1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground_2/lib/arith_2b.ma".
16 include "basic_2/rt_transition/lpr_lpr.ma".
17 include "basic_2/rt_computation/cpms_lsubr.ma".
18 include "basic_2/rt_computation/cpms_fpbg.ma".
19 include "basic_2/rt_computation/cpms_cpms.ma".
20 include "basic_2/dynamic/cnv_drops.ma".
21 include "basic_2/dynamic/cnv_preserve_far.ma".
23 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
25 (* Far diamond propery with t-bound rt-transition for terms *****************)
27 fact cnv_cpm_conf_lpr_atom_atom_aux (h) (G) (L1) (L2) (I):
28 ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
29 /2 width=3 by ex2_intro/ qed-.
31 fact cnv_cpm_conf_lpr_atom_ess_aux (h) (G) (L1) (L2) (s):
32 ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡*[1,h] T & ⦃G,L2⦄ ⊢ ⋆(next h s) ➡*[h] T.
33 /3 width=3 by cpm_cpms, ex2_intro/ qed-.
35 fact cnv_cpm_conf_lpr_atom_delta_aux (a) (h) (o) (G) (L) (i):
36 (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
39 ∀n,XV. ⦃G,K⦄ ⊢ V ➡[n,h] XV →
41 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
42 ∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
43 #a #h #o #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
44 lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
45 elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
46 elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #HV1 #H destruct
47 elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
48 elim (lpr_inv_pair_sn … H2) -H2 #K2 #V2 #HK2 #_ #H destruct
49 lapply (drops_isuni_fwd_drop2 … HLK2) -V2 // #HLK2
50 lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
51 elim (cnv_cpm_conf_lpr_far … IH … HV1 … HVX … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V
52 <minus_O_n <minus_n_O #V #HV1 #HVX
53 elim (cpms_lifts_sn … HVX … HLK2 … HXV) -XV -HLK2 #XV #HVX #HXV
54 /3 width=6 by cpms_delta_drops, ex2_intro/
57 fact cnv_cpm_conf_lpr_atom_ell_aux (a) (h) (o) (G) (L) (i):
58 (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
61 ∀n,XW. ⦃G,K⦄ ⊢ W ➡[n,h] XW →
63 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
64 ∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
65 #a #h #o #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
66 lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW
67 elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
68 elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK1 #HW1 #H destruct
69 elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
70 elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK2 #_ #H destruct
71 lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
72 lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
73 elim (cnv_cpm_conf_lpr_far … IH … HW1 … HWX … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -W
74 <minus_O_n <minus_n_O #W #HW1 #HWX
75 elim (cpms_lifts_sn … HWX … HLK2 … HXW) -XW -HLK2 #XW #HWX #HXW
76 /3 width=6 by cpms_ell_drops, ex2_intro/
79 fact cnv_cpm_conf_lpr_delta_delta_aux (a) (h) (o) (I) (G) (L) (i):
80 (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
82 ∀K1,V1. ⬇*[i]L ≘ K1.ⓑ{I}V1 → ∀K2,V2. ⬇*[i]L ≘ K2.ⓑ{I}V2 →
83 ∀n1,XV1. ⦃G,K1⦄ ⊢ V1 ➡[n1,h] XV1 → ∀n2,XV2. ⦃G,K2⦄ ⊢ V2 ➡[n2,h] XV2 →
84 ∀X1. ⬆*[↑i]XV1 ≘ X1 → ∀X2. ⬆*[↑i]XV2 ≘ X2 →
85 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
86 ∃∃T. ⦃G,L1⦄ ⊢ X1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ X2 ➡*[n1-n2,h] T.
