--- /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/cif.ma".
+include "basic_2/reducibility/cnf_lift.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
+ elim (cif_inv_delta … HLK ?) //
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+ elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+ elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
+]
+qed.
+
+lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 #HT1
+lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
+qed.
+
+lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#T1 #T2 #H elim H -T1 -T2
+[ //
+| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
+ [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ | elim (cif_inv_ri2 … H) /2 width=1/
+ ]
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
+ [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+ elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+ lapply (IHT1 … HT1) -IHT1 #H destruct
+ lapply (tps_cif_eq … HT2 ?) -HT2 //
+ | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
+ elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
+ ]
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #V1 #T1 #T #T2 #_ #_ #_ #L #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+| #V1 #T1 #T2 #_ #_ #L #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+]
+qed.
+
+lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 * #T0 #HT10 #HT02 #HT1
+lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
+qed.
+
+theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+/3 width=3/ qed.
+
+(* Note: this property is unusual *)
+lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
+#L #T #H elim H -L -T
+[ #L #K #V #i #HLK #H
+ elim (cnf_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #L #V #T #_ #IHT #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #I #L #V #T * #H1 #H2 destruct
+ [ elim (cnf_inv_zeta … H2)
+ | elim (cnf_inv_tau … H2)
+ ]
+|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+ [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
+ |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
+ ]
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+qed.
+
+theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+/2 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/static/aaa_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_aaa.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
+#L #T1 #A #HT1 #T2 * /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/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpr_bind_sn: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
+@ex2_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr2_gen_cbind *)
+lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
+elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
+lapply (tpss_lsubr_trans … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
+lapply (tpss_inv_SO2 … HT0) -HT0 #HT0
+@ex2_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
+* /2 width=1/ /3 width=1/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @@ T1 ➡ L @@ T2.
+#L elim L -L
+[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
+| normalize /3 width=1/
+].
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr2_confluence *)
+theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
+ ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
+#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
+elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
+elim (ltpr_tpr_tpss_conf ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
+elim (ltpr_tpr_tpss_conf ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
+elim (tpss_conf_eq … HU1 … HU2) -U /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/reducibility/ltpr_tpss.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning parallel unfold on terms ***************************)
+
+(* Note: we could invoke tpss_weak_full instead of ltpr_fwd_length *)
+(* Basic_1: was only: pr2_subst1 *)
+lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
+elim (ltpr_tpr_tpss_conf … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
+elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+>(ltpr_fwd_length … HL12) in HT2; #HT2
+elim (ltpr_tpr_tpss_conf … HL12 … HT2) -L1 /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
+elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
+elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
+lapply (tpss_weak_full … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
+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_full … 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-.
--- /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/ltpr_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,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2
+lapply (tpss_weak_full … 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,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-.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L1 $L2 }.
+
+include "basic_2/unfold/ltpss_sn.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+definition lfpr: relation lenv ≝
+ λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2
+.
+
+interpretation
+ "focalized parallel reduction (environment)"
+ 'FocalizedPRed L1 L2 = (lfpr L1 L2).
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄.
+/2 width=3/ qed.
+
+lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄.
+/3 width=5/ qed.
+
+lemma ltpr_lfpr: ∀L1,L2. L1 ➡ L2 → ⦃L1⦄ ➡ ⦃L2⦄.
+/3 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆.
+#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 //
+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/static/aaa_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_aaa.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_lfpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A.
+#L1 #T #A #HT #L2 * /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_sn_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
+include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
+ ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄.
+#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
+<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I
+lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2
+@(ex2_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *)
+qed.
+
+(* Properties on supclosure *************************************************)
+(*
+lamma fsub_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃L1⦄ ➡ ⦃L⦄ & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2
+elim (fsub_tpr_trans … HT12 … H1) -T2 #L #U #HL1 #HT1U #HUT
+elim (fsup_tpss_trans_full … HUT … H2) -T -HUT -H2 #L #U #HL1 #HT1U #HUT
+
+
+
+
+
+
+ #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
+elim (lift_total T d e) #U #HTU
+elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
+lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+qed-.
+*)
\ 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/reducibility/ltpr_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_ltpr.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Main properties **********************************************************)
+
+theorem lfpr_conf: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ →
+ ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
+#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2
+lapply (ltpr_fwd_length … HK01) #H
+>(ltpr_fwd_length … HK02) in H; #H
+elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2
+lapply (ltpss_sn_fwd_length … HKL1) #H1
+lapply (ltpss_sn_fwd_length … HKL2) #H2
+>H1 in HKL1 H; -H1 #HKL1
+>H2 in HKL2; -H2 #HKL2 #H
+elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1
+elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2
+elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2
+lapply (ltpr_fwd_length … HLK1) #H1
+lapply (ltpr_fwd_length … HLK2) #H2
+/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/static/aaa_ltpss_dx.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
+ ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * *
+[1,2,3:
+ #i #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+ >(tpr_inv_atom1 … H2) -Y
+|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+]
+[ >(aaa_inv_sort … H1) -X //
+| elim (aaa_inv_lref … H1) #I #K1 #V1 #HLK1 #HA
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #Y #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IH … HKV1 … HA … HK12 … HV12) -L1 -K1 -V1 /2 width=5/
+| elim (aaa_inv_gref … H1)
+| elim (aaa_inv_abbr … H1) -H1 #B #HB #HA
+ elim (tpr_inv_abbr1 … H2) -H2 *
+ [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
+ lapply (tps_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+ lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+ lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
+ | -B #T #HT1 #HXT #H destruct
+ lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
+ @(aaa_inv_lift … HA … HXT) /2 width=1/
+ ]
+| elim (aaa_inv_abst … H1) -H1 #B #A #HB #HA #H destruct
+ elim (tpr_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+ lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
+| elim (aaa_inv_appl … H1) -H1 #B #HB #HA
+ elim (tpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+ lapply (IH … HA … HL12 … HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/
+ | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+ elim (aaa_inv_abst … HA) -HA #B0 #A0 #HB0 #HA0 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+ lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
+ lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 // /2 width=1/ -T0 -L1 -V1 /4 width=7/
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
+ elim (aaa_inv_abbr … HA) -HA #B0 #HW0 #HT0
+ lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
+ lapply (IH … HB … HL12 … HV10) -HB -HV10 /width=5/ #HV0
+ lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2
+ @(aaa_abbr … HW2) -HW2
+ @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
+ ]
+| elim (aaa_inv_cast … H1) -H1 #HV1 #HT1
+ elim (tpr_inv_cast1 … H2) -H2
+ [ * #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
+ lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
+ | -HV1 #HT1X
+ lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
+ ]
+]
+qed.
+
+lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
+/2 width=5/ qed.
+
+lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
+/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/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning dx parallel unfold on local environments ***********)
+
+lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ /2 width=3/
+| #L1 #I #V1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /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_sn_alt.ma".
+include "basic_2/reducibility/ltpr_ltpss_dx.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties on sn parallel unfold on local environments *******************)
+
+(* Note: this can also be proved like ltpr_ltpss_dx_conf *)
+lemma ltpr_ltpss_sn_conf: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H
+@(ltpssa_ind … H) -K1 /2 width=3/
+#K #K1 #_ #HK1 #IHK #L2 #HL12
+elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
+elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /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/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning parallel substitution on terms *********************)
+
+(* Basic_1: was: pr0_subst1_fwd *)
+lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ /2 width=3/
+| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
+ elim (lift_total V2 0 (i+1)) #W2 #HVW2
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
+| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L1 /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: pr0_subst1_back *)
+lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 →
+ ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2.
+#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
+[ /2 width=3/
+| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
+ elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
+ elim (lift_total V1 0 (i+1)) #W1 #HVW1
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
+| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
+| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L2 /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/ltpr_tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties on partial unfold for terms ***********************************)
+
+(* Basic_1: was: pr0_subst1 *)
+lemma ltpr_tpr_tps_conf: ∀T1,T2. T1 ➡ T2 →
+ ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 →
+ ∀L2. L1 ➡ L2 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#T1 #T2 #H elim H -T1 -T2
+[ #I #L1 #d #e #U1 #H #L2 #HL12
+ elim (ltpr_tps_conf … H … HL12) -L1 /3 width=3/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV12 … HVW1 … HL12) -V1
+ elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
+ elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ lapply (tpss_lsubr_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/
+| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2
+ elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
+ elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2
+ lapply (tps_lsubr_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2
+ elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23
+ lapply (tps_lsubr_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3
+ lapply (tpss_lsubr_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23
+ lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
+ elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
+ elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (lift_total VV2 0 1) #VV #H2VV
+ lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
+ @ex2_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V
+ elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
+ elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T <minus_plus_m_m /3 width=3/
+| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+ elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
+]
+qed-.
+
+lemma tpr_tps_conf_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
+ ⋆. ⓑ{I} V1 ⊢ T1 ▶ [0, 1] U1 →
+ ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 ▶ [0, 1] U2.
+#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
+elim (ltpr_tpr_tps_conf … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ -V1 #U2 #HU12 #HTU2
+lapply (tpss_inv_SO2 … HTU2) -HTU2 /2 width=3/
+qed-.
