--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/cpm_lsubr.ma".
+include "basic_2/rt_transition/lpr_drops.ma".
+include "basic_2/dynamic/cnv_drops.ma".
+include "basic_2/dynamic/cnv_cpm_tdeq.ma".
+include "basic_2/dynamic/cnv_preserve_sub.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+definition IH_cnv_cpm_tdeq1_conf_lpr (a) (h) (o): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → L0 ≛[h, o, T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T.
+
+definition IH_cnv_cpms_tdeq1_strip_lpr (a) (h) (o): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → L0 ≛[h, o, T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T.
+
+definition IH_cnv_cpm_tdeq2_conf_lpr (a) (h) (o): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T.
+
+definition IH_cnv_cpm_tdeq2_strip_lpr (a) (h) (o): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T.
+
+include "ground_2/lib/arith_2b.ma".
+include "basic_2/rt_computation/cpms_rdeq.ma".
+include "basic_2/rt_computation/cpms_fpbg.ma".
+
+fact cnv_cpms_tdeq1_strip_lpr_sub (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_cpm_tdeq1_conf_lpr a h o G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_tdeq1_strip_lpr a h o G1 L1 T1.
+#a #h #o #G #L #T #IH2 #IH1 #G0 #L0 #T0 #H0 #HT0 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #H @(cpms_ind_dx … H) -n2 -T2
+[ #L1 #H1L01 #H2L01 #L2 #HL02 -IH2
+ elim (IH1 … H1T01 H2T01 0 T0 … L1 … L2) // -L0 -IH1 -H2T01 #T3 #HT13 #H1T03 #H2T03
+ /3 width=4 by cpm_cpms, ex3_intro/
+| #n #n2 #X2 #T2 #HX2 #IHX2 #HXT2 #L1 #H1L01 #H2L01 #L2 #HL02 -H1T01 -H2T01
+ elim (IHX2 L1 … L0) -IHX2 // #X1 #HTX1 #H1X21 #H2X21
+ elim (IH1 … H1X21 H2X21 … HXT2 L1 … L2)
+ [|*: /3 width=12 by cnv_cpms_trans_lpr_sub, fpbg_cpms_trans, cpms_rdeq_conf_sn/ ] -X2 -IH1 -IH2 #X2 #HX12 #H1TX2 #H2TX2
+ >arith_l3 @(ex3_intro … H2TX2) /2 width=3 by cpms_step_dx/ (**) (* explict constructor *)
+]
+qed-.
+
+(* Sub diamond propery with restricted rt-transition for terms **************)
+
+fact cnv_cpm_tdeq1_conf_lpr_atom_atom_aux (h) (o) (G) (L1) (L2) (I):
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[O,h] T & ⓪{I} ≛[h,o] T.
+/2 width=4 by ex3_intro/ qed-.
+
+fact cnv_cpm_tdeq1_conf_lpr_atom_ess_aux (h) (o) (G) (L1) (L2) (s):
+ ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡[1,h] T & ⦃G,L2⦄ ⊢ ⋆(next h s) ➡[h] T & ⋆(next h s) ≛[h,o] T.
+/2 width=4 by ex3_intro/ qed-.
+
+fact cnv_cpm_tdeq1_conf_lpr_ess_atom_aux (h) (o) (G) (L1) (L2) (s):
+ deg h o s 0 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⋆(next h s) ➡[h] T & ⦃G,L2⦄ ⊢⋆s ➡[1,h] T & ⋆s ≛[h,o] T.
+/4 width=6 by tdeq_sort, deg_next, ex3_intro/ qed-.
+
+fact cnv_cpm_tdeq1_conf_lpr_atom_delta_aux (a) (h) (o) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o 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 → L ≛[h,o,#i] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ #i ➡[n,h] T & ⦃G,L2⦄ ⊢ X ➡[h] T & X ≛[h,o] T.
+#a #h #o #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #H1L1 #H2L1 #L2 #HL2
+lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+elim (lpr_drops_conf … HLK … H1L1) -H1L1 // #Y1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #H1K1 #H1V1 #H destruct
+elim (rdeq_inv_lref_pair_bi … H2L1 … HLK … HLK1) -H2L1 #H2K1 #H2V1 #_
+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 (IH … H1V1 H2V1 … HVX … H1K1 H2K1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V
+<minus_O_n <minus_n_O #V #HV1 #H1VX #H2VX
+elim (cpm_lifts_sn … H1VX … HLK2 … HXV) -H1VX -HLK2 #VX #HVX #HXVX
+lapply (tdeq_lifts_bi … H2VX … HXV … HVX) -H2VX -HXV #H2VX
+/3 width=6 by cpm_delta_drops, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_conf_lpr_atom_ell_aux (a) (h) (o) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o 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 → L ≛[h,o,#i] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ #i ➡[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡[h] T & X ≛[h,o] T.