87 #a #h #o #I #G #L #i #IH #HT
88 #K #V #HLK #Y #X #HLY #n1 #XV1 #HVX1 #n2 #XV2 #HVX2 #X1 #HXV1 #X2 #HXV2
90 lapply (drops_mono … HLY … HLK) -HLY #H destruct
91 lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
92 elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
93 elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #_ #H destruct
94 lapply (drops_isuni_fwd_drop2 … HLK1) -V1 // #HLK1
95 elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
96 elim (lpr_inv_pair_sn … H2) -H2 #K2 #V2 #HK2 #_ #H destruct
97 lapply (drops_isuni_fwd_drop2 … HLK2) -V2 // #HLK2
98 lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
99 elim (cnv_cpm_conf_lpr_far … IH … HVX1 … HVX2 … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V
101 elim (cpms_lifts_sn … HVX1 … HLK1 … HXV1) -XV1 -HLK1 #W1 #HVW1 #HXW1
102 /3 width=11 by cpms_lifts_bi, ex2_intro/
105 fact cnv_cpm_conf_lpr_delta_ell_aux (L) (K1) (K2) (V) (W) (i):
106 ⬇*[i]L ≘ K1.ⓓV → ⬇*[i]L ≘ K2.ⓛW → ⊥.
107 #L #K1 #K2 #V #W #i #HLK1 #HLK2
108 lapply (drops_mono … HLK2 … HLK1) -L -i #H destruct
111 fact cnv_cpm_conf_lpr_bind_bind_aux (a) (h) (o) (p) (I) (G) (L) (V) (T):
112 (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
113 ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
114 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
115 ∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 →
116 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
117 ∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡*[n1-n2,h] T.
118 #a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0
119 #V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
121 elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
122 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
123 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
124 #T #HT1 #HT2 -L0 -V0 -T0
125 /3 width=5 by cpms_bind_dx, ex2_intro/
128 fact cnv_cpm_conf_lpr_bind_zeta_aux (a) (h) (o) (G) (L) (V) (T):
129 (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
130 ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
131 ∀V1. ⦃G,L⦄ ⊢V ➡[h] V1 →
132 ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓV⦄ ⊢ T ➡[n2,h] T2 →
133 ∀XT2. ⬆*[1]XT2 ≘ T2 →
134 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
135 ∃∃T. ⦃G,L1⦄ ⊢ +ⓓV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
136 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
137 #V1 #HV01 #n1 #T1 #HT01 #n2 #T2 #HT02 #XT2 #HXT2
139 elim (cnv_inv_bind … H0) -H0 #_ #HT0
140 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ] -L0 -T0 -V0
142 elim (cpms_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 [| /3 width=1 by drops_refl, drops_drop/ ] #XT #HXT #HXT2
143 /3 width=3 by cpms_zeta, ex2_intro/
146 fact cnv_cpm_conf_lpr_zeta_zeta_aux (a) (h) (o) (G) (L) (V) (T):
147 (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
148 ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
149 ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓV⦄ ⊢ T ➡[n2,h] T2 →
150 ∀XT1. ⬆*[1]XT1 ≘ T1 → ∀XT2. ⬆*[1]XT2 ≘ T2 →
151 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
152 ∃∃T. ⦃G,L1⦄ ⊢ XT1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
153 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
154 #n1 #T1 #HT01 #n2 #T2 #HT02 #XT1 #HXT1 #XT2 #HXT2
156 elim (cnv_inv_bind … H0) -H0 #_ #HT0
157 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ] -L0 -T0
159 elim (cpms_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #XT #HXT #HXT1
160 elim (cpms_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #X #H #HXT2
161 lapply (lifts_inj … H … HXT) -T #H destruct
162 /2 width=3 by ex2_intro/
165 fact cnv_cpm_conf_lpr_appl_appl_aux (a) (h) (o) (G) (L) (V) (T):
166 (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
167 ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
168 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
169 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
170 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
171 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡*[n1-n2,h] T.
