@(cprs_strap1 … IHT2) -IHT2 /2 width=1/
qed.
+lemma tpss_cprs: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ T1 ➡* T2.
+#L #T1 #T2 #d #e #HT12
+lapply (cpr_intro … T1 … HT12) // /2 width=1/
+qed.
+
(* Basic_1: was: pr3_pr1 *)
lemma tprs_cprs: ∀T1,T2. T1 ➡* T2 → ∀L. L ⊢ T1 ➡* T2.
#T1 #T2 #H @(tprs_ind … H) -T2 /2 width=1/ /3 width=3/
(**************************************************************************)
include "basic_2/reducibility/ltpr_tps.ma".
-include "basic_2/reducibility/cpr_ltpss.ma".
+include "basic_2/reducibility/cpr_ltpss_sn.ma".
include "basic_2/reducibility/lfpr.ma".
include "basic_2/computation/cprs.ma".
--- /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/reducibility/cpr_ltpss_dx.ma".
+include "basic_2/computation/cprs_tpss.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma cprs_ltpss_dx_conf: ∀L1,T,U1. L1 ⊢ T ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 #H @(cprs_ind … H) -U1 /2 width=3/
+#T1 #U1 #_ #HTU1 #IHT1 #L2 #d #e #HL12
+elim (IHT1 … HL12) -IHT1 #U #HTU #HT1U
+elim (ltpss_dx_cpr_conf … HTU1 … HL12) -L1 #U0 #HT1U0 #HU10
+elim (cpr_tpss_conf … HT1U0 … HT1U) -T1 #U2 #HU02 #HU2
+lapply (tpss_trans_eq … HU10 HU02) -U0 /3 width=3/
+qed-.
+
+lemma cprs_ltpss_dx_tpss_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (cprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
+elim (cprs_tpss_conf … HT1U … HT12) -T1 #T #HUT #HT2
+lapply (tpss_trans_eq … HU1 HUT) -U /2 width=3/
+qed-.
+
+lemma cprs_ltpss_dx_tpss2_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∀U2. L2 ⊢ U1 ▶* [d, e] U2 →
+ ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* [d, e] U.
+#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 #U2 #HU12
+elim (cprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -L1 -T1 #U #HT2U #HU1
+elim (tpss_conf_eq … HU12 … HU1) -U1 #U0 #HU20 #HU0
+lapply (cprs_tpss_trans … HT2U … HU0) -U /2 width=3/
+qed-.
--- /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/unfold/ltpss_sn_alt.ma".
+include "basic_2/computation/cprs_ltpss_dx.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma cprs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. L1 ⊢ T ➡* U1 →
+ ∃∃U2. L2 ⊢ T ➡* U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
+elim (cprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /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/reducibility/cpr_tpss.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties on partial unfold for terms ***********************************)
+
+lemma cprs_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
+ ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡* T2.
+#L #T1 #T #H @(cprs_ind … H) -T /2 width=3/ /3 width=5/
+qed.
+
+lemma cprs_tps_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
+ ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡* T2.
+/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *)
+
+lemma cprs_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 →
+ ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡* T.
+#L #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3/
+#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02
+elim (IHT0 … HT02) -T0 #T0 #HT0 #HT20
+elim (cpr_tpss_conf … HT1 … HT0) -T /3 width=5/
+qed-.
--- /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/unwind/sstas_ltpss_dx.ma".
+include "basic_2/computation/cprs_ltpss_dx.ma".
+include "basic_2/computation/dxprs.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma dxprs_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+elim (sstas_ltpss_dx_conf … HTU … HL12) -HTU #U0 #HTU0 #HU0
+elim (cprs_ltpss_dx_conf … HU1 … HL12) -L1 #U2 #HU2 #HU12
+elim (cprs_tpss_conf … HU2 … HU0) -U #U #HU2 #HU0
+lapply (tpss_trans_eq … HU12 HU2) -U2 /3 width=3/
+qed-.
+
+lemma dxprs_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*➡*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #d #e #HT12
+elim (sstas_tpss_conf … HTW1 … HT12) -T1 #W2 #HTW2 #HW12
+elim (cprs_tpss_conf … HWU1 … HW12) -W1 /3 width=3/
+qed-.
+
+lemma dxprs_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (dxprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
+elim (dxprs_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /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/unfold/ltpss_sn_alt.ma".
+include "basic_2/computation/dxprs_ltpss_dx.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma dxprs_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 [ /2 width=3/ ]
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
+elim (dxprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /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/dynamic/snv_lift.ma".
