X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fdynamic%2Fcnv_cpm_conf.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fdynamic%2Fcnv_cpm_conf.ma;h=160d23f887dcb2ad1931e2ebd5b1883bb68f2963;hb=ede00573e3e4cb28df7ca9a5dae6228c2b432608;hp=0000000000000000000000000000000000000000;hpb=1083ac3b1acac5f1ac1fa40a9a417dd9d268dced;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lpr_lpr.ma".
+include "basic_2/rt_computation/cpms_lsubr.ma".
+include "basic_2/rt_computation/cpms_fpbg.ma".
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/dynamic/cnv_drops.ma".
+include "basic_2/dynamic/cnv_preserve_far.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Far diamond propery with t-bound rt-transition for terms *****************)
+
+fact cnv_cpm_conf_lpr_atom_atom_aux (h) (G) (L1) (L2) (I):
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
+/2 width=3 by ex2_intro/ qed-.
+
+fact cnv_cpm_conf_lpr_atom_ess_aux (h) (G) (L1) (L2) (s):
+     ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡*[1,h] T & ⦃G,L2⦄ ⊢ ⋆(next h s) ➡*[h] T.
+/3 width=3 by cpm_cpms, ex2_intro/ qed-.
+
+fact cnv_cpm_conf_lpr_atom_delta_aux (a) (h) (o) (G) (L) (i):
+     (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄⊢#i![a,h] →
+     ∀K,V. ⬇*[i]L ≘ K.ⓓV →
+     ∀n,XV. ⦃G,K⦄ ⊢ V ➡[n,h] XV →
+     ∀X. ⬆*[↑i]XV ≘ X →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
+#a #h #o #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
+lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #HV1 #H destruct
+elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #V2 #HK2 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -V2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
+elim (cnv_cpm_conf_lpr_far … IH … HV1 … HVX … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V
+<minus_O_n <minus_n_O #V #HV1 #HVX
+elim (cpms_lifts_sn … HVX … HLK2 … HXV) -XV -HLK2 #XV #HVX #HXV
+/3 width=6 by cpms_delta_drops, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_atom_ell_aux (a) (h) (o) (G) (L) (i):
+     (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄⊢#i![a,h] →
+     ∀K,W. ⬇*[i]L ≘ K.ⓛW →
+     ∀n,XW. ⦃G,K⦄ ⊢ W ➡[n,h] XW →
+     ∀X. ⬆*[↑i]XW ≘ X →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
+#a #h #o #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
+lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW
+elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK1 #HW1 #H destruct
+elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK2 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
+elim (cnv_cpm_conf_lpr_far … IH … HW1 … HWX … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -W
+<minus_O_n <minus_n_O #W #HW1 #HWX
+elim (cpms_lifts_sn … HWX … HLK2 … HXW) -XW -HLK2 #XW #HWX #HXW
+/3 width=6 by cpms_ell_drops, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_delta_delta_aux (a) (h) (o) (I) (G) (L) (i):
+     (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄⊢#i![a,h] →
+     ∀K1,V1. ⬇*[i]L ≘ K1.ⓑ{I}V1 → ∀K2,V2. ⬇*[i]L ≘ K2.ⓑ{I}V2 →
+     ∀n1,XV1. ⦃G,K1⦄ ⊢ V1 ➡[n1,h] XV1 → ∀n2,XV2. ⦃G,K2⦄ ⊢ V2 ➡[n2,h] XV2 →
+     ∀X1. ⬆*[↑i]XV1 ≘ X1 → ∀X2. ⬆*[↑i]XV2 ≘ X2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ X1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ X2 ➡*[n1-n2,h] T.
+#a #h #o #I #G #L #i #IH #HT
+#K #V #HLK #Y #X #HLY #n1 #XV1 #HVX1 #n2 #XV2 #HVX2 #X1 #HXV1 #X2 #HXV2
+#L1 #HL1 #L2 #HL2
+lapply (drops_mono … HLY … HLK) -HLY #H destruct
+lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK1) -V1 // #HLK1
+elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #V2 #HK2 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -V2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK
+elim (cnv_cpm_conf_lpr_far … IH … HVX1 … HVX2 … HK1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V
+#V #HVX1 #HVX2
+elim (cpms_lifts_sn … HVX1 … HLK1 … HXV1) -XV1 -HLK1 #W1 #HVW1 #HXW1
+/3 width=11 by cpms_lifts_bi, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_delta_ell_aux (L) (K1) (K2) (V) (W) (i):
+     ⬇*[i]L ≘ K1.ⓓV → ⬇*[i]L ≘ K2.ⓛW → ⊥.
