon_delta
188 fact cpr_conf_lpr_flat_theta (h):
190 ∀G,L,T. ❨G0,L0,ⓐV0.ⓓ[p]W0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
192 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀T1. ❨G0,L0❩ ⊢ ⓓ[p]W0.T0 ➡[h,0] T1 →
193 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
194 ∀W2. ❨G0,L0❩ ⊢ W0 ➡[h,0] W2 → ∀T2. ❨G0,L0.ⓓW0❩ ⊢ T0 ➡[h,0] T2 →
195 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
196 ∃∃T. ❨G0,L1❩ ⊢ ⓐV1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h,0] T.
197 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
198 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
199 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
200 elim (cpm_inv_abbr1 … H) -H *
201 [ #W1 #T1 #HW01 #HT01 #H destruct
202 elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 [| /3 width=2 by drops_refl, drops_drop/ ] #U #HVU #HU2
203 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
204 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
205 /4 width=7 by cpm_bind, cpm_appl, cpm_theta, ex2_intro/
206 | #X0 #HXT0 #HX0 #H destruct
207 elim (cpm_inv_lifts_sn … HT02 (Ⓣ) … L0 … HXT0) -HT02 [| /3 width=2 by drops_refl, drops_drop/ ] #X2 #HXT2 #HX02
208 elim (IH … HX0 … HX02 … HL01 … HL02) [| /3 width=5 by fqup_strap1, fqu_drop/ ] -L0 -V0 -W0 -T0 #T #H1T #H2T
209 /4 width=8 by cpm_appl, cpm_zeta, lifts_flat, ex2_intro/
213 fact cpr_conf_lpr_beta_beta (h):
215 ∀G,L,T. ❨G0,L0,ⓐV0.ⓛ[p]W0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
217 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀W1. ❨G0,L0❩ ⊢ W0 ➡[h,0] W1 → ∀T1. ❨G0,L0.ⓛW0❩ ⊢ T0 ➡[h,0] T1 →
218 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀W2. ❨G0,L0❩ ⊢ W0 ➡[h,0] W2 → ∀T2. ❨G0,L0.ⓛW0❩ ⊢ T0 ➡[h,0] T2 →
219 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
220 ∃∃T. ❨G0,L1❩ ⊢ ⓓ[p]ⓝW1.V1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h,0] T.
221 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
222 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
223 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
224 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
225 elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
226 lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
227 lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
228 /4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
231 (* Basic_1: was: pr0_upsilon_upsilon *)
232 fact cpr_conf_lpr_theta_theta (h):
234 ∀G,L,T. ❨G0,L0,ⓐV0.ⓓ[p]W0.T0❩ ⬂+ ❨G,L,T❩ → IH_cpr_conf_lpr h G L T
236 ∀V1. ❨G0,L0❩ ⊢ V0 ➡[h,0] V1 → ∀U1. ⇧[1] V1 ≘ U1 →
237 ∀W1. ❨G0,L0❩ ⊢ W0 ➡[h,0] W1 → ∀T1. ❨G0,L0.ⓓW0❩ ⊢ T0 ➡[h,0] T1 →
238 ∀V2. ❨G0,L0❩ ⊢ V0 ➡[h,0] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
239 ∀W2. ❨G0,L0❩ ⊢ W0 ➡[h,0] W2 → ∀T2. ❨G0,L0.ⓓW0❩ ⊢ T0 ➡[h,0] T2 →
240 ∀L1. ❨G0,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G0,L0❩ ⊢ ➡[h,0] L2 →
241 ∃∃T. ❨G0,L1❩ ⊢ ⓓ[p]W1.ⓐU1.T1 ➡[h,0] T & ❨G0,L2❩ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h,0] T.
242 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
243 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
244 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
245 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
246 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
247 elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU1
248 lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -HVU2 /3 width=2 by drops_refl, drops_drop/
249 /4 width=7 by cpm_bind, cpm_appl, ex2_intro/ (**) (* full auto not tried *)
252 theorem cpr_conf_lpr (h): ∀G. lex_confluent (λL.cpm h G L 0) (λL.cpm h G L 0).
