(* *)
(**************************************************************************)
-include "basic_2/static/ssta_ltpss_dx.ma".
-include "basic_2/computation/xprs_lift.ma".
-include "basic_2/equivalence/lsubse_ssta.ma".
-include "basic_2/equivalence/fpcs_cpcs.ma".
-include "basic_2/equivalence/lfpcs_fpcs.ma".
-include "basic_2/dynamic/snv_ssta.ma".
+include "basic_2/computation/dxprs_dxprs.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 ssta_ltpr_tpr_aux: ∀h,g,n. (
- ∀L,T2. ♯{L,T2} < n →
- ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
- L ⊢ U1 ⬌* U2 ∧ l1 = l2
- ) → (
- ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
- ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
- ) → (
- ∀L1,T1. ♯{L1,T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) →
- ∀L1,T1. ♯{L1,T1} = n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 * * [|||| *]
-[ #k #_ #Y #l #H1 #L2 #HL12 #X #H2 #_ -IH3 -IH2 -IH1
- elim (ssta_inv_sort1 … H1) -H1 #Hkl #H destruct
- >(tpr_inv_atom1 … H2) -X /4 width=6/
-| #i #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- elim (ssta_inv_lref1 … H1) -H1 * #K1
+fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr 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_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ltpr_tpr h g L1 T1.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [||||*]
+[ #k #HL0 #HT0 #H1 #L2 #_ #X #H2 destruct -IH3 -IH2 -IH1 -L1
+ >(tpr_inv_atom1 … H2) -X //
+| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
>(tpr_inv_atom1 … H2) -X
- elim (snv_inv_lref … H3) -H3 #I0 #K0 #V0 #H #HV1
- [ #V1 #W1 #HLK1 #HVW1 #HWU1
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 -HV12 // -HV1 -HKV1 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #H1
- lapply (ldrop_fwd_ldrop2 … HLK2) #H2
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (fpcs_lift … HW12 … H1 H2 … HWU1 … HWU2) -H1 -H2 -W1 [ /3 width=1/ ] /3 width=6/
- | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 // -HV1 -HK12 -HKV1 #W2 #HVW2 #_ -W1
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
- ]
-| #p #Hn #U1 #l #H1 -IH3 -IH1
- elim (ssta_inv_gref1 … H1)
-| #a #I #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- elim (ssta_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
- elim (snv_inv_bind … H3) -H3 #_ #HT1
- elim (tpr_inv_bind1 … H2) -H2 *
- [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
- elim (IH1 … HTU1 (L2.ⓑ{I}V2) … HT10) -IH1 -HTU1 -HT10 // -T1 /3 width=1/ -HL12 #U0 #HTU0 #HU10
- lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
- elim (ssta_tps_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
- lapply (cpr_intro … U0 … HU02) -HU02 // #HU02
- lapply (fpcs_fpr_strap1 … HU10 (L2.ⓑ{I}V2) U2 ?) [ /2 width=1/ ] -U0 #HU12
- lapply (fpcs_fwd_shift … HU12 a) -HU12 /3 width=3/
- | #T2 #HT12 #HT2 #H1 #H2 destruct
- elim (IH1 … HTU1 (L2.ⓓV1) … HT12) -IH1 -HTU1 -HT12 // -T1 [2: /3 width=1/ ] -HL12 #U2 #HTU2 #HU12
- lapply (fpcs_fwd_shift … HU12 true) -HU12 #HU12
- elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
- lapply (fpcs_fpr_strap1 … HU12 L2 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
- ]
-| #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct
- elim (ssta_inv_appl1 … H1) -H1 #U1 #HTU1 #H destruct
- elim (snv_inv_appl … H3) -H3 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
- elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // /3 width=5/
- | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HT2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
- elim (xprs_fwd_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #H destruct
- lapply (cprs_div … HW10 … HW0) -W0 #HW1
- elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
- elim (snv_ssta … HW) #V #l1 #HWV
- elim (IH3 … HW1 … HW … HWX1 … HWV) -IH3 -HWX1 // [2: /2 width=5/ ] -IH2 #_ #H destruct -X1
- elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 #W2 #HVW2 #HW12
- elim (IH1 … HWV … HL12 W) -HWV // -HW #V0 #HWV0 #_
- elim (IH1 … HTU2 (L2.ⓛW) … HT20 HT2) -IH1 -HTU2 -HT20 -HT2 // [2: /2 width=1/ ] #U20 #HTU20 #HU20
- lapply (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
- elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
- [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
- @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
- @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
- /4 width=4 by fpcs_fwd_shift, fpcs_canc_dx, cpcs_fpcs, fpcs_fwd_abst13/
- | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
- /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
- ]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (snv_inv_bind … HT1) -HT1 #_ #HT0
- elim (IH1 … HTU0 (L2.ⓓW2) … HT02 HT0) -IH1 -HTU0 -HT02 -HT0 // -T0 [2: /2 width=1/ ] -HL12 -HW02 #U2 #HTU2 #HU02
- lapply (fpcs_fwd_shift … HU02 b) -HU02 #HU02
- lapply (fpcs_flat_dx_tpr … HU02 … HV10 Appl) -HV10 -HU02 #HU02
- lapply (fpcs_fpr_strap1 … HU02 L2 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ @ltpr_tpr_fpr // /2 width=3/ ] -V0 /4 width=3/
- ]
-| #U0 #T1 #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- lapply (ssta_inv_cast1 … H1) -H1 #HTU1
- elim (snv_inv_cast … H3) -H3 #T0 #l0 #_ #HT1 #HT10 #_
- elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
- elim (tpr_inv_cast1 … H2) -H2
- [ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // -T1 -U0 /3 width=3/
- | #HT1X
- elim (IH1 … HTU1 … HL12 … HT1X HT1) -IH1 -HTU1 -HL12 -HT1X -HT1 // -U0 -T1 /2 width=3/
- ]
-]
-qed-.