+#L #K1 #K2 #V #W #i #HLK1 #HLK2
+lapply (drops_mono … HLK2 … HLK1) -L -i #H destruct
+qed-.
+
+fact cnv_cpm_conf_lpr_bind_bind_aux (a) (h) (o) (p) (I) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡*[n1-n2,h] T.
+#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+#T #HT1 #HT2 -L0 -V0 -T0
+/3 width=5 by cpms_bind_dx, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_bind_zeta_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢V ➡[h] V1 →
+     ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓV⦄ ⊢ T ➡[n2,h] T2 →
+     ∀XT2. ⬆*[1]XT2 ≘ T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ +ⓓV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#V1 #HV01 #n1 #T1 #HT01 #n2 #T2 #HT02 #XT2 #HXT2
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_bind … H0) -H0 #_ #HT0
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ] -L0 -T0 -V0
+#T #HT1 #HT2
+elim (cpms_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 [| /3 width=1 by drops_refl, drops_drop/ ] #XT #HXT #HXT2 
+/3 width=3 by cpms_zeta, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_zeta_zeta_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+     ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓV⦄ ⊢ T ➡[n2,h] T2 →
+     ∀XT1. ⬆*[1]XT1 ≘ T1 → ∀XT2. ⬆*[1]XT2 ≘ T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ XT1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #T1 #HT01 #n2 #T2 #HT02 #XT1 #HXT1 #XT2 #HXT2
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_bind … H0) -H0 #_ #HT0
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ] -L0 -T0
+#T #HT1 #HT2
+elim (cpms_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #XT #HXT #HXT1
+elim (cpms_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #X #H #HXT2
+lapply (lifts_inj … H … HXT) -T #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_appl_appl_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #HT2 -L0 -V0 -T0
+/3 width=5 by cpms_appl_dx, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_appl_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+     ∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
+elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
+elim (cpm_inv_abst1 … HX) -HX #W1 #T1 #HW01 #HT01 #H destruct
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+#T #HT1 #HT2 -L0 -V0 -W0 -T0
+lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 [ /2 width=1 by lsubr_beta/ ] #HT2
+/4 width=5 by cpms_beta_dx, cpms_bind_dx, cpm_cast, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_appl_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+     ∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
+     ∀U2. ⬆*[1]V2 ≘ U2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02 #U2 #HVU2
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
+elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU2
+elim (cpm_inv_abbr1 … HX) -HX *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+  elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
+  elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+  #T #HT1 #HT2 -L0 -V0 -W0 -T0
+  /4 width=7 by cpms_theta_dx, cpms_appl_dx, cpms_bind_dx, ex2_intro/
+| #T1 #HT01 #HX #H destruct
+  elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+  #T #HT1 #HT2 -L0 -V0 -W0 -T0
+  elim (cpms_inv_lifts_sn … HT1 (Ⓣ) … L1 … HX) -T1 [| /3 width=1 by drops_refl, drops_drop/ ] #X0 #HXT #HX0
+  /4 width=7 by cpms_zeta, cpms_appl_dx, lifts_flat, ex2_intro/
+]
+qed-.
