(* *)
(**************************************************************************)
-include "basic_2/static/lsubd_da.ma".
-include "basic_2/computation/cpds_cpds.ma".
+include "static_2/static/lsubd_da.ma".
include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
+include "basic_2/dynamic/snv_scpes.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
(* Properties on degree assignment for terms ********************************)
-fact da_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h g G1 L1 T1.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #_ #_ #_ #_ #l #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
+fact da_cpr_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h o G1 L1 T1.
+#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #s #_ #_ #_ #_ #d #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
lapply (da_inv_sort … H2) -H2
- lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1/
-| #i #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
+| #i #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
- elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
- lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #d1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
+ lapply (drop_mono … H … HLK1) -H #H destruct
elim (cpr_inv_lref1 … H3) -H3
[1,3: #H destruct
- lapply (fsupp_lref … G1 … HLK1)
- elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=10 by da_ldef, da_ldec, fsupp_fpbg/
+ /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
|2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (fsupp_lref … G1 … HLK1)
- elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ lapply (drop_mono … H … HLK1) -H #H destruct
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -V2
- /4 width=7 by da_lift, fsupp_fpbg/
+ lapply (drop_fwd_drop2 … HLK2) -V2
+ /4 width=8 by da_lift, fqup_fpbg/
]
| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH2
elim (snv_inv_bind … H1) -H1 #_ #HT1
lapply (da_inv_bind … H2) -H2
elim (cpr_inv_bind1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=9 by da_bind, fsupp_fpbg, lpr_pair/
+ /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
| #T2 #HT12 #HT2 #H1 #H2 destruct
- /4 width=11 by da_inv_lift, fsupp_fpbg, lpr_pair, ldrop_ldrop/
+ /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, drop_drop/
]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
- lapply (da_inv_flat … H2) -H2 #Hl
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #b1 #W1 #U1 #d1 #HV1 #HT1 #HVW1 #HTU1
+ lapply (da_inv_flat … H2) -H2 #Hd
elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fsupp_fpbg/
- | #b #V2 #W #W2 #U1 #U2 #HV12 #HW2 #HU12 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HU1
- lapply (da_inv_bind … Hl) -Hl #Hl
- elim (cpds_inv_abst1 … HT10) -HT10 #W3 #U3 #HW3 #_ #H destruct -U3
- lapply (cprs_div … HW3 … HW10) -W3 #HWW1
- lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #H
- elim (snv_fwd_da … HW) #l1 #Hl1
- lapply (IH3 … HV1 … 1 … Hl0 W1 ?) /2 width=2 by fsupp_fpbg, ssta_lsstas/ #HW1
- lapply (da_cpcs_aux … IH2 IH1 … Hl1 … H … HWW1) -H
- /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, ssta_fpbs/ #H destruct
- lapply (IH1 … HV1 … Hl0 … HV12 … HL12) -HV1 -Hl0 -HV12 [ /2 by fsupp_fpbg/ ] #Hl0
- lapply (IH1 … Hl1 … HW2 … HL12) -Hl1 // /2 width=1 by fsupp_fpbg/ -HW
- lapply (IH1 … HU1 … Hl … HU12 (L2.ⓛW2) ?) -IH1 -HU1 -Hl -HU12 [1,2: /2 by fsupp_fpbg, lpr_pair/ ] -HL12 -HW2
- /4 width=6 by da_bind, lsubd_da_trans, lsubd_abbr/
- | #b #V #V2 #W #W2 #U1 #U2 #HV1 #HV2 #HW2 #HU12 #H1 #H2 destruct -IH3 -IH2 -V -W0 -T0 -l0 -HV1 -HVW1
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fqup_fpbg/
+ | #b #V2 #W #W2 #U #U2 #HV12 #HW2 #HU2 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HU
+ lapply (da_inv_bind … Hd) -Hd #Hd
+ elim (scpds_inv_abst1 … HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -d1
+ elim (snv_fwd_da … HV1) #d1 #Hd1
+ elim (snv_fwd_da … HW) #d0 #Hd0
+ lapply (cprs_scpds_div … HW3 … Hd0 … 1 HVW1) -W3 #H
+ elim (da_scpes_aux … IH3 IH2 IH1 … Hd0 … Hd1 … H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
+ <minus_n_O #H destruct >(plus_minus_k_k d1 1) in Hd1; // -H1 #Hd1
+ lapply (IH1 … HV1 … Hd1 … HV12 … HL12) -HV1 -Hd1 -HV12 [ /2 by fqup_fpbg/ ]
+ lapply (IH1 … Hd0 … HW2 … HL12) -Hd0 /2 width=1 by fqup_fpbg/ -HW
+ lapply (IH1 … HU … Hd … HU2 (L2.ⓛW2) ?) -IH1 -HU -Hd -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
+ /4 width=6 by da_bind, lsubd_da_trans, lsubd_beta/
+ | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -d1 -HV1
elim (snv_inv_bind … HT1) -HT1 #_
- lapply (da_inv_bind … Hl) -Hl
- /5 width=9 by da_bind, da_flat, fsupp_fpbg, lpr_pair/
+ lapply (da_inv_bind … Hd) -Hd
+ /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
- lapply (da_inv_flat … H2) -H2 #Hl
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
+ lapply (da_inv_flat … H2) -H2 #Hd
elim (cpr_inv_cast1 … H3) -H3
- [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fsupp_fpbg/
- | /3 width=7 by fsupp_fpbg/
+ [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
+ | /3 width=7 by fqup_fpbg/
]
]
qed-.
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