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_sub.ma".
23 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
25 (* Sub 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_sub … 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_sub … 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_sub … 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_sub … 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 → ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 →
132 ∀T2. ⬆*[1]T2 ≘ T → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
133 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
134 ∃∃T. ⦃G,L1⦄ ⊢ +ⓓV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
135 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
136 #V1 #HV01 #n1 #T1 #HT01 #T2 #HT20 #n2 #XT2 #HXT2
138 elim (cnv_inv_bind … H0) -H0 #_ #HT0
139 lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HT20) -HT0
140 [ /3 width=3 by drops_refl, drops_drop/ ] #HT2
141 elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01
142 [| /3 width=1 by drops_refl, drops_drop/ ] #XT1 #HXT1 #HXT12
143 elim (cnv_cpm_conf_lpr_sub … IH … HXT12 … HXT2 … HL01 … HL02)
144 [|*: /3 width=1 by fqup_fpbg, fqup_zeta/ ] -L0 -T0 -V0 #T #HT1 #HT2
145 /3 width=3 by cpms_zeta, ex2_intro/
148 fact cnv_cpm_conf_lpr_zeta_zeta_aux (a) (h) (o) (G) (L) (V) (T):
149 (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
150 ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
151 ∀T1. ⬆*[1]T1 ≘ T → ∀T2. ⬆*[1]T2 ≘ T →
152 ∀n1,XT1. ⦃G,L⦄ ⊢ T1 ➡[n1,h] XT1 → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
153 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
154 ∃∃T. ⦃G,L1⦄ ⊢ XT1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
155 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
156 #T1 #HT10 #T2 #HT20 #n1 #XT1 #HXT1 #n2 #XT2 #HXT2
158 elim (cnv_inv_bind … H0) -H0 #_ #HT0
159 lapply (lifts_inj … HT10 … HT20) -HT10 #H destruct
160 lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HT20) -HT0
161 [ /3 width=3 by drops_refl, drops_drop/ ] #HT2
162 elim (cnv_cpm_conf_lpr_sub … IH … HXT1 … HXT2 … HL01 … HL02)
163 [|*: /3 width=1 by fqup_fpbg, fqup_zeta/ ] -L0 -T0 #T #HT1 #HT2
164 /2 width=3 by ex2_intro/
167 fact cnv_cpm_conf_lpr_appl_appl_aux (a) (h) (o) (G) (L) (V) (T):
168 (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
169 ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
170 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
171 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
172 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
173 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡*[n1-n2,h] T.
174 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
175 #V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
177 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
178 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
179 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
180 #T #HT1 #HT2 -L0 -V0 -T0
181 /3 width=5 by cpms_appl_dx, ex2_intro/
184 fact cnv_cpm_conf_lpr_appl_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
185 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
186 ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
187 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
188 ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
189 ∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
190 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
191 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
192 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
193 #V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02
195 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
196 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
197 elim (cpm_inv_abst1 … HX) -HX #W1 #T1 #HW01 #HT01 #H destruct
198 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
199 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
200 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
201 #T #HT1 #HT2 -L0 -V0 -W0 -T0
202 lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 [ /2 width=1 by lsubr_beta/ ] #HT2
203 /4 width=5 by cpms_beta_dx, cpms_bind_dx, cpm_cast, ex2_intro/
206 fact cnv_cpm_conf_lpr_appl_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
207 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
208 ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
209 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
210 ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
211 ∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
213 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
214 ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
215 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
216 #V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02 #U2 #HVU2
218 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
219 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
220 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
221 elim (cpm_inv_abbr1 … HX) -HX *
222 [ #W1 #T1 #HW01 #HT01 #H destruct
223 elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU2
224 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
225 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
226 #T #HT1 #HT2 -L0 -V0 -W0 -T0
227 /4 width=7 by cpms_theta_dx, cpms_appl_dx, cpms_bind_dx, ex2_intro/
228 | #X0 #HXT0 #H1X0 #H destruct
229 lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HXT0) -HT0 [ /3 width=3 by drops_refl, drops_drop/ ] #H2X0
230 elim (cpm_inv_lifts_sn … HT02 (Ⓣ) … L0 … HXT0) -HT02 [| /3 width=1 by drops_refl, drops_drop/ ] #X2 #HXT2 #HX02
231 elim (cnv_cpm_conf_lpr_sub … IH … H1X0 … HX02 … HL01 … HL02)
232 [|*: /4 width=5 by fqup_fpbg, fqup_strap1, fqu_drop/ ] #T #HT1 #HT2 -L0 -V0 -W0 -T0
233 /4 width=8 by cpms_zeta, cpms_appl_dx, lifts_flat, ex2_intro/
237 fact cnv_cpm_conf_lpr_beta_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
238 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
239 ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
240 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
241 ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
242 ∀n1,T1. ⦃G,L.ⓛW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
243 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
244 ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
245 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
246 #V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02
248 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
249 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
250 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
251 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
252 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
253 #T #HT1 #HT2 -L0 -V0 -W0 -T0
254 lapply (lsubr_cpms_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/ #HT1
255 lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ #HT2
256 /4 width=5 by cpms_bind_dx, cpm_eps, ex2_intro/
259 fact cnv_cpm_conf_lpr_theta_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
260 (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
261 ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
262 ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
263 ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
264 ∀n1,T1. ⦃G,L.ⓓW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
265 ∀U1. ⬆*[1]V1 ≘ U1 → ∀U2. ⬆*[1]V2 ≘ U2 →
266 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
267 ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
268 #a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
269 #V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02 #U1 #HVU1 #U2 #HVU2
271 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
272 elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
273 elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
274 elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
275 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
276 #T #HT1 #HT2 -L0 -V0 -W0 -T0
277 elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -V1 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU1
278 lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -V2 -V [ /3 width=1 by drops_refl, drops_drop/ ] #HU2
279 /4 width=7 by cpms_appl_dx, cpms_bind_dx, ex2_intro/
282 fact cnv_cpm_conf_lpr_cast_cast_aux (a) (h) (o) (G) (L) (V) (T):
283 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
284 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
285 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
286 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
287 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
288 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡*[n1-n2,h] T.