+#a #h #o #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #H1L1 #H2L1 #L2 #HL2
+lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW
+elim (lpr_drops_conf … HLK … H1L1) -H1L1 // #Y1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #H1K1 #H1W1 #H destruct
+elim (rdeq_inv_lref_pair_bi … H2L1 … HLK … HLK1) -H2L1 #H2K1 #H2W1 #_
+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 (IH … H1W1 H2W1 … HWX … H1K1 H2K1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -W
+<minus_O_n <minus_n_O #W #HW1 #H1WX #H2WX
+elim (cpm_lifts_sn … H1WX … HLK2 … HXW) -H1WX -HLK2 #WX #HWX #HXWX
+lapply (tdeq_lifts_bi … H2WX … HXW … HWX) -H2WX -HXW #H2WX
+/3 width=6 by cpm_ell_drops, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+ ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+ ∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 →
+ ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓑ{p,I}V.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[n1-n2,h] T & ⓑ{p,I}V2.T2 ≛[h,o] T.
+#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0
+#V2 #HV02 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #HT02
+#L1 #H1L01 #H2L01 #L2 #HL02
+elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
+elim (rdeq_inv_bind … H2L01) -H2L01 #H0L01 #H2L01
+elim (IH … V0 … 0 V0 … HV02 … H1L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-H0L01 <minus_n_O #V #HV1 #H1V2 #H2V2
+elim (IH … H1T01 H2T01 … HT02 (L1.ⓑ{I}V0) … (L2.ⓑ{I}V2)) [|*: /2 width=1 by fqup_fpbg, lpr_pair/ ]
+#T #HT1 #H1T2 #H2T2 -L0 -T0
+/3 width=6 by cpm_bind, tdeq_pair, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+ ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 →
+ ∀T2. ⬆*[1]T2 ≘ T → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,+ⓓV.T]L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ +ⓓV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡[n1-n2,h] T & XT2 ≛[h,o] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #T1 #H1T01 #H2T01 #T2 #HT20 #n2 #XT2 #HXT2
+#L1 #H1L01 #H2L01 #L2 #HL02
+elim (cnv_inv_bind … H0) -H0 #_ #HT0
+elim (rdeq_inv_bind … H2L01) -H2L01 #_ #H2L01
+lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HT20) -HT0
+[ /3 width=3 by drops_refl, drops_drop/ ] #HT2
+elim (cpm_inv_lifts_sn … H1T01 (Ⓣ) … L0 … HT20) -H1T01
+[| /3 width=1 by drops_refl, drops_drop/ ] #XT1 #HXT1 #H1XT12
+lapply (tdeq_inv_lifts_bi … H2T01 … HT20 … HXT1) -H2T01 #H2XT12
+lapply (rdeq_inv_lifts_bi … H2L01 (Ⓣ) … L0 L1 … HT20) -H2L01
+[4:|*: /3 width=1 by drops_refl, drops_drop/ ] #H2L01
+elim (IH … H1XT12 H2XT12 … HXT2 … H1L01 … HL02)
+[|*: /3 width=1 by fqup_fpbg, fqup_zeta/ ] -L0 -T0 #T #HT1 #H1T2 #H2T2
+/3 width=4 by cpm_zeta, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+ ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
+ ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡[n1-n2,h] T & ⓐV2.T2 ≛[h,o] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#V2 #HV02 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #HT02
+#L1 #H1L01 #H2L01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
+elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01
+elim (IH … V0 … 0 V0 … HV02 … H1L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-H0L01 <minus_n_O #V #HV1 #H1V2 #H2V2
+elim (IH … H1T01 H2T01 … HT02 … H1L01 H2L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #H1T2 #H2T2 -L0 -T0
+/3 width=6 by cpm_appl, tdeq_pair, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+ ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+ ∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ⓛ{p}W.T ≛[h,o] T1 →
+ ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.ⓛ{p}W.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[n1-n2,h] T & ⓓ{p}ⓝW2.V2.T2 ≛[h,o] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V2 #HV02 #W2 #HW02 #n1 #X #H1X #H2X #n2 #T2 #HT02
+#L1 #H1L01 #H2L01 #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 … H1X) -H1X #W1 #T1 #H1W01 #H1T01 #H destruct
+elim (tdeq_inv_pair … H2X) -H2X #_ #H2W01 #H2T01
+elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01
+elim (rdeq_inv_bind … H2L01) -H2L01 #H3L01 #H2L01
+elim (IH … V0 … 0 V0 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-HV0 -HV02 -H0L01 <minus_n_O #V #HV1 #H1V2 #H2V2
+elim (IH … H1W01 H2W01 … HW02 … H1L01 H3L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-HW0 <minus_n_O #W #HW1 #H1W2 #H2W2
+elim (IH … H1T01 H2T01 … HT02 (L1.ⓛW1) … (L2.ⓛW2))
+[|*: /3 width=2 by fqup_fpbg, lpr_pair, rdeq_bind_repl_dx, ext2_pair/ ]
+#T #HT1 #H1T2 #H2T02 -L0 -W0 -T0
+lapply (lsubr_cpm_trans … H1T2 (L2.ⓓⓝW2.V2) ?) -H1T2 [ /2 width=1 by lsubr_beta/ ] #H1T2
+/4 width=6 by cpm_beta, cpm_cast, cpm_bind, tdeq_pair, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+ ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
+ ∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ⓓ{p}W.T ≛[h,o] T1 →
+ ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 → ∀U2. ⬆*[1]V2 ≘ U2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.ⓓ{p}W.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[n1-n2,h] T & ⓓ{p}W2.ⓐU2.T2 ≛[h,o] T.