+
+fact cnv_cpm_conf_lpr_beta_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+     ∀n1,T1. ⦃G,L.ⓛW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
+elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+#T #HT1 #HT2 -L0 -V0 -W0 -T0
+lapply (lsubr_cpms_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/ #HT1
+lapply (lsubr_cpms_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ #HT2
+/4 width=5 by cpms_bind_dx, cpm_eps, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_theta_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+     ∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+     ∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+     ∀n1,T1. ⦃G,L.ⓓW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
+     ∀U1. ⬆*[1]V1 ≘ U1 → ∀U2. ⬆*[1]V2 ≘ U2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02 #U1 #HVU1 #U2 #HVU2
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
+elim (cnv_inv_bind … H0) -H0 #HW0 #HT0
+elim (cpr_conf_lpr … HV01 … HV02 … HL01 … HL02) #V #HV1 #HV2
+elim (cpr_conf_lpr … HW01 … HW02 … HL01 … HL02) #W #HW1 #HW2
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+#T #HT1 #HT2 -L0 -V0 -W0 -T0
+elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -V1 [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU1
+lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -V2 -V [ /3 width=1 by drops_refl, drops_drop/ ] #HU2
+/4 width=7 by cpms_appl_dx, cpms_bind_dx, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_cast_cast_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+     ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
+elim (cnv_cpm_conf_lpr_far … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #HT2 #V #HV1 #HV2 -L0 -V0 -T0
+/3 width=5 by cpms_cast, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_cast_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 →
+     ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #HT2 -L0 -V0 -T0
+/3 width=3 by cpms_eps, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_cast_ee_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+     ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01
+#L1 #HL01 #L2 #HL02 -HV01
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
+lapply (cnv_cpms_trans_lpr_far … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
+elim (cnv_cpms_strip_lpr_far … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+elim (cnv_cpms_strip_lpr_far … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
+-HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
+elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
+-L0 -V0 -T0 -X0 #U #HVU #HTU
+lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
+lapply (cpms_trans … HT1 … HTU) -T <arith_l #H1
+/3 width=3 by cpms_eps, ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #T1 #HT01 #n2 #T2 #HT02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #_ #HT0 #_ #_ -X0
+elim (cnv_cpm_conf_lpr_far … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #HT2 -L0 -V0 -T0
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_epsilon_ee_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#n1 #T1 #HT01 #n2 #V2 #HV02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
+lapply (cnv_cpms_trans_lpr_far … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
+elim (cnv_cpms_strip_lpr_far … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+elim (cnv_cpms_strip_lpr_far … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ]
+-HV02 -HTX0 -HT01 <minus_O_n <minus_n_O #T #HT2 #HT1 #V #HV1 #HV2
+elim (IH1 … HV1 … HT2 … HL02 … HL01) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ]
+-L0 -V0 -T0 -X0 #U #HVU #HTU
+lapply (cpms_trans … HV2 … HVU) -V <plus_O_n >minus_plus #H2
+lapply (cpms_trans … HT1 … HTU) -T <arith_l #H1
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_ee_ee_aux (a) (h) (o) (G) (L) (V) (T):
+     (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
+     ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+     ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+     ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+     ∃∃T. ⦃G,L1⦄ ⊢ V1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-n2,h] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #V1 #HV01 #n2 #V2 #HV02
+#L1 #HL01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #_ #_ #_ -X0
+elim (cnv_cpm_conf_lpr_far … IH … HV01 … HV02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#V #HV1 #HV2 -L0 -V0 -T0
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cnv_cpm_conf_lpr_aux (a) (h) (o):
+                          ∀G0,L0,T0.
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+                          ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
+#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
+  elim (cpm_inv_atom1_drops … HX1) -HX1 *
+  elim (cpm_inv_atom1_drops … HX2) -HX2 *
+  [ #H21 #H22 #H11 #H12 destruct -L -a -o
+    <minus_O_n
+    /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