+include "basic_2/dynamic/snv_fpr_ssta.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for closures ***************)
+
+fact snv_ltpr_tpr_aux: ∀h,g,n. (
+ ∀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 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) →
+ ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
+#h #g #n #IH2 #IH1 #L1 * * [||||*]
+[ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
+ >(tpr_inv_atom1 … H2) -X //
+| #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ >(tpr_inv_atom1 … H2) -X
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
+ lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 //
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] -HV12 /2 width=5/
+| #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -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 //
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] #HV2
+ lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct
+ lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /2 width=2/
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] -HT12 -HL12 #HT2
+ lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+ ]
+| #V1 #T1 #Hn #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 ?)
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ | //
+ ] #HV2
+ lapply (IH1 … HT1 … HL12 T2 ?)
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ | //
+ ] #HT2
+ lapply (IH1 … HT1 … HTU1) -IH1 // #H
+ elim (snv_inv_bind … H) -H #HW1 #HU1
+ elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
+ lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
+ @(snv_appl … HV2 HT2 HVW2)
--- /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/static/ssta_ltpss_sn.ma".
+include "basic_2/computation/dxprs_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".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for closures ***************)
+
+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
+ >(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 (dxprs_inv_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 ssta_ltpr_cpr_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. L2 ⊢ 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. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #HL12 #T2 * #T0 #HT10 #HT02 #HT1
+elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1 … HL12 … HT10 HT1)
+-T1 -IH3 -IH2 -HL12 [2: /3 width=5/ ] -n #U0 #HTU0 #HU10
+elim (ssta_tpss_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
+lapply (fpcs_fpr_strap1 … HU10 L2 U2 ?) -HU10 /2 width=3/ -HTU2 /3 width=3/
+qed-.
+
+fact ssta_fpr_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,T2. ⦃L1, T1⦄ ➡ ⦃L2, 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,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #T2 #H12 #HT1
+elim (fpr_inv_all … H12) -H12 #L #HL1 #HT12 #HL2
+elim (ssta_ltpr_cpr_aux … IH3 IH2 … Hn … HTU1 … HL1 … HT12 HT1)
+-T1 -IH3 -IH2 -HL1 [2: /3 width=5/ ] -n #U #HTU #HU1
+elim (ssta_ltpss_sn_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
+lapply (fpcs_fpr_strap1 … HU1 L2 U2 ?) -HU1 /2 width=3/ -HTU2 /3 width=4/
+qed-.
--- /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/dynamic/snv_fpr.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel computation for closures *************)
+
+fact ssta_fprs_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. ⦃h, L⦄ ⊢ T •*➡*[g] U → ⦃h, L⦄ ⊩ U :[g]
+ ) → (
+ ∀L1,T1. ♯{L1,T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, 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,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #T2 #H12 #HT1
+@(fprs_ind … H12) -L2 -T2 [-IH1 /2 width=3/ ] (**) (* auto fails without -IH1 *)
+#L #L2 #T #T2 #HT1 #HT2 * #U #HTU #HU1
+(*
+lapply (IH2 …
+
+elim (ssta_fpr_aux … IH3 … Hn … HTU1 … HT2 HT1)
+-T1 -IH3 -IH2 -HL1 [2: /3 width=5/ ] -n #U #HTU #HU1
+*)
\ No newline at end of file
+++ /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/static/ssta_ltpss_dx.ma".
-include "basic_2/computation/dxprs_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".
-
-(* 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
- >(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 (dxprs_inv_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-.
+++ /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/dynamic/snv_ltpr_1.ma".
-include "basic_2/dynamic/snv_lift.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact snv_ltpr_tpr_aux: ∀h,g,n. (
- ∀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 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
- ) →
- ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-#h #g #n #IH2 #IH1 #L1 * * [||||*]
-[ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
- >(tpr_inv_atom1 … H2) -X //
-| #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- >(tpr_inv_atom1 … H2) -X
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
- lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 //
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] -HV12 /2 width=5/
-| #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -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 //
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] #HV2
- lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct
- lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /2 width=2/
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] -HT12 -HL12 #HT2
- lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
- ]
-| #V1 #T1 #Hn #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 ?)
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- | //
- ] #HV2
- lapply (IH1 … HT1 … HL12 T2 ?)
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- | //
- ] #HT2
- lapply (IH1 … HT1 … HTU1) -IH1 // #H
- elim (snv_inv_bind … H) -H #HW1 #HU1
- elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
- lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
- @(snv_appl … HV2 HT2 HVW2)
--- /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/computation/dxprs_ltpss_dx.ma".
+include "basic_2/equivalence/cpcs_ltpss_dx.ma".
+include "basic_2/dynamic/snv_lift.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma snv_ltpss_dx_tpss_conf: ∀h,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊩ T2 :[g].