+#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#V2 #HV02 #W2 #HW02 #n1 #X #H1X #H2X #n2 #T2 #HT02 #U2 #HVU2
+#L1 #H1L01 #H2L01 #L2 #HL02
+elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
+elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T1 #HW0 #HT0 #H1T01 #H2T01 #H destruct
+elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01
+elim (rdeq_inv_bind … H2L01) -H2L01 #H3L01 #H2L01
+elim (IH … V0 … 0 V0 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-HV0 -HV02 -H0L01 <minus_n_O #V #HV1 #H1V2 #H2V2
+elim (IH … W0 … 0 W0 … HW02 … H1L01 H3L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+-HW0 <minus_n_O #W #HW1 #H1W2 #H2W2
+elim (IH … H1T01 H2T01 … HT02 (L1.ⓓW0) … (L2.ⓓW2))
+[|*: /3 width=2 by fqup_fpbg, lpr_bind_refl_dx, lpr_bind, ext2_pair/ ]
+#T #HT1 #H1T2 #H2T02 -L0 -T0
+elim (cpm_lifts_sn … H1V2 (Ⓣ) … (L2.ⓓW2) … HVU2) -H1V2
+[| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #H1U2
+lapply (tdeq_lifts_bi … H2V2 … HVU2 … HVU) -H2V2 #H2U2
+/4 width=8 by cpm_theta, cpm_appl, cpm_bind, tdeq_pair, ex3_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+ ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 →
+ ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+ ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2≛[h,o]T.
+#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#n1 #V1 #H1V01 #H2V01 #n2 #V2 #HV02 #T1 #H1T01 #H2T01 #T2 #HT02
+#L1 #H1L01 #H2L01 #L2 #HL02
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
+elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01
+elim (IH … H1V01 H2V01 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+elim (IH … H1T01 H2T01 … HT02 … H1L01 H2L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+#T #HT1 #H1T2 #H2T2 #V #HV1 #H1V2 #H2V2 -L0 -V0 -T0
+/3 width=6 by cpm_cast, tdeq_pair, ex3_intro/
+qed-.
+(*
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+ ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 →
+ ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] 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_sub … 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_tdeq1_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_cpm_tdeq1_conf_lpr a h o G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+ ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 →
+ ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
+ ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o]T1 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T]L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡[n1-↑n2,h] T & V2 ≛[h,o] T.
+#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#n1 #V1 #H1V01 #H2V01 #n2 #V2 #HV02 #T1 #H1T01 #H2T01
+#L1 #H1L01 #H2L01 #L2 #HL02
+(* n2=0 n1=1 *)
+ -H1V01 -H2V01
+elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
+elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01
+lapply (cnv_cpms_trans_lpr_sub … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0
+elim (cnv_cpms_strip_lpr_sub … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
+elim (cnv_cpms_strip_lpr_sub … 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_l2 #H1
+/3 width=3 by cpms_eps, ex2_intro/
+qed-.
+
+fact cnv_cpm_tdeq1_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_cpm_tdeq1_conf_lpr a h o G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_tdeq1_conf_lpr a h o G1 L1 T1.