289 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
290 #n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01 #T2 #HT02
292 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
293 elim (cnv_cpm_conf_lpr_sub … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
294 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
295 #T #HT1 #HT2 #V #HV1 #HV2 -L0 -V0 -T0
296 /3 width=5 by cpms_cast, ex2_intro/
299 fact cnv_cpm_conf_lpr_cast_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
300 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
301 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
302 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 →
303 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
304 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
305 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
306 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
307 #n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02
309 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
310 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
311 #T #HT1 #HT2 -L0 -V0 -T0
312 /3 width=3 by cpms_eps, ex2_intro/
315 fact cnv_cpm_conf_lpr_cast_ee_aux (a) (h) (o) (G) (L) (V) (T):
316 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
317 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
318 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
319 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
320 ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 →
321 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
322 ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
323 #a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
324 #n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01
325 #L1 #HL01 #L2 #HL02 -HV01
326 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
327 lapply (cnv_cpms_trans_lpr_sub … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
328 elim (cnv_cpms_strip_lpr_sub … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
329 elim (cnv_cpms_strip_lpr_sub … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
330 -HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
331 elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
332 -L0 -V0 -T0 -X0 #U #HVU #HTU
333 lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
334 lapply (cpms_trans … HT1 … HTU) -T <arith_l2 #H1
335 /3 width=3 by cpms_eps, ex2_intro/
338 fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
339 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
340 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
341 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
342 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
343 ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
344 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
345 #n1 #T1 #HT01 #n2 #T2 #HT02
347 elim (cnv_inv_cast … H0) -H0 #X0 #_ #HT0 #_ #_ -X0
348 elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
349 #T #HT1 #HT2 -L0 -V0 -T0
350 /2 width=3 by ex2_intro/
353 fact cnv_cpm_conf_lpr_epsilon_ee_aux (a) (h) (o) (G) (L) (V) (T):
354 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
355 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
356 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
357 ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
358 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
359 ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
360 #a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
361 #n1 #T1 #HT01 #n2 #V2 #HV02
363 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
364 lapply (cnv_cpms_trans_lpr_sub … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
365 elim (cnv_cpms_strip_lpr_sub … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
366 elim (cnv_cpms_strip_lpr_sub … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
367 -HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
368 elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
369 -L0 -V0 -T0 -X0 #U #HVU #HTU
370 lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
371 lapply (cpms_trans … HT1 … HTU) -T <arith_l2 #H1
372 /2 width=3 by ex2_intro/
375 fact cnv_cpm_conf_lpr_ee_ee_aux (a) (h) (o) (G) (L) (V) (T):
376 (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
377 ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
378 ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
379 ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
380 ∃∃T. ⦃G,L1⦄ ⊢ V1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-n2,h] T.