253 #h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓣ) … G0 L0 T0) -G0 -L0 -T0
254 #G #L #T #IH #G0 #L0 * [| * ]
255 [ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
256 elim (cpr_inv_atom1_drops … H1) -H1
257 elim (cpr_inv_atom1_drops … H2) -H2
259 @cpr_conf_lpr_atom_atom
260 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
261 @(cpr_conf_lpr_atom_delta … IH) -IH /width=6 by/
262 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
263 @ex2_commute @(cpr_conf_lpr_atom_delta … IH) -IH /width=6 by/
264 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
265 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
266 @(cpr_conf_lpr_delta_delta … IH) -IH /width=6 by/
268 | #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
269 elim (cpm_inv_bind1 … H1) -H1 *
270 [ #V1 #T1 #HV01 #HT01 #H1
271 | #T1 #HT10 #HTX1 #H11 #H12
273 elim (cpm_inv_bind1 … H2) -H2 *
274 [1,3: #V2 #T2 #HV02 #HT02 #H2
275 |2,4: #T2 #HT20 #HTX2 #H21 #H22
277 [ @(cpr_conf_lpr_bind_bind … IH) -IH /width=1 by/
278 | @ex2_commute @(cpr_conf_lpr_bind_zeta … IH) -IH /width=3 by/
279 | @(cpr_conf_lpr_bind_zeta … IH) -IH /width=3 by/
280 | @(cpr_conf_lpr_zeta_zeta … IH) -IH /width=3 by/
282 | #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
283 elim (cpr_inv_flat1 … H1) -H1 *
284 [ #V1 #T1 #HV01 #HT01 #H1
286 | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
287 | #p1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
289 elim (cpr_inv_flat1 … H2) -H2 *
290 [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
292 |3,7,11,15: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
293 |4,8,12,16: #p2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
295 [ @(cpr_conf_lpr_flat_flat … IH) -IH /width=1 by/
296 | @ex2_commute @(cpr_conf_lpr_flat_eps … IH) -IH /width=1 by/
297 | @ex2_commute @(cpr_conf_lpr_flat_beta … IH) -IH /width=1 by/
298 | @ex2_commute @(cpr_conf_lpr_flat_theta … IH) -IH /width=3 by/
299 | @(cpr_conf_lpr_flat_eps … IH) -IH /width=1 by/
300 | @(cpr_conf_lpr_eps_eps … IH) -IH /width=1 by/
301 | @(cpr_conf_lpr_flat_beta … IH) -IH /width=1 by/
302 | @(cpr_conf_lpr_beta_beta … IH) -IH /width=1 by/
303 | @(cpr_conf_lpr_flat_theta … IH) -IH /width=3 by/
304 | @(cpr_conf_lpr_theta_theta … IH) -IH /width=3 by/
309 (* Properties with context-sensitive parallel reduction for terms ***********)
311 lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ❨G,L0❩ ⊢ T0 ➡[h,0] T1 → ∀L1. ❨G,L0❩ ⊢ ➡[h,0] L1 →
312 ∃∃T. ❨G,L1❩ ⊢ T0 ➡[h,0] T & ❨G,L1❩ ⊢ T1 ➡[h,0] T.
313 #h #G #L0 #T0 #T1 #HT01 #L1 #HL01
314 elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01
315 /2 width=3 by ex2_intro/
318 lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ❨G,L0❩ ⊢ T0 ➡[h,0] T1 → ∀L1. ❨G,L0❩ ⊢ ➡[h,0] L1 →
319 ∃∃T. ❨G,L1❩ ⊢ T0 ➡[h,0] T & ❨G,L0❩ ⊢ T1 ➡[h,0] T.
320 #h #G #L0 #T0 #T1 #HT01 #L1 #HL01
321 elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01
322 /2 width=3 by ex2_intro/
325 (* Main properties **********************************************************)
327 theorem lpr_conf (h) (G): confluent … (lpr h 0 G).
328 /3 width=6 by lex_conf, cpr_conf_lpr/