+#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #HX2 #L1 #H1L1 #H2L1 #L2 #HL2 destruct
+ elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 *
+ elim (cpm_inv_atom1_drops … HX2) -HX2 *
+ [ #H21 #H22 #H11 #H12 destruct -a -L
+ <minus_O_n
+ /2 width=1 by cnv_cpm_tdeq1_conf_lpr_atom_atom_aux/
+ | #s2 #H21 #H22 #H23 #H11 #H12 destruct -a -L
+ <minus_O_n <minus_n_O
+ /2 width=1 by cnv_cpm_tdeq1_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_tdeq1_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_tdeq1_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
+ | #H21 #H22 #s1 #H11 #H12 #H13 #H14 destruct -a -L
+ <minus_O_n <minus_n_O
+ /2 width=1 by cnv_cpm_tdeq1_conf_lpr_ess_atom_aux/
+ | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 #_ destruct -a -L
+ <minus_n_n
+ /2 width=1 by cnv_cpm_tdeq1_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
+ ]
+| #p #I #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #HX2 #L1 #H1L1 #H2L1 #L2 #HL2 destruct
+ elim (cpm_tdeq_inv_bind_sn … HT … H1X1 H2X1) -H1X1 -H2X1
+ elim (cpm_inv_bind1 … HX2) -HX2 *
+ [ #V2 #T2 #HV2 #HT2 #H21 #T1 #_ #_ #H1T1 #H2T1 #H11 destruct -IH2
+ @(cnv_cpm_tdeq1_conf_lpr_bind_bind_aux … IH1) -IH1 /1 width=1 by/
+ | #T2 #HT2 #HTX2 #H21 #H22 #T1 #_ #_ #H1T1 #H2T1 #H11 destruct -IH2
+ @(cnv_cpm_tdeq1_conf_lpr_bind_zeta_aux … IH1) -IH1 /1 width=3 by/
+ ]
+| #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #HX2 #L1 #H1L1 #H2L1 #L2 #HL2 destruct
+ elim (cpm_tdeq_inv_appl_sn … HT … H1X1 H2X1) -H1X1 -H2X1
+ elim (cpm_inv_appl1 … HX2) -HX2 *
+ [ #V2 #T2 #HV2 #HT2 #H21 #m #q #W1 #U1 #T1 #_ #_ #_ #_ #_ #H1T1 #H2T1 #H11 destruct -IH2 -m -q -W1 -U1
+ @(cnv_cpm_tdeq1_conf_lpr_appl_appl_aux … IH1) -IH1 /1 width=1 by/
+ | #p2 #V2 #XW2 #W2 #XT2 #T2 #HV2 #HW2 #HT2 #H21 #H22 #m #q #W1 #U1 #T1 #_ #_ #_ #_ #_ #H1T1 #H2T1 #H11 destruct -IH2 -m -q -W1 -U1
+ @(cnv_cpm_tdeq1_conf_lpr_appl_beta_aux … IH1) -IH1 /1 width=1 by/
+ | #p2 #V2 #XV2 #XW2 #W2 #XT2 #T2 #HV2 #HXV2 #HW2 #HT2 #H21 #H22 #m #q #W1 #U1 #T1 #_ #_ #_ #_ #_ #H1T1 #H2T1 #H11 destruct -IH2 -m -q -W1 -U1
+ @(cnv_cpm_tdeq1_conf_lpr_appl_theta_aux … IH1) -IH1 /1 width=3 by/
+ ]
+| #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #HX2 #L1 #H1L1 #H2L1 #L2 #HL2 destruct
+ elim (cpm_tdeq_inv_cast_sn … HT … H1X1 H2X1) -H1X1 -H2X1
+ elim (cpm_inv_cast1 … HX2) -HX2 [ * || * ]
+ [ #V2 #T2 #HV2 #HT2 #H21 #U1 #V1 #T1 #_ #_ #_ #H1V1 #H2V1 #_ #H1T1 #H2T1 #H11 destruct -IH2 -U1
+ @(cnv_cpm_tdeq1_conf_lpr_cast_cast_aux … IH1) -IH1 /1 width=1 by/
+ | #HT2 #U1 #V1 #T1 #_ #_ #_ #H1V1 #H2V1 #_ #H1T1 #H2T1 #H11 destruct -IH2 -U1
+(* @(cnv_cpm_tdeq1_conf_lpr_cast_epsilon_aux … IH1) -IH1 /1 width=1 by/ *)
+ | #m2 #HV2 #H21 #U1 #V1 #T1 #_ #_ #_ #H1V1 #H2V1 #_ #H1T1 #H2T1 #H11 destruct -U1
+ @(cnv_cpm_tdeq1_conf_lpr_cast_ee_aux … IH2 IH1) -IH2 -IH1 /1 width=1 by/
+ ]
+]
projects / helm.git / commitdiff