+
+lemma ltpr_tpr_tpss_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
+[ /2 width=3/
+| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
+ elim (ltpr_tpr_tps_conf … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
+ lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
+]
+qed-.
+
+lemma tpr_tpss_conf: ∀T1,T2. T1 ➡ T2 →
+ ∀L,U1,d,e. L ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L ⊢ T2 ▶* [d, e] U2.
+/2 width=5 by ltpr_tpr_tpss_conf/ 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/thin_delift.ma".
+include "basic_2/reducibility/tpr_delift.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+(* Basic_1: was only: pr2_gen_cabbr *)
+lemma thin_cpr_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
+ ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∃∃T2. K ⊢ T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
+#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1
+elim (tpr_delift_conf … HU1 … HTU1) -U1 #T #HT1 #HUT
+elim (le_or_ge (|L|) d) #Hd
+[ elim (thin_delift_tpss_conf_le … HU2 … HUT … HLK ?)
+| elim (le_or_ge (|L|) (d+e)) #Hde
+ [ elim (thin_delift_tpss_conf_le_up … HU2 … HUT … HLK ? ? ?)
+ | elim (thin_delift_tpss_conf_be … HU2 … HUT … HLK ? ?)
+ ]
+] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2
+lapply (cpr_intro … HT1 HT2) -T /2 width=3/
+qed.
(* *)
(**************************************************************************)
+notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $L $T1 $d $e $T2 }.
+
include "basic_2/unfold/tpss.ma".
(* INVERSE BASIC TERM RELOCATION *******************************************)
(* *)
(**************************************************************************)
+notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubstAlt $L $T1 $d $e $T2 }.
+
include "basic_2/unfold/delift_lift.ma".
(* INVERSE BASIC TERM RELOCATION *******************************************)
(* *)
(**************************************************************************)
+notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $T1 $d $e $T2 }.
+
include "basic_2/unfold/ltpss_sn.ma".
(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
--- /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/delift.ma".
+include "basic_2/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
+#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1
+elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21
+elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+definition cfpr: lenv → bi_relation lenv term ≝
+ λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
+
+interpretation
+ "context-sensitive parallel reduction (closure)"
+ 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
+/2 width=1/ qed.
+
+lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 * /3 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
+#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V2. V1 ➡ V2 &
+ ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
+ L2 = ⋆.ⓑ{I}V2@@K2.
+#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
+elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
+>shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
+elim (tpr_inv_bind1 … H) -H *
+[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
+| #T0 #_ #_ #H destruct
+]
+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_aaa.ma".
+include "basic_2/reducibility/cfpr_cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_fpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #HT1 #L2 #T2 #H
+elim (fpr_inv_all … H) -H
+/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/
+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_cpr.ma".
+include "basic_2/reducibility/cfpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Main properties **********************************************************)
+
+theorem cfpr_conf: ∀L. bi_confluent … (cfpr L).
+#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
+elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
+elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /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/reducibility/cpr_tpss.ma".
+include "basic_2/reducibility/cpr_cpr.ma".
+include "basic_2/reducibility/cfpr_ltpss.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 →
+ L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+#L1 #L #H elim H -L1 -L
+[ #L2 #T1 #T2 #HT12 #HL2
+ lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct
+ lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/
+| #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) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cfpr_inv_all: ∀L1,L2,L0,T1,T2. L0 ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃L. L0 @@ L1 ➡ L0 @@ L & L0 @@ L ⊢ T1 ➡ T2 &
+ L0 @@ L ⊢ ▶* [0, |L0| + |L|] L0 @@ L2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L2 #L0 #T1 #T2 #H
+ elim (cfpr_inv_atom1 … H) -H #HT12 #H destruct /3 width=4/
+| #I #L1 #V1 #IH #X #L0 #T1 #T2 #H
+ elim (cfpr_inv_pair1 … H) -H #L2 #V #V2 #HV1 #HV2 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L #HL1 #HT12 #HL2
+ elim (ltpr_inv_append1 … HL1) -HL1 #X #Y #HX #HY #H
+ lapply (ltpr_fwd_length … HX) -HX #HX
+ elim (append_inj_dx … H ?) -H // -HX #_ #H destruct -X
+ lapply (ltpss_sn_fwd_length … HL2) >append_length >append_length #H
+ lapply (injective_plus_r … H) -H #H
+ @(ex3_intro … (⋆.ⓑ{I}V@@Y)) <append_assoc // -HT12
+ <append_assoc [ /3 width=1/ ] -HV1 -HY
+ >append_length <associative_plus
+ @(ltpss_sn_dx_trans_eq … HL2) -HL2 >H -H >commutative_plus /3 width=1/
+]
+qed-.
+
+lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2.
+#L1 #L2 #T1 #T2 #H
+lapply (fpr_cfpr … H) -H #H
+elim (cfpr_inv_all … H) -H /2 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/reducibility/cpr_lift.ma".
+include "basic_2/reducibility/cpr_ltpss_sn.ma".
+include "basic_2/reducibility/cfpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 &
+ L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
+ L2 = ⋆.ⓑ{I}V2@@K2.
+* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H
+elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
+>shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/
+ | #T0 #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T * #V0 #HV10 #HV0 #HT1 #H destruct
+ lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12
+ /3 width=7/
+]
+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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+definition fpr: bi_relation lenv term ≝
+ λL1,T1,L2,T2. |L1| = |L2| ∧ L1 @@ T1 ➡ L2 @@ T2.
+
+interpretation
+ "context-free parallel reduction (closure)"
+ 'FocalizedPRed L1 T1 L2 T2 = (fpr L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpr_refl: bi_reflexive … fpr.
+/2 width=1/ qed.
+
+lemma fpr_shift: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2.ⓑ{I2}V2, T2⦄.
+#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12
+@conj // normalize // (**) (* explicit constructor *)
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fpr_inv_atom1: ∀L2,T1,T2. ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → T1 ➡ T2 ∧ L2 = ⋆.
+#L2 #T1 #T2 * #H
+lapply (length_inv_zero_sn … H) -H #H destruct /2 width=1/
+qed-.
+
+lemma fpr_inv_atom3: ∀L1,T1,T2. ⦃L1,T1⦄ ➡ ⦃⋆,T2⦄ → T1 ➡ T2 ∧ L1 = ⋆.
+#L1 #T1 #T2 * #H
+lapply (length_inv_zero_dx … H) -H #H destruct /2 width=1/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpr_fwd_pair1: ∀I1,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃I2,K2,V2. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
+ L2 = K2.ⓑ{I2}V2.
+#I1 #K1 #L2 #V1 #T1 #T2 * #H
+elim (length_inv_pos_sn … H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/
+qed-.
+
+lemma fpr_fwd_pair3: ∀I2,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I2}V2, T2⦄ →
+ ∃∃I1,K1,V1. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
+ L1 = K1.ⓑ{I1}V1.
+#I2 #L1 #K2 #V2 #T1 #T2 * #H
+elim (length_inv_pos_dx … H) -H #I1 #K1 #V1 #HK12 #H destruct /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/reducibility/cfpr_cpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
+/2 width=4/ qed.
+
+lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
+ ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
+ ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
+#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
+elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
+elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
+lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
+elim (le_or_ge (|K|) d) #Hd
+[ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
+| elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
+] // -Hd #L2 #HL2 #HLK2
+lapply (ltpss_sn_weak_full … HL2) -K /3 width=4/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
+ ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 #HT12
+elim (fpr_inv_all … HT12) -HT12 /4 width=4/
+qed.
+
+lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
+ ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
+elim (fpr_inv_all … H) /3 width=4/
+qed.
+
+lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
+ ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
+ U2 = -ⓑ{I}V2.T2.
+#I1 #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b
+elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
+elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
+qed-.
+
+lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
+* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
+ | #T #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T #HV1 #_ #H destruct /3 width=4/
+]
+qed-.
+
+lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
+ U2 = ⓛ{a}V2.T2.
+#a #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
+elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I1 #K1 #X #V1 #T1 #T2 #H
+elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+
+lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #X #K2 #V2 #T1 #T2 #H
+elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /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/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+(* Main properties **********************************************************)
+
+theorem fpr_conf: bi_confluent … fpr.
+#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
+elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
+elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 ⊢ ➡ ➡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSnAlt $L1 $L2 }.
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRedAlt $L1 $L2 }.
+
+include "basic_2/grammar/lenv_px_bi.ma".
+include "basic_2/reducibility/fpr_cpr.ma".
+include "basic_2/reducibility/lfpr_fpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* alternative definition *)
+definition lfpra: relation lenv ≝ lpx_bi fpr.
+
+interpretation
+ "focalized parallel reduction (environment) alternative"
+ 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lfpra_refl: reflexive … lfpra.
+/2 width=1/ qed.
+
+lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
+#L1 elim L1 -L1
+[ #L2 #T1 #T2 #H
+ elim (fpr_inv_atom1 … H) -H #_ #H destruct //
+| #L1 #I #V1 #IH #L2 #T1 #T2 #H
+ elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆.
+/2 width=2 by lpx_bi_inv_atom1/ qed-.
+
+lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ →
+ ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_bi_inv_pair1/ qed-.
+
+lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆.
+/2 width=2 by lpx_bi_inv_atom2/ qed-.