+#h #g #L1 #T1 #H elim H -L1 -T1
+[ #L1 #k #L2 #d #e #_ #X #H
+ >(tpss_inv_sort1 … H) -X //
+| #I #L1 #K1 #V1 #i #HLK1 #HV1 #IHV1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ | elim (lt_or_ge i (d + e)) #Hide ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 …) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
+ | elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=5/
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H1 #H2
+ elim (ltpss_dx_inv_tpss21 … H1 …) -H1 /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … H2 … HLK2) -H2 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ lapply (IHV1 … HK12 … HV1W2) -IHV1 -HK12 -HV1W2 /2 width=7/
+ ]
+| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
+ lapply (IHT1 … HT12) -IHT1 -HT12 /2 width=1/
+| #a #L1 #V1 #W1 #W0 #T1 #U1 #l #_ #_ #HVW1 #HW10 #HTU1 #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
+ lapply (IHT1 … HT12) -IHT1 /2 width=1/ #HT2
+ elim (ssta_ltpss_dx_tpss_conf … HVW1 … HL12 … HV12) -V1 #W2 #HVW2 #HW12
+ elim (dxprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #X #HTU2 #H
+ elim (tpss_inv_bind1 … H) -H #W #U2 #HW0 #_ #H -U1 destruct
+ elim (cprs_ltpss_dx_tpss2_conf … HW10 … HL12 … HW12 … HW0) -L1 -W1 -W0 #W0 #HW20 #HW0
+ lapply (dxprs_strap1 … (ⓛ{a}W0.U2) HTU2 ?) -HTU2 /3 width=3/ -HW0 /2 width=8/
+| #L1 #W1 #T1 #U1 #l #_ #_ #HTU1 #HUW1 #IHW1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (IHW1 … HL12 … HW12) -IHW1 #HW2
+ lapply (IHT1 … HL12 … HT12) -IHT1 #HT2
+ elim (ssta_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #U2 #HTU2 #HU12
+ lapply (cpcs_ltpss_dx_tpss2_conf … HL12 … HUW1 … HU12 … HW12) -L1 -W1 -U1 /2 width=4/
+]
+qed-.
+
+lemma snv_ltpss_dx_conf: ∀h,g,L1,T. ⦃h, L1⦄ ⊩ T :[g] →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊩ T :[g].
+#h #g #L1 #T #HT #L2 #d #e #HL12
+@(snv_ltpss_dx_tpss_conf … HT … HL12) //
+qed-.
+
+lemma snv_tpss_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊩ T1 :[g] →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊩ T2 :[g].
+#h #g #L #T1 #HT1 #T2 #d #e #HT12
+@(snv_ltpss_dx_tpss_conf … HT1 … HT12) //
+qed-.
+
+lemma snv_tps_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊩ T1 :[g] →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊩ T2 :[g].
+#h #g #L #T1 #HT1 #T2 #d #e #HT12
+@(snv_tpss_conf … HT1 T2) /2 width=3/
+qed-.
--- /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/dynamic/snv_ltpss_dx.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma snv_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T. ⦃h, L1⦄ ⊩ T :[g] → ⦃h, L2⦄ ⊩ T :[g].
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T1 #HT1
+lapply (IHL1 … HT1) -IHL1 -HT1 #HT1
+lapply (snv_ltpss_dx_conf … HT1 … HL2) -HT1 -HL2 //
+qed-.
lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
/3 width=3/ qed.
+lemma cpcs_tpss_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T →
+ ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ⬌* T2.
+#L #T1 #T #HT1 #T2 #d #e #HT2
+@(cpcs_cpr_strap1 … HT1) -T1 /2 width=3/
+qed-.
+
+lemma cpcs_tpss_conf: ∀L,T,T1,d,e. L ⊢ T ▶* [d, e] T1 →
+ ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+#L #T #T1 #d #e #HT1 #T2 #HT2
+@(cpcs_cpr_conf … HT2) -T2 /2 width=3/
+qed-.
+
(* Basic_1: removed theorems 9:
clear_pc3_trans pc3_ind_left
pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0
- Basic_1: removed local theorems 6:
+*)
+(* Basic_1: removed local theorems 6:
pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
pc3_wcpr0_t_aux
*)
+++ /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/equivalence/cpcs_cpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Properties concerning partial unfold on local environments ***************)
-
-lemma ltpss_dx_cpr_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 *
-lapply (ltpss_dx_weak_all … HL12)
->(ltpss_dx_fwd_length … HL12) -HL12 #HL12 #T #HT1 #HT2
-elim (ltpss_dx_tpss_conf … HT2 … HL12) -L1 #T0 #HT0 #HT20
-@(cprs_div … T0) /3 width=3/ (**) (* /4/ is too slow *)
-qed.
-
-lemma ltpss_dx_cprs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-@(cpcs_trans … IHT1) -T1 /2 width=5/
-qed.
-
-lemma ltpss_dx_cpcs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
-@(cpcs_canc_dx … T) /2 width=5/
-qed.