+  | #s2 #H21 #H22 #H23 #H11 #H12 destruct -L -a -o
+    <minus_O_n <minus_n_O
+    /2 width=1 by cnv_cpm_conf_lpr_atom_ess_aux/
+  | #K2 #V2 #XV2 #i #HLK2 #HVX2 #HXV2 #H21 #H11 #H12 destruct -IH2
+    <minus_O_n <minus_n_O
+    @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
+  | #m2 #K2 #W2 #XW2 #i #HLK2 #HWX2 #HXW2 #H21 #H22 #H11 #H12 destruct -IH2
+    <minus_O_n <minus_n_O
+    @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
+  | #H21 #H22 #s1 #H11 #H12 #H13 destruct -L -a -o
+    <minus_O_n <minus_n_O
+    /3 width=1 by cnv_cpm_conf_lpr_atom_ess_aux, ex2_commute/
+  | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -L -a -o
+    <minus_n_n
+    /2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
+  | #K2 #V2 #XV2 #i2 #_ #_ #_ #H21 #s1 #H11 #H12 #H13 destruct
+  | #m2 #K2 #W2 #XW2 #i2 #_ #_ #_ #H21 #H22 #s1 #H11 #H12 #H13 destruct
+  | #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
+    <minus_O_n <minus_n_O
+    @ex2_commute @(cnv_cpm_conf_lpr_atom_delta_aux … IH1) -IH1 /1 width=6 by/
+  | #s2 #H21 #H22 #H23 #K1 #V1 #XV1 #i1 #_ #_ #_ #H11 destruct
+  | #K2 #V2 #XV2 #i2 #HLK2 #HVX2 #HXV2 #H21 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
+  | #m2 #K2 #W2 #XW2 #i2 #HLK2 #_ #_ #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #_ #_ #H11 destruct -a -o -XW2 -XV1 -HL2 -HL1
+    elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
+  | #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2 
+    <minus_O_n <minus_n_O
+    @ex2_commute @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
+  | #s2 #H21 #H22 #H23 #m1 #K1 #W1 #XW1 #i1 #_ #_ #_ #H11 #H12 destruct
+  | #K2 #V2 #XV2 #i2 #HLK2 #_ #_ #H21 #m1 #K1 #W1 #XW1 #i1 #HLK1 #_ #_ #H11 #H12 destruct -a -o -XV2 -XW1 -HL2 -HL1
+    elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
+  | #m2 #K2 #W2 #XW2 #i2 #HLK2 #HWX2 #HXW2 #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
+    >minus_S_S >minus_S_S
+    @(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
+  ]
+| #p #I #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
+  elim (cpm_inv_bind1 … HX1) -HX1 *
+  elim (cpm_inv_bind1 … HX2) -HX2 *
+  [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_bind_bind_aux … IH1) -IH1 /1 width=1 by/
+  | #T2 #HT2 #HXT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
+  | #V2 #T2 #HV2 #HT2 #H21 #T1 #HT1 #HXT1 #H11 #H12 destruct -IH2
+    @ex2_commute @(cnv_cpm_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
+  | #T2 #HT2 #HXT2 #H21 #H22 #T1 #HT1 #HXT1 #H11 #H12 destruct -IH2
+    @(cnv_cpm_conf_lpr_zeta_zeta_aux … IH1) -IH1 /1 width=3 by/
+  ]
+| #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
+  elim (cpm_inv_appl1 … HX1) -HX1 *
+  elim (cpm_inv_appl1 … HX2) -HX2 *
+  [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_appl_appl_aux … IH1) -IH1 /1 width=1 by/
+  | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
+  | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
+  | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
+    @ex2_commute @(cnv_cpm_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
+  | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #p1 #V1 #XW1 #W1 #XT1 #T1 #HV1 #HW1 #HT1 #H11 #H12 destruct -IH2
+    @(cnv_cpm_conf_lpr_beta_beta_aux … IH1) -IH1 /1 width=1 by/
+  | #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
+  | #V2 #T2 #HV2 #HT2 #H21 #p1 #V1 #XV1 #XW1 #W1 #XT1 #T1 #HV1 #HXV1 #HW1 #HT1 #H11 #H12 destruct -IH2
+    @ex2_commute @(cnv_cpm_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
+  | #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
+  | #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
+    @(cnv_cpm_conf_lpr_theta_theta_aux … IH1) -IH1 /1 width=3 by/
+  ]
+| #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
+  elim (cpm_inv_cast1 … HX1) -HX1 [ * || * ]
+  elim (cpm_inv_cast1 … HX2) -HX2 [ * || * | * || * | * || * ]
+  [ #V2 #T2 #HV2 #HT2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_cast_cast_aux … IH1) -IH1 /1 width=1 by/
+  | #HT2 #V1 #T1 #HV1 #HT1 #H11 destruct -IH2
+    @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
+  | #m2 #HV2 #H21 #V1 #T1 #HV1 #HT1 #H11 destruct
+    @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
+  | #V2 #T2 #HV2 #HT2 #H21 #HT1 destruct -IH2
+    @ex2_commute @(cnv_cpm_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/
+  | #HT2 #HT1 -IH2
+    @(cnv_cpm_conf_lpr_epsilon_epsilon_aux … IH1) -IH1 /1 width=1 by/
+  | #m2 #HV2 #H21 #HT1 destruct
+    @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
+  | #V2 #T2 #HV2 #HT2 #H21 #m1 #HV1 #H11 destruct
+    @ex2_commute @(cnv_cpm_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
+  | #HT2 #m1 #HV1 #H11 destruct
+    @ex2_commute @(cnv_cpm_conf_lpr_epsilon_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
+  | #m2 #HV2 #H21 #m1 #HV1 #H11 destruct -IH2
+    >minus_S_S >minus_S_S
+    @(cnv_cpm_conf_lpr_ee_ee_aux … IH1) -IH1 /1 width=1 by/
+  ]
+]
+qed-.