+
+lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ →
+ ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_bi_inv_pair2/ qed-.
+
+lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄.
+#L1 #L2 * -L1 -L2 // /3 width=1/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|.
+/2 width=2 by lpx_bi_fwd_length/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
+#L1 #L2 #H
+lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/
+qed.
+
+theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
+#L1 #L2 #H
+lapply (lfpra_inv_fpr … H (⋆0)) -H /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/lfpr.ma".
+include "basic_2/reducibility/cfpr_cpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Properties on context-free parallel reduction for closures ***************)
+
+lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
+#L1 #L2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H /2 width=3/
+qed.
+
+(* Inversion lemmas on context-free parallel reduction for closures *********)
+
+lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄.
+#L1 #L2 * /2 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/reducibility/cpr.ma".
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-definition cfpr: lenv → bi_relation lenv term ≝
- λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
-
-interpretation
- "context-sensitive parallel reduction (closure)"
- 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
-/2 width=1/ qed.
-
-lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 * /3 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
-#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V2. V1 ➡ V2 &
- ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
- L2 = ⋆.ⓑ{I}V2@@K2.
-#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
-elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
->shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
-elim (tpr_inv_bind1 … H) -H *
-[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
-| #T0 #_ #_ #H destruct
-]
-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_aaa.ma".
-include "basic_2/reducibility/cfpr_cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_fpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #HT1 #L2 #T2 #H
-elim (fpr_inv_all … H) -H
-/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/
-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_cpr.ma".
-include "basic_2/reducibility/cfpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Main properties **********************************************************)
-
-theorem cfpr_conf: ∀L. bi_confluent … (cfpr L).
-#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
-elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
-elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /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/reducibility/cpr_tpss.ma".
-include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/reducibility/cfpr_ltpss.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 →
- L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-#L1 #L #H elim H -L1 -L
-[ #L2 #T1 #T2 #HT12 #HL2
- lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct
- lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/
-| #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) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cfpr_inv_all: ∀L1,L2,L0,T1,T2. L0 ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃L. L0 @@ L1 ➡ L0 @@ L & L0 @@ L ⊢ T1 ➡ T2 &
- L0 @@ L ⊢ ▶* [0, |L0| + |L|] L0 @@ L2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L2 #L0 #T1 #T2 #H
- elim (cfpr_inv_atom1 … H) -H #HT12 #H destruct /3 width=4/
-| #I #L1 #V1 #IH #X #L0 #T1 #T2 #H
- elim (cfpr_inv_pair1 … H) -H #L2 #V #V2 #HV1 #HV2 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L #HL1 #HT12 #HL2
- elim (ltpr_inv_append1 … HL1) -HL1 #X #Y #HX #HY #H
- lapply (ltpr_fwd_length … HX) -HX #HX
- elim (append_inj_dx … H ?) -H // -HX #_ #H destruct -X
- lapply (ltpss_sn_fwd_length … HL2) >append_length >append_length #H
- lapply (injective_plus_r … H) -H #H
- @(ex3_intro … (⋆.ⓑ{I}V@@Y)) <append_assoc // -HT12
- <append_assoc [ /3 width=1/ ] -HV1 -HY
- >append_length <associative_plus
- @(ltpss_sn_dx_trans_eq … HL2) -HL2 >H -H >commutative_plus /3 width=1/
-]
-qed-.
-
-lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2.
-#L1 #L2 #T1 #T2 #H
-lapply (fpr_cfpr … H) -H #H
-elim (cfpr_inv_all … H) -H /2 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/reducibility/cpr_lift.ma".
-include "basic_2/reducibility/cpr_ltpss_sn.ma".
-include "basic_2/reducibility/cfpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 &
- L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
- L2 = ⋆.ⓑ{I}V2@@K2.
-* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H
-elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
->shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H
-[ elim (cpr_inv_abbr1 … H) -H *
- [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/
- | #T0 #_ #_ #H destruct
- ]
-| elim (cpr_inv_abst1 … H Abst V2) -H
- #V #T * #V0 #HV10 #HV0 #HT1 #H destruct
- lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12
- /3 width=7/
-]
-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/crf.ma".
-
-(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
-
-definition cif: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥.
-
-interpretation "context-sensitive irreducibility (term)"
- 'NotReducible L T = (cif L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cif_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥.
-/3 width=3/ qed-.
-
-lemma cif_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
-/3 width=1/ qed-.
-
-lemma cif_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
- L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
-/4 width=1/ qed-.
-
-lemma cif_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
- ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
-#a * [ elim a -a ]
-[ #L #V #T #H elim (H ?) -H /3 width=1/
-|*: #L #V #T #H elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=1/
-]
-qed-.
-
-lemma cif_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
-#L #V #T #HVT @and3_intro /3 width=1/
-generalize in match HVT; -HVT elim T -T //
-* // #a * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
-qed-.
-
-lemma cif_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
- ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-* #L #V #T #H
-[ elim (cif_inv_appl … H) -H /2 width=1/
-| elim (cif_inv_ri2 … H) -H /2 width=1/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tif_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄.
-/2 width=2 by trf_inv_atom/ qed.
-
-lemma cif_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
-#a #I #L #V #T #HI #HV #HT #H
-elim (crf_inv_ib2 … HI H) -HI -H /2 width=1/
-qed.
-
-lemma cif_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄.
-#L #V #T #HV #HT #H1 #H2
-elim (crf_inv_appl … H2) -H2 /2 width=1/
-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/crf_append.ma".
-include "basic_2/reducibility/cif.ma".
-
-(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
-
-(* Advanved properties ******************************************************)
-
-lemma cif_labst_last: ∀L,T,W. L ⊢ 𝐈⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄.
-/3 width=2 by crf_inv_labst_last/ qed.
-
-lemma cif_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄.
-/3 width=2 by crf_inv_trf/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cif_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃T⦄.
-/3 width=1/ qed-.
-
-lemma cif_inv_tif: ∀T,W. ⋆.ⓛW ⊢ 𝐈⦃T⦄ → ⋆ ⊢ 𝐈⦃T⦄.
-/3 width=1/ 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.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
-
-interpretation
- "context-sensitive normality (term)"
- 'Normal L T = (cnf L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cnf_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#L #V1 #T1 #HVT1 @and3_intro
-[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
-| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
- [ elim (lift_total V1 0 1) #V2 #HV12
- lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct
- | lapply (H (ⓓ{a}V1.U1) ?) -H /3 width=1/ #H destruct
-]
-qed-.
-
-lemma cnf_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
-#L #V #T #H elim (is_lift_dec T 0 1)
-[ * #U #HTU
- lapply (H U ?) -H /3 width=3 by cpr_tpr, tpr_zeta/ #H destruct (**) (* auto too slow without trace *)
- elim (lift_inv_pair_xy_y … HTU)
-| #HT
- elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
- lapply (H (+ⓓV.T2) ?) -H /3 width=3 by cpr_tpr, tpr_delta/ -HT2 #H destruct /3 width=2/ (**) (* auto too slow without trace *)
-]
-qed.
-
-lemma cnf_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
-#L #V #T #H lapply (H T ?) -H /2 width=1/ #H
-@discr_tpair_xy_y //
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: nf2_sort *)
-lemma cnf_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄.
-#L #k #X #H
->(cpr_inv_sort1 … H) //
-qed.
-
-(* Basic_1: was: nf2_dec *)
-axiom cnf_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨
- ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
-
-(* Basic_1: removed theorems 1: nf2_abst_shift *)
+++ /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/cif.ma".
-include "basic_2/reducibility/cnf_lift.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
- elim (cif_inv_delta … HLK ?) //
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
- elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
- lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
- elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
-]
-qed.
-
-lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 #HT1
-lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
-qed.
-
-lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#T1 #T2 #H elim H -T1 -T2
-[ //
-| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
- [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
- >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
- | elim (cif_inv_ri2 … H) /2 width=1/
- ]
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
- elim (cif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
- [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
- elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
- lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
- lapply (IHT1 … HT1) -IHT1 #H destruct
- lapply (tps_cif_eq … HT2 ?) -HT2 //
- | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
- elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
- ]
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
- elim (cif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #V1 #T1 #T #T2 #_ #_ #_ #L #H
- elim (cif_inv_ri2 … H) /2 width=1/
-| #V1 #T1 #T2 #_ #_ #L #H
- elim (cif_inv_ri2 … H) /2 width=1/
-]
-qed.
-
-lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 * #T0 #HT10 #HT02 #HT1
-lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
-qed.
-
-theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
-/3 width=3/ qed.
-
-(* Note: this property is unusual *)
-lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
-#L #T #H elim H -L -T
-[ #L #K #V #i #HLK #H
- elim (cnf_inv_delta … HLK H)
-| #L #V #T #_ #IHV #H
- elim (cnf_inv_appl … H) -H /2 width=1/
-| #L #V #T #_ #IHT #H
- elim (cnf_inv_appl … H) -H /2 width=1/
-| #I #L #V #T * #H1 #H2 destruct
- [ elim (cnf_inv_zeta … H2)
- | elim (cnf_inv_tau … H2)
- ]
-|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
- [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
- |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
- ]
-| #a #L #V #W #T #H
- elim (cnf_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #a #L #V #W #T #H
- elim (cnf_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-]
-qed.