172 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
173 #V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
175 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
176 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
177 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
178 #T #HT1 #HT2 -L0 -V0 -T0
179 /3 width=5 by cpms_appl_dx, ex2_intro/
182 fact cnv_cpm_conf_lpr_appl_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
183 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
184 ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
185 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
186 ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
187 ∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
188 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
189 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
190 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
191 #V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02
193 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
194 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
195 elim (cpm_inv_abst1 … HX) -HX #W1 #T1 #HW01 #HT01 #H destruct
196 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
197 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
198 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
199 #T #HT1 #HT2 -L0 -V0 -W0 -T0
200 lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 [ /2 width=1 by lsubr_beta/ ] #HT2
201 /4 width=5 by cpms_beta_dx, cpms_bind_dx, cpm_cast, ex2_intro/
204 fact cnv_cpm_conf_lpr_appl_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
205 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
206 ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
207 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
208 ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
209 ∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
211 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
212 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
213 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
214 #V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02 #U2 #HVU2
216 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
217 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
218 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
219 elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU2
220 elim (cpm_inv_abbr1 … HX) -HX *
221 [ #W1 #T1 #HW01 #HT01 #H destruct
222 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
223 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
224 #T #HT1 #HT2 -L0 -V0 -W0 -T0
225 /4 width=7 by cpms_theta_dx, cpms_appl_dx, cpms_bind_dx, ex2_intro/
226 | #T1 #HT01 #HX #H destruct
227 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
228 #T #HT1 #HT2 -L0 -V0 -W0 -T0
229 elim (cpms_inv_lifts_sn … HT1 (Ⓣ) … L1 … HX) -T1 [| /3 width=1 by drops_refl, drops_drop/ ] #X0 #HXT #HX0
230 /4 width=7 by cpms_zeta, cpms_appl_dx, lifts_flat, ex2_intro/
234 fact cnv_cpm_conf_lpr_beta_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
235 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
236 ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
237 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
238 ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
239 ∀n1,T1. ⦃G,L.ⓛW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
240 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
241 ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
242 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
243 #V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02
245 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
246 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
247 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
248 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
249 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
250 #T #HT1 #HT2 -L0 -V0 -W0 -T0
251 lapply (lsubr_cpms_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/ #HT1
252 lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ #HT2
253 /4 width=5 by cpms_bind_dx, cpm_eps, ex2_intro/
256 fact cnv_cpm_conf_lpr_theta_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
257 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
258 ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
259 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
260 ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
261 ∀n1,T1. ⦃G,L.ⓓW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
262 ∀U1. ⬆*[1]V1 ≘ U1 → ∀U2. ⬆*[1]V2 ≘ U2 →
263 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
264 ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
265 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
266 #V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02 #U1 #HVU1 #U2 #HVU2
268 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
269 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
270 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
271 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
272 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
273 #T #HT1 #HT2 -L0 -V0 -W0 -T0
274 elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -V1 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU1
275 lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -V2 -V [ /3 width=1 by drops_refl, drops_drop/ ] #HU2
276 /4 width=7 by cpms_appl_dx, cpms_bind_dx, ex2_intro/
279 fact cnv_cpm_conf_lpr_cast_cast_aux (a) (h) (o) (G) (L) (V) (T):
280 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
281 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
282 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
283 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
284 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
285 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡*[n1-n2,h] T.