-(*
-fact snv_ltpr_tpr_aux: ∀h,g,L,T. (
- ∀L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- h ⊢ ⦃L, T⦄ •⭃*[g] ⦃L1 ,T1⦄ →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) → (
- ∀L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 →
- h ⊢ ⦃L, T⦄ •⭃*[g] ⦃L1 ,T1⦄ → ⦃h, L2⦄ ⊩ T2 :[g]
- ) →
- ∀L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
- L = L1 → T = T1 → ⦃h, L2⦄ ⊩ T2 :[g].
-#h #g #L #T #IH2 #IH1 #L1 #T1 * -L1 -T1
-[ #L1 #k #L2 #_ #X #H #H1 #H2 destruct -IH2 -IH1 -L1
- >(tpr_inv_atom1 … H) -X //
-| #I #L1 #K1 #V1 #i #HLK1 #HV1 #L2 #HL12 #X #H #H1 #H2 destruct -IH2
- >(tpr_inv_atom1 … H) -X
elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IH1 … HV1 … HK12 V2 ? ?) -IH1 -HV1 -HK12 /4 width=1/ -HV12 /3 width=4/ -HLK1 /3 width=5/
-| #a #I #L1 #V1 #T1 #HV1 #HT1 #L2 #HL12 #X #H #H1 #H2 destruct -IH2
- elim (tpr_inv_bind1 … H) -H *
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
+ lapply (IH1 … HK12 … HV12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HLK1 /2 width=5/
+| #p #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+ elim (tpr_inv_bind1 … H2) -H2 *
[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
- lapply (cpr_intro (L2.ⓑ{I}V2) … T2 0 1 HT10 ?) -HT10 /2 width=1/ -HT02 #HT12
- lapply (IH1 … HV1 … HL12 V2 ? ?) -HV1 /4 width=1/ #HV2
- lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ? ?) -IH1 -HT1 /3 width=1/
+ lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
+ lapply (snv_ltpr_cpr_aux … HT1 (L2.ⓑ{I}V2) … T2 ?) -HT1
+ [ /3 width=5 by cpr_intro, tps_tpss/ | /2 width=1/ | /3 width=1/ ] -IH1 -T0 /2 width=1/
| #T2 #HT12 #HXT2 #H1 #H2 destruct
- lapply (IH1 … HT1 (L2.ⓓV1) ? T2 ? ?) -IH1 -HT1 /4 width=1/ -HT12 -HL12 #HT2
+ lapply (IH1 … HT1 (L2.ⓓV1) … HT12) -IH1 -HT1 -HT12 [1,2: /2 width=1/ ] -HL12 #HT2
lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
]
-| #a #L1 #V1 #W1 #W10 #T1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1 #L2 #HL12 #X #H #H1 #H2 destruct
- elim (tpr_inv_appl1 … H) -H *
+| #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+ elim (tpr_inv_appl1 … H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH1 … HV1 … HL12 V2 ? ?) /4 width=1/ #HV2
- lapply (IH1 … HT1 … HL12 T2 ? ?) /4 width=1/ #HT2
- lapply (IH1 … HT1 … HTU1 ?) -IH1 -HT1 // /2 width=1/ #H
- elim (snv_inv_bind … H) -H #HW10 #HU1
- elim (IH2 … HVW1 … HL12 … HV12 HV1 ?) -IH2 -HVW1 -HL12 -HV12 -HV1 /2 width=1/ #W2 #HVW2 #HW12
- lapply (fpcs_canc_sn L1 L1 … W10 … HW12) -HW12 /3 width=1/ -W1 #HW102
- @(snv_appl … HV2 HT2 HVW2)
-*)
+ lapply (IH1 … HV1 … HL12 … HV12) [ /2 width=1/ ] #HV2
+ lapply (IH1 … HT1 … HL12 … HT12) [ /2 width=1/ ] #HT2
+ elim (IH3 … HVW1 … HL12 … HV12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+ elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU1 … HL12 T2) // [2: /3 width=1/ |3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
+ elim (cprs_inv_abst1 Abst W1 … H) -H #W20 #U2 #HW120 #_ #H destruct
+ lapply (ltpr_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 /2 width=8/
+ | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
+ elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
+ lapply (cprs_div … HW230 … HW10) -W30 #HW210
+ lapply (ltpr_cpcs_conf … HL12 … HW210) -HW210 #HW210
+ lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
+ lapply (IH1 … HW20 … HL12 W20 ?) // [ /2 width=1/ ] -HW20 #HW20
+ lapply (IH1 … HT20 … (L2.ⓛW20) … HT202) [1,2: /2 width=1/ ] -IH1 -HT20 -HT202 #HT2
+ elim (IH3 … HVW1 … HL12 … HV12) // [2: /2 width=1/ ] -HV1 -HVW1 -HV12 #W200 #HVW200 #H
+ lapply (cpcs_trans … HW210 … H) -W10 #HW200
+ lapply (lsubsv_snv_trans … HT2 (L2.ⓓV2) ?) -L1 -HT2 /2 width=1/ /2 width=4/
+ |
\ No newline at end of file
projects / helm.git / blobdiff