-
-theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
-/2 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/reducibility/cpr_lift.ma".
-include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/reducibility/cnf.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cnf_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥.
-#L #K #V #i #HLK #H
-elim (lift_total V 0 (i+1)) #W #HVW
-lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct
-elim (lift_inv_lref2_be … HVW ? ?) -HVW //
-qed-.
-
-lemma cnf_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄.
-#a #L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
-]
-qed-.
-
-lemma cnf_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄.
-#L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was only: nf2_csort_lref *)
-lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍⦃#i⦄.
-#L #i #HLK #X #H
-elim (cpr_inv_lref1 … H) // *
-#K0 #V0 #V1 #HLK0 #_ #_ #_
-lapply (ldrop_mono … HLK … HLK0) -L #H destruct
-qed.
-
-(* Basic_1: was: nf2_lref_abst *)
-lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍⦃#i⦄.
-#L #K #V #i #HLK #X #H
-elim (cpr_inv_lref1 … H) // *
-#K0 #V0 #V1 #HLK0 #_ #_ #_
-lapply (ldrop_mono … HLK … HLK0) -L #H destruct
-qed.
-
-(* Basic_1: was: nf2_abst *)
-lemma cnf_abst: ∀a,I,L,V,W,T. L ⊢ 𝐍⦃W⦄ → L. ⓑ{I} V ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄.
-#a #I #L #V #W #T #HW #HT #X #H
-elim (cpr_inv_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct
->(HW … HW0) -W0 >(HT … HT0) -T0 //
-qed.
-
-(* Basic_1: was only: nf2_appl_lref *)
-lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄.
-#L #V #T #HV #HT #HS #X #H
-elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct
->(HV … HV0) -V0 >(HT … HT0) -T0 //
-qed.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: nf2_lift *)
-lemma cnf_lift: ∀L0,L,T,T0,d,e.
- L ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄.
-#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
-elim (cpr_inv_lift1 … HL0 … HT0 … H) -L0 #T1 #HT10 #HT1
-<(HLT … HT1) in HT0; -L #HT0
->(lift_mono … HT10 … HT0) -T1 -X //
-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/tpss.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Basic_1: includes: pr2_delta1 *)
-definition cpr: lenv → relation term ≝
- λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2.
-
-interpretation
- "context-sensitive parallel reduction (term)"
- 'PRed L T1 T2 = (cpr L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed-.
-
-(* Basic_1: was by definition: pr2_free *)
-lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
-/2 width=3/ qed.
-
-lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed.
-
-lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
-/2 width=1/ qed.
-
-(* Note: new property *)
-(* Basic_1: was only: pr2_thin_dx *)
-lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
- L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-lemma cpr_cast: ∀L,V,T1,T2.
- L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2.
-#L #V #T1 #T2 * /3 width=3/
-qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-(* Basic_1: was only: pr2_change *)
-lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
-lapply (tpss_lsubr_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: pr2_gen_csort *)
-lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
-#T1 #T2 * #T #HT normalize #HT2
-<(tpss_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-(* Basic_1: was: pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
-#L #T2 #k * #X #H
->(tpr_inv_atom1 … H) -H #H
->(tpss_inv_sort1 … H) -H //
-qed-.
-
-(* Basic_1: was: pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → (
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U2 = ⓝV2. T2
- ) ∨ L ⊢ T1 ➡ U2.
-#L #V1 #T1 #U2 * #X #H #HU2
-elim (tpr_inv_cast1 … H) -H /3 width=3/
-* #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #L #V1 #T1 #T * #X #H1 #H2 #b
-elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
-qed-.
-
-lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L #L1 #T1 #T * #X #H1 #H2
-elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct
-elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/
-qed-.
-
-(* Basic_1: removed theorems 6:
- pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
- pr2_gen_ctail pr2_ctail
- Basic_1: removed local theorems 3:
- pr2_free_free pr2_free_delta pr2_delta_delta
-*)
+++ /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/aaa_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_aaa.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
-#L #T1 #A #HT1 #T2 * /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/reducibility/tpr_tpr.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_bind_sn: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
- L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
-@ex2_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr2_gen_cbind *)
-lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 →
- L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
-elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
-lapply (tpss_lsubr_trans … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
-lapply (tpss_inv_SO2 … HT0) -HT0 #HT0
-@ex2_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr2_head_1 *)
-lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
- L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
-* /2 width=1/ /3 width=1/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @@ T1 ➡ L @@ T2.
-#L elim L -L
-[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
-| normalize /3 width=1/
-].
-qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: pr2_confluence *)
-theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
- ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
-#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
-elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
-elim (ltpr_tpr_tpss_conf ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
-elim (ltpr_tpr_tpss_conf ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
-elim (tpss_conf_eq … HU1 … HU2) -U /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/thin_delift.ma".
-include "basic_2/reducibility/tpr_delift.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-(* Basic_1: was only: pr2_gen_cabbr *)
-lemma thin_cpr_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
- ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∃∃T2. K ⊢ T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
-#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1
-elim (tpr_delift_conf … HU1 … HTU1) -U1 #T #HT1 #HUT
-elim (le_or_ge (|L|) d) #Hd
-[ elim (thin_delift_tpss_conf_le … HU2 … HUT … HLK ?)
-| elim (le_or_ge (|L|) (d+e)) #Hde
- [ elim (thin_delift_tpss_conf_le_up … HU2 … HUT … HLK ? ? ?)
- | elim (thin_delift_tpss_conf_be … HU2 … HUT … HLK ? ?)
- ]
-] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2
-lapply (cpr_intro … HT1 HT2) -T /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/tpss_lift.ma".
-include "basic_2/reducibility/tpr_lift.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
- ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
- ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
-#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
-lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
-@ex2_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
-qed.
-
-lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
- ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
-#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a #I
-lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
-lapply (tpss_lsubr_trans … HT02 (L.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
-@(ex2_intro … (ⓑ{a,I}V0.T0)) /2 width=1/ (* explicit constructors *)
-qed.
-
-lemma cpr_beta: ∀a,L,V1,V2,W,T1,T2.
- L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
-#a #L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
-lapply (tpss_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
-@(ex2_intro … (ⓓ{a}V.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
-qed.
-
-lemma cpr_beta_dx: ∀a,L,V1,V2,W,T1,T2.
- V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
-/3 width=1/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
- T2 = #i ∨
- ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
- ⇧[0, i + 1] T1 ≡ T2 &
- i < |L|.
-#L #T2 #i * #X #H
->(tpr_inv_atom1 … H) -H #H
-elim (tpss_inv_lref1 … H) -H /2 width=1/
-* /3 width=6/
-qed-.
-
-(* Basic_1: was pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 →
- (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 &
- L. ⓓV ⊢ T1 ➡ T2 &
- U2 = ⓓ{a}V2. T2
- ) ∨
- ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true.
-#a #L #V1 #T1 #Y * #X #H1 #H2
-elim (tpr_inv_abbr1 … H1) -H1 *
-[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
- lapply (tps_weak_full … HT0) -HT0 #HT0
- lapply (tpss_lsubr_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02
- lapply (tpss_weak_full … HT02) -HT02 #HT02
- lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/
-| #T2 #HT12 #HXT2 #H destruct
- elim (lift_total Y 0 1) #Z #HYZ
- lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H
- lapply (cpr_intro … HT12 … H) -T2 /3 width=3/
-]
-qed-.
-
-(* Basic_1: was: pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W.
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
-#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W
-elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
-lapply (tpss_lsubr_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
-qed-.
-
-(* Basic_1: was pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2
- | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2.
-#L #V1 #U0 #Y * #X #H1 #H2
-elim (tpr_inv_appl1 … H1) -H1 *
-[ #V #U #HV1 #HU0 #H destruct
- elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
-| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/
-| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
- elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
- elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=13/
-]
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U = ⓐV2. T2.
-#L #V1 #T1 #U #H #HT1
-elim (cpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
- ∃∃V2,T2. L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
- U2 = ⓛ{a}V2.T2.
-#a #L #V1 #T1 #U2 * #U #H #HU2 #b #I #W
-elim (tpr_fwd_abst1 … H b I W) -H #V #T #HT1 #H destruct
-elim (tpss_inv_bind1 … HU2) -HU2 #V2 #T2 #_ #HT2
-lapply (tpss_lsubr_trans … HT2 (L.ⓑ{I}W) ?) -HT2 /2 width=1/ /4 width=5/
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr2_lift *)
-lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
-elim (lt_or_ge (|K|) d) #HKd
-[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
-| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
-]
-qed.
-
-(* Basic_1: was: pr2_gen_lift *)
-lemma cpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
-elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #T #HTU #T1
-elim (lt_or_ge (|L|) d) #HLd
-[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK /2 width=2/
- /3 width=7 by ex2_intro, cpr_intro/
-| elim (lt_or_ge (|L|) (d + e)) #HLde
- [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // /2 width=2/
- /3 width=7 by ex2_intro, cpr_intro/
- | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK //
- /3 width=7 by ex2_intro, cpr_intro/
- ]
-]
-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/ltpr_tpss.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning parallel unfold on terms ***************************)
-
-(* Note: we could invoke tpss_weak_full instead of ltpr_fwd_length *)
-(* Basic_1: was only: pr2_subst1 *)
-lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
-elim (ltpr_tpr_tpss_conf … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
-elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
->(ltpr_fwd_length … HL12) in HT2; #HT2
-elim (ltpr_tpr_tpss_conf … HL12 … HT2) -L1 /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
-elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
-elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
-lapply (tpss_weak_full … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
-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_full … 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-.