286 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
287 #n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01 #T2 #HT02
289 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
290 elim (cnv_cpm_conf_lpr_far … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
291 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
292 #T #HT1 #HT2 #V #HV1 #HV2 -L0 -V0 -T0
293 /3 width=5 by cpms_cast, ex2_intro/
296 fact cnv_cpm_conf_lpr_cast_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
297 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
298 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
299 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 →
300 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
301 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
302 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
303 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
304 #n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02
306 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
307 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
308 #T #HT1 #HT2 -L0 -V0 -T0
309 /3 width=3 by cpms_eps, ex2_intro/
312 fact cnv_cpm_conf_lpr_cast_ee_aux (a) (h) (o) (G) (L) (V) (T):
313 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
314 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
315 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
316 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
317 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 →
318 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
319 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
320 #a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
321 #n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01
322 #L1 #HL01 #L2 #HL02 -HV01
323 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
324 lapply (cnv_cpms_trans_lpr_far … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
325 elim (cnv_cpms_strip_lpr_far … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
326 elim (cnv_cpms_strip_lpr_far … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
327 -HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
328 elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
329 -L0 -V0 -T0 -X0 #U #HVU #HTU
330 lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
331 lapply (cpms_trans … HT1 … HTU) -T <arith_l2 #H1
332 /3 width=3 by cpms_eps, ex2_intro/
335 fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
336 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
337 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
338 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
339 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
340 ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
341 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
342 #n1 #T1 #HT01 #n2 #T2 #HT02
344 elim (cnv_inv_cast … H0) -H0 #X0 #_ #HT0 #_ #_ -X0
345 elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
346 #T #HT1 #HT2 -L0 -V0 -T0
347 /2 width=3 by ex2_intro/
350 fact cnv_cpm_conf_lpr_epsilon_ee_aux (a) (h) (o) (G) (L) (V) (T):
351 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
352 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
353 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
354 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
355 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
356 ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
357 #a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
358 #n1 #T1 #HT01 #n2 #V2 #HV02
360 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
361 lapply (cnv_cpms_trans_lpr_far … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
362 elim (cnv_cpms_strip_lpr_far … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
363 elim (cnv_cpms_strip_lpr_far … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
364 -HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
365 elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
366 -L0 -V0 -T0 -X0 #U #HVU #HTU
367 lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
368 lapply (cpms_trans … HT1 … HTU) -T <arith_l2 #H1
369 /2 width=3 by ex2_intro/
372 fact cnv_cpm_conf_lpr_ee_ee_aux (a) (h) (o) (G) (L) (V) (T):
373 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
374 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
375 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
376 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
377 ∃∃T. ⦃G,L1⦄ ⊢ V1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-n2,h] T.
378 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
379 #n1 #V1 #HV01 #n2 #V2 #HV02
381 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #_ #_ #_ -X0
382 elim (cnv_cpm_conf_lpr_far … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
383 #V #HV1 #HV2 -L0 -V0 -T0
384 /2 width=3 by ex2_intro/
387 fact cnv_cpm_conf_lpr_aux (a) (h) (o):
389 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
390 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
391 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
392 #a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
393 [ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
394 elim (cpm_inv_atom1_drops … HX1) -HX1 *
395 elim (cpm_inv_atom1_drops … HX2) -HX2 *
396 [ #H21 #H22 #H11 #H12 destruct -L -a -o
