(* *)
(**************************************************************************)
-include "basic_2/computation/dxprs_dxprs.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/dynamic/snv_lift.ma".
include "basic_2/dynamic/snv_cpcs.ma".
+include "basic_2/dynamic/lsubsv_snv.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
(* Properties on context-free parallel reduction for local environments *****)
-fact snv_cpr_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g L1 T1.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
-[ #k #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
+fact snv_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_lsstas_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, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
>(cpr_inv_sort1 … H2) -X //
-| #i #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKL
+ lapply (fqup_lref … G1 … HLK1) #HKL
elim (cpr_inv_lref1 … H2) -H2
- [ #H destruct -HLK1
- lapply (IH1 … HV12 … HK12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HKL /2 width=5/
+ [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
| * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
- lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (IH1 … HVW0 … HK12) -IH1 -HVW0 -HK12 // -HV1 [ /2 width=1/ ] -HKL /3 width=7/
+ lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 /4 width=7 by fqup_fpbg, snv_lift/
]
-| #p #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
+| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
elim (cpr_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH1 … HV12 … HL12) // -HV1 [ /2 width=1/ ] #HV2
- lapply (IH1 … HT12 (L2.ⓑ{I}V2) ?) // -IH1 /2 width=1/
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpr_pair/
| #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
- lapply (IH1 … HT1 … HT12 (L2.ⓓV1) ?) -IH1 [1,2: /2 width=1/ ] -L1 -T1 #HT2
- lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+ /4 width=10 by fqup_fpbg, snv_inv_lift, lpr_pair, ldrop_ldrop/
]
-| #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1
elim (cpr_inv_appl1 … H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct -IH4
- lapply (IH1 … HV1 … HV12 … HL12) [ /2 width=1/ ] #HV2
- lapply (IH1 … HT1 … HT12 … HL12) [ /2 width=1/ ] #HT2
- elim (IH3 … HVW1 … HV12 … HL12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
- elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU1 … T2 … HL12) // [2,3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
- elim (cprs_fwd_abst1 … H Abst W1) -H #W20 #U2 #HW120 #_ #H destruct
- lapply (cprs_lpr_conf … HL12 … HW10) -L1 #HW10
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #H2l0
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbg/ -HV1 #W2 #HVW2 #HW12
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HT12 -HTU1 #X #HTU2 #H
+ elim (cprs_inv_abst1 … H) -H #W20 #U2 #HW120 #_ #H destruct
+ lapply (lpr_cprs_conf … HL12 … HW10) -L1 #HW10
lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
- lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 -HTU2 /2 width=8/
- | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
+ lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?)
+ /2 width=7 by snv_appl, cprs_bind/
+ | #b #V2 #W20 #W2 #T20 #T2 #HV12 #HW202 #HT202 #H1 #H2 destruct
elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
- elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
+ elim (cpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
lapply (cprs_div … HW10 … HW230) -W30 #HW120
- lapply (cpcs_lpr_conf … HL12 … HW120) -HW120 #HW120
- elim (IH3 … HVW1 … HV12 … HL12) // [2: /2 width=1/ ] -HVW1 #W2 #HVW2 #HW102
- lapply (cpcs_canc_sn … HW102 … HW120) -W10 #HW220
- lapply (IH1 … HV12 … HL12) // [ /2 width=1/ ] #HV2 -HV1
- lapply (IH1 … HW20 … W20 … HL12) // [ /2 width=1/ ] -HW20 #HW20
- lapply (IH1 … HT20 … HT202 … (L2.ⓛW20) ?) [1,2: /2 width=1/ ] -HT20 #HT2
- lapply (IH2 … HVW2) //
- [ @(ygt_yprs_trans … L1 L1 … V1) (**) (* auto /4 width=5/ is a bit slow even with trace *)
- [ /2 width=1 by fw_ygt/
- | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
- ]
- ] #HW2
- elim (snv_fwd_ssta … HW20) #l0 #U20 #HWU20
- elim (ssta_fwd_correct … HVW2) <minus_plus_m_m #U2 #HWU2
- elim (ssta_cpcs_lpr_aux … IH1 IH3 … HWU2 … HWU20 … HW220 … L2) // -IH3
- [2: /4 width=4 by ygt_yprs_trans, fw_ygt, ypr_yprs, ypr_lpr/
- |3: @(ygt_yprs_trans … L1 L2 … V2) (**) (* auto not tried *)
- [ @(ygt_yprs_trans … L1 L1 … V1)
- [ /2 width=1 by fw_ygt/
- | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
- ]
- | /3 width=2 by ypr_ssta, ypr_yprs/
- ]
- ] #H #_ destruct -IH2 -U2
- lapply (IH4 … HT2 (L2.ⓓV2) ?)