+++ /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/ltpr_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,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2
-lapply (tpss_weak_full … 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,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-.
+++ /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/substitution/ldrop.ma".
-
-(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
-
-(* reducible binary items *)
-definition ri2: item2 → Prop ≝
- λI. I = Bind2 true Abbr ∨ I = Flat2 Cast.
-
-(* irreducible binary binders *)
-definition ib2: bool → bind2 → Prop ≝
- λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr.
-
-(* reducible terms *)
-inductive crf: lenv → predicate term ≝
-| crf_delta : ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → crf L (#i)
-| crf_appl_sn: ∀L,V,T. crf L V → crf L (ⓐV. T)
-| crf_appl_dx: ∀L,V,T. crf L T → crf L (ⓐV. T)
-| crf_ri2 : ∀I,L,V,T. ri2 I → crf L (②{I}V. T)
-| crf_ib2_sn : ∀a,I,L,V,T. ib2 a I → crf L V → crf L (ⓑ{a,I}V. T)
-| crf_ib2_dx : ∀a,I,L,V,T. ib2 a I → crf (L.ⓑ{I}V) T → crf L (ⓑ{a,I}V. T)
-| crf_beta : ∀a,L,V,W,T. crf L (ⓐV. ⓛ{a}W. T)
-| crf_theta : ∀a,L,V,W,T. crf L (ⓐV. ⓓ{a}W. T)
-.
-
-interpretation
- "context-sensitive reducibility (term)"
- 'Reducible L T = (crf L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact trf_inv_atom_aux: ∀I,L,T. L ⊢ 𝐑⦃T⦄ → L = ⋆ → T = ⓪{I} → ⊥.
-#I #L #T * -L -T
-[ #L #K #V #i #HLK #H1 #H2 destruct
- lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
-| #L #V #T #_ #_ #H destruct
-| #L #V #T #_ #_ #H destruct
-| #J #L #V #T #_ #_ #H destruct
-| #a #J #L #V #T #_ #_ #_ #H destruct
-| #a #J #L #V #T #_ #_ #_ #H destruct
-| #a #L #V #W #T #_ #H destruct
-| #a #L #V #W #T #_ #H destruct
-]
-qed.
-
-lemma trf_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥.
-/2 width=6/ qed-.
-
-fact trf_inv_lref_aux: ∀L,T. L ⊢ 𝐑⦃T⦄ → ∀i. T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
-#L #T * -L -T
-[ #L #K #V #j #HLK #i #H destruct /2 width=3/
-| #L #V #T #_ #i #H destruct
-| #L #V #T #_ #i #H destruct
-| #J #L #V #T #_ #i #H destruct
-| #a #J #L #V #T #_ #_ #i #H destruct
-| #a #J #L #V #T #_ #_ #i #H destruct
-| #a #L #V #W #T #i #H destruct
-| #a #L #V #W #T #i #H destruct
-]
-qed.
-
-lemma trf_inv_lref: ∀L,i. L ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
-/2 width=3/ qed-.
-
-fact crf_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W. U →
- L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄.
-#a #I #L #W #U #T #HI * -T
-[ #L #K #V #i #_ #H destruct
-| #L #V #T #_ #H destruct
-| #L #V #T #_ #H destruct
-| #J #L #V #T #H1 #H2 destruct
- elim H1 -H1 #H destruct
- elim HI -HI #H destruct
-| #b #J #L #V #T #_ #HV #H destruct /2 width=1/
-| #b #J #L #V #T #_ #HT #H destruct /2 width=1/
-| #b #L #V #W #T #H destruct
-| #b #L #V #W #T #H destruct
-]
-qed.
-
-lemma crf_inv_ib2: ∀a,I,L,W,T. ib2 a I → L ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ →
- L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄.
-/2 width=5/ qed-.
-
-fact crf_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW. U →
- ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
-#L #W #U #T * -T
-[ #L #K #V #i #_ #H destruct
-| #L #V #T #HV #H destruct /2 width=1/
-| #L #V #T #HT #H destruct /2 width=1/
-| #J #L #V #T #H1 #H2 destruct
- elim H1 -H1 #H destruct
-| #a #I #L #V #T #_ #_ #H destruct
-| #a #I #L #V #T #_ #_ #H destruct
-| #a #L #V #W0 #T #H destruct
- @or3_intro2 #H elim (simple_inv_bind … H)
-| #a #L #V #W0 #T #H destruct
- @or3_intro2 #H elim (simple_inv_bind … H)
-]
-qed.
-
-lemma crf_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
-/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/substitution/ldrop_append.ma".
-include "basic_2/reducibility/crf.ma".
-
-(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
-
-(* Advanved properties ******************************************************)
-
-lemma crf_labst_last: ∀L,T,W. L ⊢ 𝐑⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄.
-#L #T #W #H elim H -L -T /2 width=1/
-#L #K #V #i #HLK
-lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
-lapply (ldrop_O1_append_sn_le … HLK … (⋆.ⓛW)) -HLK /2 width=2/ -Hi /2 width=3/
-qed.
-
-lemma crf_trf: ∀T,W. ⋆ ⊢ 𝐑⦃T⦄ → ⋆.ⓛW ⊢ 𝐑⦃T⦄.
-#T #W #H lapply (crf_labst_last … W H) //
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact crf_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄ →
- ∀L2. L1 = ⋆.ⓛW @@ L2 → L2 ⊢ 𝐑⦃T⦄.
-#L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
-[ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
- lapply (ldrop_fwd_ldrop2_length … HLK1)
- >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
- elim (le_to_or_lt_eq i (|L2|) ?) /2 width=1/ -H #Hi destruct
- [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
- lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
- normalize #H destruct /2 width=3/
- | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
- lapply (ldrop_inv_refl … H) -H #H destruct
- ]
-| #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
-]
-qed.
-
-lemma crf_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐑⦃T⦄.
-/2 width=4/ qed-.
-
-lemma crf_inv_trf: ∀T,W. ⋆.ⓛW ⊢ 𝐑⦃T⦄ → ⋆ ⊢ 𝐑⦃T⦄.
-/2 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/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-definition fpr: bi_relation lenv term ≝
- λL1,T1,L2,T2. |L1| = |L2| ∧ L1 @@ T1 ➡ L2 @@ T2.
-
-interpretation
- "context-free parallel reduction (closure)"
- 'FocalizedPRed L1 T1 L2 T2 = (fpr L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpr_refl: bi_reflexive … fpr.
-/2 width=1/ qed.
-
-lemma fpr_shift: ∀I1,I2,L1,L2,V1,V2,T1,T2.
- ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
- ⦃L1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2.ⓑ{I2}V2, T2⦄.
-#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12
-@conj // normalize // (**) (* explicit constructor *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fpr_inv_atom1: ∀L2,T1,T2. ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → T1 ➡ T2 ∧ L2 = ⋆.
-#L2 #T1 #T2 * #H
-lapply (length_inv_zero_sn … H) -H #H destruct /2 width=1/
-qed-.
-
-lemma fpr_inv_atom3: ∀L1,T1,T2. ⦃L1,T1⦄ ➡ ⦃⋆,T2⦄ → T1 ➡ T2 ∧ L1 = ⋆.
-#L1 #T1 #T2 * #H
-lapply (length_inv_zero_dx … H) -H #H destruct /2 width=1/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpr_fwd_pair1: ∀I1,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃I2,K2,V2. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
- L2 = K2.ⓑ{I2}V2.
-#I1 #K1 #L2 #V1 #T1 #T2 * #H
-elim (length_inv_pos_sn … H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/
-qed-.
-
-lemma fpr_fwd_pair3: ∀I2,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I2}V2, T2⦄ →
- ∃∃I1,K1,V1. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
- L1 = K1.ⓑ{I1}V1.
-#I2 #L1 #K2 #V2 #T1 #T2 * #H
-elim (length_inv_pos_dx … H) -H #I1 #K1 #V1 #HK12 #H destruct /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/reducibility/cfpr_cpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-/3 width=4/ qed.
-
-lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
-/2 width=4/ qed.
-
-lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
- ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
- ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
-#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
-elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
-elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
-lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
-elim (le_or_ge (|K|) d) #Hd
-[ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
-| elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
-] // -Hd #L2 #HL2 #HLK2
-lapply (ltpss_sn_weak_full … HL2) -K /3 width=4/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 #HT12
-elim (fpr_inv_all … HT12) -HT12 /4 width=4/
-qed.
-
-lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
- ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
-#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
-elim (fpr_inv_all … H) /3 width=4/
-qed.
-
-lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
- ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
- U2 = -ⓑ{I}V2.T2.
-#I1 #L1 #L2 #V1 #T1 #U2 #H
-elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
-elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
- T = -ⓑ{I}V2.T2.