381 #a #h #o #G0 #L0 #V0 #T0 #IH #H0
382 #n1 #V1 #HV01 #n2 #V2 #HV02
384 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #_ #_ #_ -X0
385 elim (cnv_cpm_conf_lpr_sub … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
386 #V #HV1 #HV2 -L0 -V0 -T0
387 /2 width=3 by ex2_intro/
390 fact cnv_cpm_conf_lpr_aux (a) (h) (o):
392 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
393 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
394 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
395 #a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
396 [ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
397 elim (cpm_inv_atom1_drops … HX1) -HX1 *
398 elim (cpm_inv_atom1_drops … HX2) -HX2 *
399 [ #H21 #H22 #H11 #H12 destruct -L -a -o
401 /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
402 | #s2 #H21 #H22 #H23 #H11 #H12 destruct -L -a -o
403 <minus_O_n <minus_n_O
404 /2 width=1 by cnv_cpm_conf_lpr_atom_ess_aux/
405 | #K2 #V2 #XV2 #i #HLK2 #HVX2 #HXV2 #H21 #H11 #H12 destruct -IH2
406 <minus_O_n <minus_n_O
407 @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
408 | #m2 #K2 #W2 #XW2 #i #HLK2 #HWX2 #HXW2 #H21 #H22 #H11 #H12 destruct -IH2
409 <minus_O_n <minus_n_O
410 @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
411 | #H21 #H22 #s1 #H11 #H12 #H13 destruct -L -a -o
412 <minus_O_n <minus_n_O
413 /3 width=1 by cnv_cpm_conf_lpr_atom_ess_aux, ex2_commute/
414 | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -L -a -o
416 /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
417 | #K2 #V2 #XV2 #i2 #_ #_ #_ #H21 #s1 #H11 #H12 #H13 destruct
418 | #m2 #K2 #W2 #XW2 #i2 #_ #_ #_ #H21 #H22 #s1 #H11 #H12 #H13 destruct
419 | #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
420 <minus_O_n <minus_n_O
421 @ex2_commute @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
422 | #s2 #H21 #H22 #H23 #K1 #V1 #XV1 #i1 #_ #_ #_ #H11 destruct
423 | #K2 #V2 #XV2 #i2 #HLK2 #HVX2 #HXV2 #H21 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
424 @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
425 | #m2 #K2 #W2 #XW2 #i2 #HLK2 #_ #_ #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #_ #_ #H11 destruct -a -o -XW2 -XV1 -HL2 -HL1
426 elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
427 | #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
428 <minus_O_n <minus_n_O
429 @ex2_commute @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
430 | #s2 #H21 #H22 #H23 #m1 #K1 #W1 #XW1 #i1 #_ #_ #_ #H11 #H12 destruct
431 | #K2 #V2 #XV2 #i2 #HLK2 #_ #_ #H21 #m1 #K1 #W1 #XW1 #i1 #HLK1 #_ #_ #H11 #H12 destruct -a -o -XV2 -XW1 -HL2 -HL1
432 elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
433 | #m2 #K2 #W2 #XW2 #i2 #HLK2 #HWX2 #HXW2 #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
434 >minus_S_S >minus_S_S
435 @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
437 | #p #I #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
438 elim (cpm_inv_bind1 … HX1) -HX1 *
439 elim (cpm_inv_bind1 … HX2) -HX2 *
440 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
441 @(cnv_cpm_conf_lpr_bind_bind_aux … IH1) -IH1 /1 width=1 by/
442 | #T2 #HT2 #HTX2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
443 @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
444 | #V2 #T2 #HV2 #HT2 #H21 #T1 #HT1 #HTX1 #H11 #H12 destruct -IH2
445 @ex2_commute @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
446 | #T2 #HT2 #HTX2 #H21 #H22 #T1 #HT1 #HTX1 #H11 #H12 destruct -IH2
447 @(cnv_cpm_conf_lpr_zeta_zeta_aux … IH1) -IH1 /1 width=3 by/
449 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
450 elim (cpm_inv_appl1 … HX1) -HX1 *
451 elim (cpm_inv_appl1 … HX2) -HX2 *
452 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
453 @(cnv_cpm_conf_lpr_appl_appl_aux … IH1) -IH1 /1 width=1 by/
454 | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
455 @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
456 | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
457 @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
458 | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
459 @ex2_commute @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
460 | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
461 @(cnv_cpm_conf_lpr_beta_beta_aux … IH1) -IH1 /1 width=1 by/
462 | #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
463 | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XV1 #XW1 #W1 #XT1 #T1 #HV1 #HXV1 #HW1 #HT1 #H11 #H12 destruct -IH2
464 @ex2_commute @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
465 | #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
466 | #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
467 @(cnv_cpm_conf_lpr_theta_theta_aux … IH1) -IH1 /1 width=3 by/
469 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
470 elim (cpm_inv_cast1 … HX1) -HX1 [ * || * ]
471 elim (cpm_inv_cast1 … HX2) -HX2 [ * || * | * || * | * || * ]
472 [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
473 @(cnv_cpm_conf_lpr_cast_cast_aux … IH1) -IH1 /1 width=1 by/
474 | #HT2 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
475 @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
476 | #m2 #HV2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct
477 @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
478 | #V2 #T2 #HV2 #HT2 #H21 #HT1 destruct -IH2
479 @ex2_commute @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
481 @(cnv_cpm_conf_lpr_epsilon_epsilon_aux … IH1) -IH1 /1 width=1 by/
482 | #m2 #HV2 #H21 #HT1 destruct
483 @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
484 | #V2 #T2 #HV2 #HT2 #H21 #m1 #HV1 #H11 destruct
485 @ex2_commute @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
486 | #HT2 #m1 #HV1 #H11 destruct
487 @ex2_commute @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
488 | #m2 #HV2 #H21 #m1 #HV1 #H11 destruct -IH2
489 >minus_S_S >minus_S_S
490 @(cnv_cpm_conf_lpr_ee_ee_aux … IH1) -IH1 /1 width=1 by/