398 /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
399 | #s2 #H21 #H22 #H23 #H11 #H12 destruct -L -a -o
400 <minus_O_n <minus_n_O
401 /2 width=1 by cnv_cpm_conf_lpr_atom_ess_aux/
402 | #K2 #V2 #XV2 #i #HLK2 #HVX2 #HXV2 #H21 #H11 #H12 destruct -IH2
403 <minus_O_n <minus_n_O
404 @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
405 | #m2 #K2 #W2 #XW2 #i #HLK2 #HWX2 #HXW2 #H21 #H22 #H11 #H12 destruct -IH2
406 <minus_O_n <minus_n_O
407 @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
408 | #H21 #H22 #s1 #H11 #H12 #H13 destruct -L -a -o
409 <minus_O_n <minus_n_O
410 /3 width=1 by cnv_cpm_conf_lpr_atom_ess_aux, ex2_commute/
411 | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -L -a -o
413 /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
414 | #K2 #V2 #XV2 #i2 #_ #_ #_ #H21 #s1 #H11 #H12 #H13 destruct
415 | #m2 #K2 #W2 #XW2 #i2 #_ #_ #_ #H21 #H22 #s1 #H11 #H12 #H13 destruct
416 | #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
417 <minus_O_n <minus_n_O
418 @ex2_commute @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
419 | #s2 #H21 #H22 #H23 #K1 #V1 #XV1 #i1 #_ #_ #_ #H11 destruct
420 | #K2 #V2 #XV2 #i2 #HLK2 #HVX2 #HXV2 #H21 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
421 @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
422 | #m2 #K2 #W2 #XW2 #i2 #HLK2 #_ #_ #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #_ #_ #H11 destruct -a -o -XW2 -XV1 -HL2 -HL1
423 elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
424 | #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
425 <minus_O_n <minus_n_O
426 @ex2_commute @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
427 | #s2 #H21 #H22 #H23 #m1 #K1 #W1 #XW1 #i1 #_ #_ #_ #H11 #H12 destruct
428 | #K2 #V2 #XV2 #i2 #HLK2 #_ #_ #H21 #m1 #K1 #W1 #XW1 #i1 #HLK1 #_ #_ #H11 #H12 destruct -a -o -XV2 -XW1 -HL2 -HL1
429 elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
430 | #m2 #K2 #W2 #XW2 #i2 #HLK2 #HWX2 #HXW2 #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
431 >minus_S_S >minus_S_S
432 @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
434 | #p #I #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
435 elim (cpm_inv_bind1 … HX1) -HX1 *
436 elim (cpm_inv_bind1 … HX2) -HX2 *
437 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
438 @(cnv_cpm_conf_lpr_bind_bind_aux … IH1) -IH1 /1 width=1 by/
439 | #T2 #HT2 #HXT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
440 @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
441 | #V2 #T2 #HV2 #HT2 #H21 #T1 #HT1 #HXT1 #H11 #H12 destruct -IH2
442 @ex2_commute @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
443 | #T2 #HT2 #HXT2 #H21 #H22 #T1 #HT1 #HXT1 #H11 #H12 destruct -IH2
444 @(cnv_cpm_conf_lpr_zeta_zeta_aux … IH1) -IH1 /1 width=3 by/
446 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
447 elim (cpm_inv_appl1 … HX1) -HX1 *
448 elim (cpm_inv_appl1 … HX2) -HX2 *
449 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
450 @(cnv_cpm_conf_lpr_appl_appl_aux … IH1) -IH1 /1 width=1 by/
451 | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
452 @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
453 | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
454 @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
455 | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
456 @ex2_commute @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
457 | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
458 @(cnv_cpm_conf_lpr_beta_beta_aux … IH1) -IH1 /1 width=1 by/
459 | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct
460 | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XV1 #XW1 #W1 #XT1 #T1 #HV1 #HXV1 #HW1 #HT1 #H11 #H12 destruct -IH2
461 @ex2_commute @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
462 | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XV1 #XW1 #W1 #XT1 #T1 #HV1 #HXV1 #HW1 #HT1 #H11 #H12 destruct
463 | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XV1 #XW1 #W1 #XT1 #T1 #HV1 #HXV1 #HW1 #HT1 #H11 #H12 destruct -IH2
464 @(cnv_cpm_conf_lpr_theta_theta_aux … IH1) -IH1 /1 width=3 by/
466 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
467 elim (cpm_inv_cast1 … HX1) -HX1 [ * || * ]
468 elim (cpm_inv_cast1 … HX2) -HX2 [ * || * | * || * | * || * ]
469 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
470 @(cnv_cpm_conf_lpr_cast_cast_aux … IH1) -IH1 /1 width=1 by/
471 | #HT2 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
472 @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
473 | #m2 #HV2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct
474 @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
475 | #V2 #T2 #HV2 #HT2 #H21 #HT1 destruct -IH2
476 @ex2_commute @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
478 @(cnv_cpm_conf_lpr_epsilon_epsilon_aux … IH1) -IH1 /1 width=1 by/
479 | #m2 #HV2 #H21 #HT1 destruct
480 @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
481 | #V2 #T2 #HV2 #HT2 #H21 #m1 #HV1 #H11 destruct
482 @ex2_commute @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
483 | #HT2 #m1 #HV1 #H11 destruct
484 @ex2_commute @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
485 | #m2 #HV2 #H21 #m1 #HV1 #H11 destruct -IH2
486 >minus_S_S >minus_S_S
487 @(cnv_cpm_conf_lpr_ee_ee_aux … IH1) -IH1 /1 width=1 by/