-#I #L1 #L #V1 #T1 #T #H1 #b
-elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
-elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
-qed-.
-
-lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
- ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
- ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
-* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
-elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
-[ elim (cpr_inv_abbr1 … H) -H *
- [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
- | #T #_ #_ #H destruct
- ]
-| elim (cpr_inv_abst1 … H Abst V2) -H
- #V #T #HV1 #_ #H destruct /3 width=4/
-]
-qed-.
-
-lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
- U2 = ⓛ{a}V2.T2.
-#a #L1 #L2 #V1 #T1 #U2 #H
-elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
-elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I1 #K1 #X #V1 #T1 #T2 #H
-elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
-
-lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
- ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L1 = K1.ⓑ{I}V1.
-#I2 #X #K2 #V2 #T1 #T2 #H
-elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
-
-(* More advanced forward lemmas *********************************************)
-
-lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I #K1 #L2 #V1 #T1 #T2 #H #b
-elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
-elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /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/reducibility/tpr_tpr.ma".
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-(* Main properties **********************************************************)
-
-theorem fpr_conf: bi_confluent … fpr.
-#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
-elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
-elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /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_sn.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-definition lfpr: relation lenv ≝
- λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2
-.
-
-interpretation
- "focalized parallel reduction (environment)"
- 'FocalizedPRed L1 L2 = (lfpr L1 L2).
-
-(* Basic properties *********************************************************)
-
-(* Note: lemma 250 *)
-lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄.
-/2 width=3/ qed.
-
-lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄.
-/3 width=5/ qed.
-
-lemma ltpr_lfpr: ∀L1,L2. L1 ➡ L2 → ⦃L1⦄ ➡ ⦃L2⦄.
-/3 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆.
-#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 //
-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/static/aaa_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_aaa.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_lfpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A.
-#L1 #T #A #HT #L2 * /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/grammar/lenv_px_bi.ma".
-include "basic_2/reducibility/fpr_cpr.ma".
-include "basic_2/reducibility/lfpr_fpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* alternative definition *)
-definition lfpra: relation lenv ≝ lpx_bi fpr.
-
-interpretation
- "focalized parallel reduction (environment) alternative"
- 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lfpra_refl: reflexive … lfpra.
-/2 width=1/ qed.
-
-lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
-#L1 elim L1 -L1
-[ #L2 #T1 #T2 #H
- elim (fpr_inv_atom1 … H) -H #_ #H destruct //
-| #L1 #I #V1 #IH #L2 #T1 #T2 #H
- elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆.
-/2 width=2 by lpx_bi_inv_atom1/ qed-.
-
-lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ →
- ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- L2 = K2. ⓑ{I} V2.
-/2 width=1 by lpx_bi_inv_pair1/ qed-.
-
-lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆.
-/2 width=2 by lpx_bi_inv_atom2/ qed-.
-
-lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ →
- ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- L1 = K1. ⓑ{I} V1.
-/2 width=1 by lpx_bi_inv_pair2/ qed-.
-
-lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄.
-#L1 #L2 * -L1 -L2 // /3 width=1/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|.
-/2 width=2 by lpx_bi_fwd_length/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
-#L1 #L2 #H
-lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/
-qed.
-
-theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
-#L1 #L2 #H
-lapply (lfpra_inv_fpr … H (⋆0)) -H /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_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_ldrop.ma".
-include "basic_2/reducibility/cpr.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
- ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄.
-#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
-<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I
-lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2
-@(ex2_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *)
-qed.
-
-(* Properties on supclosure *************************************************)
-(*
-lamma fsub_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃L1⦄ ➡ ⦃L⦄ & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2
-elim (fsub_tpr_trans … HT12 … H1) -T2 #L #U #HL1 #HT1U #HUT
-elim (fsup_tpss_trans_full … HUT … H2) -T -HUT -H2 #L #U #HL1 #HT1U #HUT
-
-
-
-
-
-
- #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
-lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-qed-.
-*)
\ 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/reducibility/lfpr.ma".
-include "basic_2/reducibility/cfpr_cpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Properties on context-free parallel reduction for closures ***************)
-
-lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
-#L1 #L2 #T1 #T2 #H
-elim (fpr_inv_all … H) -H /2 width=3/
-qed.
-
-(* Inversion lemmas on context-free parallel reduction for closures *********)
-
-lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄.
-#L1 #L2 * /2 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/reducibility/ltpr_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_ltpr.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Main properties **********************************************************)
-
-theorem lfpr_conf: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ →
- ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
-#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2
-lapply (ltpr_fwd_length … HK01) #H
->(ltpr_fwd_length … HK02) in H; #H
-elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2
-lapply (ltpss_sn_fwd_length … HKL1) #H1
-lapply (ltpss_sn_fwd_length … HKL2) #H2
->H1 in HKL1 H; -H1 #HKL1
->H2 in HKL2; -H2 #HKL2 #H
-elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1
-elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2
-elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2
-lapply (ltpr_fwd_length … HLK1) #H1
-lapply (ltpr_fwd_length … HLK2) #H2
-/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/grammar/lenv_px.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-definition ltpr: relation lenv ≝ lpx tpr.
-
-interpretation
- "context-free parallel reduction (environment)"
- 'PRed L1 L2 = (ltpr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ltpr_refl: reflexive … ltpr.
-/2 width=1/ qed.
-
-lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2.
-/2 width=1/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: wcpr0_gen_sort *)
-lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
-/2 width=2 by lpx_inv_atom1/ qed-.
-
-(* Basic_1: was: wcpr0_gen_head *)
-lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
- ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-/2 width=1 by lpx_inv_pair1/ qed-.
-
-lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
-/2 width=2 by lpx_inv_atom2/ qed-.
-
-lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=1 by lpx_inv_pair2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
-/2 width=2 by lpx_fwd_length/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L →
- ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2.
-/2 width=1 by lpx_inv_append1/ qed-.
-
-lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 →
- ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1.
-/2 width=1 by lpx_inv_append2/ qed-.
-
-(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
+++ /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/aaa_ltpss_dx.ma".
-include "basic_2/static/lsuba_aaa.ma".
-include "basic_2/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
- ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * *
-[1,2,3:
- #i #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
- >(tpr_inv_atom1 … H2) -Y
-|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
-]
-[ >(aaa_inv_sort … H1) -X //
-| elim (aaa_inv_lref … H1) #I #K1 #V1 #HLK1 #HA
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #Y #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IH … HKV1 … HA … HK12 … HV12) -L1 -K1 -V1 /2 width=5/
-| elim (aaa_inv_gref … H1)
-| elim (aaa_inv_abbr … H1) -H1 #B #HB #HA
- elim (tpr_inv_abbr1 … H2) -H2 *
- [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
- lapply (tps_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
- lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
- lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
- | -B #T #HT1 #HXT #H destruct
- lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
- @(aaa_inv_lift … HA … HXT) /2 width=1/
- ]
-| elim (aaa_inv_abst … H1) -H1 #B #A #HB #HA #H destruct
- elim (tpr_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
- lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
-| elim (aaa_inv_appl … H1) -H1 #B #HB #HA
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
- lapply (IH … HA … HL12 … HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/
- | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
- elim (aaa_inv_abst … HA) -HA #B0 #A0 #HB0 #HA0 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
- lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
- lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 // /2 width=1/ -T0 -L1 -V1 /4 width=7/
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
- elim (aaa_inv_abbr … HA) -HA #B0 #HW0 #HT0
- lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
- lapply (IH … HB … HL12 … HV10) -HB -HV10 /width=5/ #HV0
- lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2
- @(aaa_abbr … HW2) -HW2
- @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
- ]
-| elim (aaa_inv_cast … H1) -H1 #HV1 #HT1
- elim (tpr_inv_cast1 … H2) -H2
- [ * #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
- lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
- | -HV1 #HT1X
- lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
- ]
-]
-qed.
-
-lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
-/2 width=5/ qed.
-
-lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
-/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/substitution/ldrop_lpx.ma".
-include "basic_2/substitution/fsup.ma".
-include "basic_2/reducibility/tpr_lift.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properies on local environment slicing ***********************************)
-
-(* Basic_1: was: wcpr0_drop *)
-lemma ltpr_ldrop_conf: dropable_sn ltpr.
-/3 width=3 by lpx_deliftable_dropable, tpr_inv_lift1/ qed.
-
-(* Basic_1: was: wcpr0_drop_back *)
-lemma ldrop_ltpr_trans: dedropable_sn ltpr.
-/2 width=3/ qed.
-
-lemma ltpr_ldrop_trans_O1: dropable_dx ltpr.
-/2 width=3/ qed.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsub_tpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. T2 ➡ U2 →
- ∃∃L,U1. L1 ➡ L & T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
-lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-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/tpr_tpr.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Main properties **********************************************************)
-
-theorem ltpr_conf: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 →
- ∃∃L. L1 ➡ L & L2 ➡ L.
-#L0 #L1 #H elim H -L0 -L1 /2 width=3/
-#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H
-elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
-elim (IHK01 … HK02) -K0 #K #HK1 #HK2
-elim (tpr_conf … HV01 HV02) -V0 /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/reducibility/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties concerning dx parallel unfold on local environments ***********)
-
-lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ /2 width=3/
-| #L1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /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_sn_alt.ma".