--- /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/computation/cprs_ltpss_dx.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma cpcs_ltpss_dx_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #HL12 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
+elim (cprs_ltpss_dx_conf … HT1 … HL12) -HT1 #U1 #HTU1 #H1
+elim (cprs_ltpss_dx_conf … HT2 … HL12) -L1 #U2 #HTU2 #H2
+elim (tpss_conf_eq … H1 … H2) -T #U #HU1 #HU2
+lapply (cprs_tpss_trans … HTU1 … HU1) -U1
+lapply (cprs_tpss_trans … HTU2 … HU2) -U2 /2 width=3/
+qed-.
+
+lemma cpcs_ltpss_dx_tpss_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T,T2. L1 ⊢ T ⬌* T2 →
+ ∀T1. L2 ⊢ T ▶* [d, e] T1 →
+ L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #HL12 #T #T2 #HT2 #T1 #HT1
+lapply (cpcs_ltpss_dx_conf … HL12 … HT2) -L1 #HT2
+lapply (cpcs_tpss_conf … HT1 … HT2) -T //
+qed-.
+
+lemma cpcs_ltpss_dx_tpss2_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀T3. L2 ⊢ T1 ▶* [d, e] T3 →
+ ∀T4. L2 ⊢ T2 ▶* [d, e] T4 →
+ L2 ⊢ T3 ⬌* T4.
+#L1 #L2 #d #e #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24
+lapply (cpcs_ltpss_dx_tpss_conf … HL12 … HT12 … HT13) -L1 -T1 #HT32
+lapply (cpcs_tpss_strap1 … HT32 … HT24) -T2 //
+qed-.
--- /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/equivalence/cpcs_ltpss_dx.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma cpcs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T1 #U1 #HTU1
+lapply (IHL1 … HTU1) -IHL1 -HTU1 #HTU1
+lapply (cpcs_ltpss_dx_conf … HL2 … HTU1) -HTU1 -HL2 //
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/unfold/ltpss_sn_alt.ma".
include "basic_2/reducibility/cpr_tpss.ma".
include "basic_2/reducibility/cpr_cpr.ma".
include "basic_2/reducibility/cfpr_ltpss.ma".
| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H
elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
lapply (cpr_bind_dx false … HV1 HT12) -HV1 -HT12 #HT12
- lapply (cpr_tpss_trans … HT12 (-ⓑ{I}V2.T2) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
+ lapply (cpr_tpss_trans … HT12 (-ⓑ{I}V2.T2) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
]
qed.
(**************************************************************************)
include "basic_2/reducibility/cpr_lift.ma".
-include "basic_2/reducibility/cpr_ltpss.ma".
+include "basic_2/reducibility/cpr_ltpss_sn.ma".
include "basic_2/reducibility/cfpr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+++ /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/unfold/ltpss_sn_ltpss_sn.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning partial unfold on local environments ***************)
-
-lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 *
-lapply (ltpss_sn_weak_all … HL12)
-<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/
-qed.
--- /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/unfold/ltpss_dx_ltpss_dx.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma ltpss_dx_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡ U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+lapply (ltpss_dx_fwd_length … HL12) #H >H in HU1; -H #HU1
+elim (ltpss_dx_tpss_conf … HU1 … HL12) -L1 /3 width=3/
+qed-.
--- /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/unfold/ltpss_sn_alt.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
+#L1 #L2 #d #e #HL12 #T1 #T2 *
+lapply (ltpss_sn_weak_all … HL12)
+<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/
+qed.
+
+lemma ltpss_sn_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡ U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+lapply (ltpss_sn_fwd_length … HL12) #H >H in HU1; -H #HU1
+elim (ltpss_sn_tpss_conf … HU1 … HL12) -HU1 -HL12 /3 width=3/
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/reducibility/tpr_tpss.ma".
include "basic_2/reducibility/cpr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
(* Properties on partial unfold for terms ***********************************)
lemma cpr_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
- ∀T2. L ⊢ T ▶* [O, |L|] T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T * #T0 #HT10 #HT0 #T2 #HT2
+ ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2
+lapply (tpss_weak_all … HT2) -HT2 #HT2
lapply (tpss_trans_eq … HT0 HT2) -T /2 width=3/
qed.
lemma cpr_tps_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
- ∀T2. L ⊢ T ▶ [O, |L|] T2 → L ⊢ T1 ➡ T2.
-/3 width=3/ qed.
+ ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed.
+
+lemma cpr_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡ T1 →
+ ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡ T.
+#L #T0 #T1 * #U0 #HTU0 #HU0T1 #T2 #d #e #HT02
+elim (tpr_tpss_conf … HTU0 … HT02) -T0 #T0 #HT20 #HUT0
+elim (tpss_conf_eq … HU0T1 … HUT0) -U0 /3 width=5/
+qed-.
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