- [ /2 width=6/
- | @(ygt_yprs_trans … (L1.ⓛW20) … T2) (**) (* auto /5 width=5/ is too slow even with trace timeout=35 *)
- [ /4 width=4 by ygt_yprs_trans, fw_ygt, ypr_yprs, ypr_cpr/
- | /4 width=1 by ypr_yprs, ypr_lpr, lpr_pair/
- ]
- ] -L1 -V1 -W2 -T20 -U20 -W20 -l0 /2 width=1/
+ lapply (snv_ssta_aux … IH4 … Hl0 … HVW1) /2 width=1 by fqup_fpbg/ #HW10
+ lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #HlW10
+ elim (snv_fwd_da … HW20) #l #Hl
+ lapply (da_cpcs_aux … IH1 IH2 … HlW10 … Hl … HW120) // -HlW10
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, ssta_fpbs/ #H destruct
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HlV2
+ lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fqup_fpbg/ #HlW2
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #W3 #HV2W3 #HW103
+ lapply (ssta_da_conf … HV2W3 … HlV2) <minus_plus_m_m #HlW3
+ lapply (cpcs_cpr_strap1 … HW120 … HW202) -HW120 #HW102
+ lapply (lpr_cpcs_conf … HL12 … HW102) -HW102 #HW102
+ lapply (cpcs_canc_sn … HW103 … HW102) -W10 #HW32
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 #HV2
+ lapply (IH1 … HW202 … HL12) /2 width=1 by fqup_fpbg/ -HW20 #HW2
+ lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT20 #HT2
+ lapply (snv_ssta_aux … IH4 … HlV2 … HV2W3)
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW3
+ lapply (lsubsv_snv_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /3 width=3 by snv_bind, snv_cast/
+ @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/ #W #W0 #l0 #Hl0 #HV2W #HW20
+ lapply (lsstas_ssta_conf_pos … HV2W3 … HV2W) -HV2W #HW3W
+ @(lsstas_cpcs_lpr_aux … IH1 IH2 IH3 … HlW3 … HW3W … HlW2 … HW20 … HW32) //
+ [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_ssta_fpbs/
+ | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
+ ]
| #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
- elim (dxprs_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+ elim (cpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
- lapply (cprs_lpr_conf … HL12 … HW10) -HW10 #HW10
- elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU0 T2 … (L2.ⓓW2) ?) // [2,3,4: /2 width=1/ ] -IH2 -HTU0 #X #HTU2 #H
- elim (cprs_fwd_abst1 … H Abst W3) -H #W #U2 #HW1 #_ #H destruct -U3
- elim (IH3 … HVW1 … HV10 … HL12) // /2 width=1/ -IH3 -HVW1 #X #H1 #H2
+ lapply (lpr_cprs_conf … HL12 … HW10) -HW10 #HW10
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HTU2 #H
+ elim (cprs_inv_abst1 … H) -H #W #U2 #HW1 #_ #H destruct -U3
+ elim (ssta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HVW1 #X #H1 #H2
lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
elim (lift_total X 0 1) #W20 #H3
- lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
- lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
+ lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1 by ldrop_ldrop/ -H1 #HVW20
+ lapply (cpcs_lift … (L2.ⓓW2) … H3 … HW13 H2) /2 width=1 by ldrop_ldrop/ -HW13 -H3 -H2 #HW320
lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
- lapply (dxprs_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
- lapply (IH1 … HW02 … HL12) // [ /2 width=1/ ] -HW0 #HW2
- lapply (IH1 … HV10 … HL12) // [ /2 width=1/ ] -HV1 -HV10 #HV0
- lapply (IH1 … HT02 (L2.ⓓW2) ?) // [1,2: /2 width=1/ ] -L1 #HT2
- lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
+ lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) /2 width=1 by cprs_bind/ -HW3 -HTU2 #HTU2
+ lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -Hl0 #Hl0
+ lapply (da_lift … Hl0 (L2.ⓓW2) … HV02) /2 width=1 by ldrop_ldrop/ -Hl0 #Hl0
+ lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ -HW0 #HW2
+ lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV10 #HV0
+ lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
+ lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /3 width=7 by snv_bind, snv_appl, ldrop_ldrop/
]
-| #W1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
+ elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
elim (cpr_inv_cast1 … H2) -H2
[ * #W2 #T2 #HW12 #HT12 #H destruct
- lapply (cpcs_cprs_strap1 … HUW1 W2 ?) [ /2 width=1/ ] -HUW1 #H1
- lapply (IH1 … HW12 … HL12) // /2 width=1/ -HW1 -HW12 #HW2
- lapply (IH1 … HT12 … HL12) // /2 width=1/ -IH1 #HT2
- elim (IH3 … HTU1 … HT12 … HL12) // /2 width=1/ -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
- lapply (cpcs_lpr_conf … HL12 … H1) -L1 #H1
- lapply (cpcs_canc_sn … HU12 H1) -U1 /2 width=4/
- | #H -IH3 -HW1 -HTU1 -HUW1
- lapply (IH1 … H … HL12) // /2 width=1/
+ lapply (cpcs_cprs_strap1 … HUW1 W2 ?) /2 width=1 by cpr_cprs/ -HUW1 #H1
+ lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/ -HW1 -HW12 #HW2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH1 #HT2
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HTU1 #U2 #HTU2 #HU12
+ lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -HT1 -HT12 -Hl0 #Hl0
+ /4 width=7 by snv_cast, lpr_cpcs_conf, cpcs_canc_sn/
+ | #H -IH3 -IH2 -HW1 -HTU1 -HUW1
+ lapply (IH1 … H … HL12) /2 width=1 by fqup_fpbg/
]
]
qed-.
projects / helm.git / blobdiff