-include "basic_2/reducibility/ltpr_ltpss_dx.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties on sn parallel unfold on local environments *******************)
-
-(* Note: this can also be proved like ltpr_ltpss_dx_conf *)
-lemma ltpr_ltpss_sn_conf: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H
-@(ltpssa_ind … H) -K1 /2 width=3/
-#K #K1 #_ #HK1 #IHK #L2 #HL12
-elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
-elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /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/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties concerning parallel substitution on terms *********************)
-
-(* Basic_1: was: pr0_subst1_fwd *)
-lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ /2 width=3/
-| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
- elim (lift_total V2 0 (i+1)) #W2 #HVW2
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
-| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
-| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L1 /3 width=5/
-]
-qed-.
-
-(* Basic_1: was: pr0_subst1_back *)
-lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 →
- ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2.
-#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
-[ /2 width=3/
-| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
- elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
- elim (lift_total V1 0 (i+1)) #W1 #HVW1
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
-| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
-| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L2 /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/ltpr_tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties on partial unfold for terms ***********************************)
-
-(* Basic_1: was: pr0_subst1 *)
-lemma ltpr_tpr_tps_conf: ∀T1,T2. T1 ➡ T2 →
- ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 →
- ∀L2. L1 ➡ L2 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#T1 #T2 #H elim H -T1 -T2
-[ #I #L1 #d #e #U1 #H #L2 #HL12
- elim (ltpr_tps_conf … H … HL12) -L1 /3 width=3/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV12 … HVW1 … HL12) -V1
- elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
- elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- lapply (tpss_lsubr_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/
-| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2
- elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
- elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2
- lapply (tps_lsubr_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2
- elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23
- lapply (tps_lsubr_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3
- lapply (tpss_lsubr_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23
- lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
- elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
- elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- elim (lift_total VV2 0 1) #VV #H2VV
- lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
- @ex2_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V
- elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
- elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T <minus_plus_m_m /3 width=3/
-| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
- elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
-]
-qed-.
-
-lemma tpr_tps_conf_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
- ⋆. ⓑ{I} V1 ⊢ T1 ▶ [0, 1] U1 →
- ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 ▶ [0, 1] U2.
-#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
-elim (ltpr_tpr_tps_conf … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ -V1 #U2 #HU12 #HTU2
-lapply (tpss_inv_SO2 … HTU2) -HTU2 /2 width=3/
-qed-.
-
-lemma ltpr_tpr_tpss_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
-[ /2 width=3/
-| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
- elim (ltpr_tpr_tps_conf … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
- lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
-]
-qed-.
-
-lemma tpr_tpss_conf: ∀T1,T2. T1 ➡ T2 →
- ∀L,U1,d,e. L ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. U1 ➡ U2 & L ⊢ T2 ▶* [d, e] U2.
-/2 width=5 by ltpr_tpr_tpss_conf/ 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/grammar/tshf.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************)
-
-definition thnf: predicate term ≝ NF … tpr tshf.
-
-interpretation
- "context-free head normality (term)"
- 'HdNormal T = (thnf T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma thnf_inv_tshf: ∀T. 𝐇𝐍⦃T⦄ → T ≈ T.
-normalize /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tpr_tshf: ∀T1,T2. T1 ➡ T2 → T1 ≈ T1 → T1 ≈ T2.
-#T1 #T2 #H elim H -T1 -T2 //
-[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
- lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2
- lapply (simple_tshf_repl_dx … HUT2 HT1) /2 width=1/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
- elim (simple_inv_bind … H)
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct //
-| #a #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
- elim (simple_inv_bind … H)
-| #V #T #T1 #T2 #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct
-| #V #T1 #T2 #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
-]
-qed.
-
-lemma thnf_tshf: ∀T. T ≈ T → 𝐇𝐍⦃T⦄.
-/3 width=1/ 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/substitution/tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Basic_1: includes: pr0_delta1 *)
-inductive tpr: relation term ≝
-| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
-| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| tpr_beta : ∀a,V1,V2,W,T1,T2.
- tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
-| tpr_delta: ∀a,I,V1,V2,T1,T,T2.
- tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
- tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
- tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
- tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
-| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
-| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
-.
-
-interpretation
- "context-free parallel reduction (term)"
- 'PRed T1 T2 = (tpr T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-/2 width=3/ qed.
-
-(* Basic_1: was by definition: pr0_refl *)
-lemma tpr_refl: reflexive … tpr.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
-#U1 #U2 * -U1 -U2
-[ //
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
-| #V #T1 #T #T2 #_ #_ #k #H destruct
-| #V #T1 #T2 #_ #k #H destruct
-]
-qed.
-
-(* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
-lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#U1 #U2 * -U1 -U2
-[ #J #a #I #V #T #H destruct
-| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
-| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
-| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_abbr *)
-lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓓ{a}V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #V1 #T1 #U2 #H
-elim (tpr_inv_bind1 … H) -H * /3 width=7/
-qed-.
-
-fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-#U1 #U2 * -U1 -U2
-[ #I #J #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
-| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
-| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
-| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_appl *)
-lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2.
-#V1 #U0 #U2 #H
-elim (tpr_inv_flat1 … H) -H *
-/3 width=5/ /3 width=9/ /3 width=13/
-#_ #H destruct
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
- U = ⓐV2. T2.
-#V1 #T1 #U #H #HT1
-elim (tpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Basic_1: was: pr0_gen_cast *)
-lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
- (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
- ∨ T1 ➡ U2.
-#V1 #T1 #U2 #H
-elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
-[ #_ #_ #_ #_ #H destruct
-| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-#T1 #T2 * -T1 -T2
-[ #I #i #H destruct /2 width=1/
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
-| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
-]
-qed.
-
-lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-/2 width=3/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
-| #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #T1 #X
- >shift_append_assoc normalize #H
- elim (tpr_inv_bind1 … H) -H *
- [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
- elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
- elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
- >append_length >HL1 >HL2 -L1 -L
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
-
-(* Basic_1: removed theorems 3:
- pr0_subst0_back pr0_subst0_fwd pr0_subst0
-*)
-(* Basic_1: removed local theorems: 1: pr0_delta_tau *)
+++ /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/delift.ma".
-include "basic_2/reducibility/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
-#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1
-elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21
-elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /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/substitution/tps_lift.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr0_lift *)
-lemma tpr_lift: t_liftable tpr.
-#T1 #T2 #H elim H -T1 -T2
-[ * #i #U1 #d #e #HU1 #U2 #HU2
- lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
- [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
- | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
- | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
- ]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
- elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
- elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
- elim (lift_total T (d + 1) e) #U #HTU
- @tpr_delta
- [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
- elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
- elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
- elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
- elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
-| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
-]
-qed.
-
-(* Basic_1: was: pr0_gen_lift *)
-lemma tpr_inv_lift1: t_deliftable_sn tpr.
-#T1 #T2 #H elim H -T1 -T2
-[ * #i #X #d #e #HX
- [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
- | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
- ]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
- elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
- elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
- elim (IHT1 … HUT1) -T1 #U #HUT #HU1
- elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
- @ex2_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
- elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
- elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
- elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
- @ex2_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
- elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
- elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
- elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
-| #V #T1 #T2 #_ #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
- elim (IHT12 … HT01) -T1 /3 width=3/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr0_gen_abst *)
-lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
-#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T #T2 #HV12 #HT1 #HT2
- lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /2 width=5/
-| #T2 #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tpr_fwd_abst1: ∀a,V1,T1,U2. ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
- ∃∃V2,T2. ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
- U2 = ⓛ{a}V2.T2.
-#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T #T2 #HV12 #HT1 #HT2
- lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /3 width=4/
-| #T2 #_ #_ #_ #H destruct
-]
-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/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Confluence lemmas ********************************************************)
-
-fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
-/2 width=3/ qed.
-
-fact tpr_conf_flat_flat:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0.
-#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
-qed.
-
-fact tpr_conf_flat_beta:
- ∀a,V0,V1,T1,V2,W0,U0,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 →
- U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X.
-#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
-elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
-qed.
-
-(* Basic-1: was:
- pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
- pr0_cong_upsilon_cong pr0_cong_upsilon_delta
-*)
-fact tpr_conf_flat_theta:
- ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V →
- W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X.
-#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
-elim (lift_total VV 0 1) #VVV #HVV
-lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
-elim (tpr_inv_abbr1 … H) -H *
-(* case 1: delta *)
-[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct
- elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
- elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
- elim (tpr_tps_conf_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
- @ex2_intro
- [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
- |1:skip
- |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
- ] (**) (* /5 width=14/ is too slow *)
-(* case 3: zeta *)
-| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1
- elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2
- elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct
- @(ex2_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *)
-]
-qed.
-
-fact tpr_conf_flat_cast:
- ∀X2,V0,V1,T0,T1. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
- ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X.
-#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
-qed.
-
-fact tpr_conf_beta_beta:
- ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X.
-#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
-qed.
-
-(* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
-fact tpr_conf_delta_delta:
- ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 →
- ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 →
- ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X.
-#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
-elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
-elim (tpr_tps_conf_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
-elim (tpr_tps_conf_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
-elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
-@ex2_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
-qed.
-
-fact tpr_conf_delta_zeta:
- ∀X2,V0,V1,T0,T1,TT1,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 →
- T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 →
- ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X.
-#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2
-elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2
-elim (tpr_tps_conf_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT
-elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2
-lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/
-qed.
-
-(* Basic_1: was: pr0_upsilon_upsilon *)
-fact tpr_conf_theta_theta:
- ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 →
- ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 →
- ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X.
-#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
-elim (lift_total V 0 1) #VV #HVV
-lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
-lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2
-@ex2_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
-qed.
-
-fact tpr_conf_zeta_zeta:
- ∀V0:term. ∀X2,TT0,T0,T1,TT2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{TT0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 →
- TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 →
- ∃∃X. T1 ➡ X & X2 ➡ X.
-#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2
-elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2
-elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10
-elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2
-lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/
-qed.
-
-fact tpr_conf_tau_tau:
- ∀V0,T0:term. ∀X2,T1. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- T0 ➡ T1 → T0 ➡ X2 →
- ∃∃X. T1 ➡ X & X2 ➡ X.
-#V0 #T0 #X2 #T1 #IH #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
-qed.
-
-(* Confluence ***************************************************************)
-
-(* Basic_1: was: pr0_confluence *)
-theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
- ∃∃T. T1 ➡ T & T2 ➡ T.
-#T0 @(f_ind … tw … T0) -T0 #n #IH *
-[ #I #_ #X1 #X2 #H1 #H2 -n
- >(tpr_inv_atom1 … H1) -X1
- >(tpr_inv_atom1 … H2) -X2
-(* case 1: atom, atom *)
- //
-| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2
- [ elim (tpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (tpr_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #U2 #HV02 #HT02 #HTU2 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
-(* case 2: delta, delta *)
- [ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
-(* case 3: zeta, delta (repeated) *)
- | @ex2_commute /3 width=10 by tpr_conf_delta_zeta/
-(* case 4: delta, zeta *)
- | /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
-(* case 5: zeta, zeta *)
- | /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
- ]
- | elim (tpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #b1 #V1 #W1 #U1 #T1 #HV01 #HUT1 #H11 #H12 #H13
- | #b1 #V1 #Y1 #W1 #Z1 #U1 #T1 #HV01 #HWZ1 #HUT1 #HVY1 #H11 #H12 #H13
- | #HX1 #H1
- ]
- elim (tpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #b2 #V2 #W2 #U2 #T2 #HV02 #HUT2 #H21 #H22 #H23
- |3,7,11,15: #b2 #V2 #Y2 #W2 #Z2 #U2 #T2 #HV02 #HWZ2 #HUT2 #HVY2 #H21 #H22 #H23
- |4,8,12,16: #HX2 #H2
- ] destruct
-(* case 6: flat, flat *)
- [ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
-(* case 7: beta, flat (repeated) *)
- | @ex2_commute /3 width=8 by tpr_conf_flat_beta/
-(* case 8: theta, flat (repeated) *)
- | @ex2_commute /3 width=11 by tpr_conf_flat_theta/
-(* case 9: tau, flat (repeated) *)
- | @ex2_commute /3 width=6 by tpr_conf_flat_cast/
-(* case 10: flat, beta *)
- | /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 11: beta, beta *)
- | /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 12: flat, theta *)
- | /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
-(* case 13: theta, theta *)
- | /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
-(* case 14: flat, tau *)
- | /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
-(* case 15: tau, tau *)
- | /3 width=5 by tpr_conf_tau_tau/
- ]
- ]
-]
-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/static/ssta.ma".
-include "basic_2/reducibility/cpr.ma".
-
-lemma arith1: ∀x,y,z,w. z < y → x + (y + w) - z = x + y - z + w.
-#x #y #z #w #H <le_plus_minus_comm // /3 width=1 by lt_to_le, le_plus_a/
-qed-.
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-notation "hvbox( ⦃term 46 h, break term 46 L⦄ ⊢ break term 46 T1 ➡ break [ term 46 g ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'XPRed $h $g $L $T1 $T2 }.
-
-inductive xpr (h) (g): lenv → relation term ≝
-| xpr_cpr : ∀I,L,U2. L ⊢ ⓪{I} ➡ U2 → xpr h g L (⓪{I}) U2
-| xpr_ssta: ∀I,L,U2,l. ⦃h, L⦄ ⊢ ⓪{I} •[g] ⦃l+1, U2⦄ → xpr h g L (⓪{I}) U2
-| xpr_bind: ∀a,I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g (L.ⓑ{I}V2) T1 T2 →
- L ⊢ ⓑ{a,I}V2.T2 ➡ U2 → xpr h g L (ⓑ{a,I}V1.T1) U2
-| xpr_flat: ∀I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g L T1 T2 →
- L ⊢ ⓕ{I}V2.T2 ➡ U2 → xpr h g L (ⓕ{I}V1.T1) U2
-.
-
-interpretation
- "context-sensitive extended parallel reduction (term)"
- 'XPRed h g L T1 T2 = (xpr h g L T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Note: this is "∀h,g,L. reflexive … (xpr h g L)" *)
-lemma xpr_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
-#h #g #T elim T -T /2 width=1/ * /2 width=5/
-qed.
-
-lemma cpr_xpr: ∀h,g,T1,L,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=1/ * /2 width=5/
-qed.
-
-lemma ssta_xpr: ∀h,g,T1,L,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=2/ * [|*]
-[ #a #I #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_bind1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_appl1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- lapply (ssta_inv_cast1 … H) -H /3 width=5/
-]
-qed.
-
-include "basic_2/substitution/lift_lift.ma".
-include "basic_2/substitution/fsup.ma".
-include "basic_2/unfold/ltpss_dx_ldrop.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/reduction/crf.ma".
+
+(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
+
+definition cif: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥.
+
+interpretation "context-sensitive irreducibility (term)"
+ 'NotReducible L T = (cif L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cif_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥.
+/3 width=3/ qed-.
+
+lemma cif_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
+/3 width=1/ qed-.
+
+lemma cif_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+ L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
+/4 width=1/ qed-.
+
+lemma cif_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+ ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
+#a * [ elim a -a ]
+[ #L #V #T #H elim (H ?) -H /3 width=1/
+|*: #L #V #T #H elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=1/
+]
+qed-.
+
+lemma cif_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#L #V #T #HVT @and3_intro /3 width=1/
+generalize in match HVT; -HVT elim T -T //
+* // #a * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
+qed-.
+
+lemma cif_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
+ ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+* #L #V #T #H
+[ elim (cif_inv_appl … H) -H /2 width=1/
+| elim (cif_inv_ri2 … H) -H /2 width=1/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tif_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄.
+/2 width=2 by trf_inv_atom/ qed.
+
+lemma cif_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
+#a #I #L #V #T #HI #HV #HT #H
+elim (crf_inv_ib2 … HI H) -HI -H /2 width=1/
+qed.
+
+lemma cif_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄.
+#L #V #T #HV #HT #H1 #H2
+elim (crf_inv_appl … H2) -H2 /2 width=1/
+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/reduction/crf_append.ma".
+include "basic_2/reduction/cif.ma".
+
+(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
+
+(* Advanved properties ******************************************************)
+
+lemma cif_labst_last: ∀L,T,W. L ⊢ 𝐈⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄.
+/3 width=2 by crf_inv_labst_last/ qed.
+
+lemma cif_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄.
+/3 width=2 by crf_inv_trf/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cif_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+/3 width=1/ qed-.
+
+lemma cif_inv_tif: ∀T,W. ⋆.ⓛW ⊢ 𝐈⦃T⦄ → ⋆ ⊢ 𝐈⦃T⦄.
+/3 width=1/ 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/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
+
+interpretation
+ "context-sensitive normality (term)"
+ 'Normal L T = (cnf L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cnf_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥.
+#L #K #V #i #HLK #H
+elim (lift_total V 0 (i+1)) #W #HVW
+lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct
+elim (lift_inv_lref2_be … HVW ? ?) -HVW //
+qed-.
+
+lemma cnf_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄.
+#a #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnf_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄.
+#L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnf_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
+#L #V #T #H elim (is_lift_dec T 0 1)
+[ * #U #HTU
+ lapply (H U ?) -H /2 width=3/ #H destruct
+ elim (lift_inv_pair_xy_y … HTU)
+| #HT
+ elim (cpss_delift (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
+ lapply (H (+ⓓV.T2) ?) -H /4 width=1/ -HT2 #H destruct /3 width=2/
+]
+qed-.
+
+lemma cnf_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#L #V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
+| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
+ [ elim (lift_total V1 0 1) #V2 #HV12
+ lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct
+ | lapply (H (ⓓ{a}V1.U1) ?) -H /3 width=1/ #H destruct
+]
+qed-.
+
+lemma cnf_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
+#L #V #T #H lapply (H T ?) -H /2 width=1/ #H
+@discr_tpair_xy_y //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_sort *)
+lemma cnf_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄.
+#L #k #X #H
+>(cpr_inv_sort1 … H) //
+qed.
+
+(* Basic_1: was: nf2_abst *)
+lemma cnf_abst: ∀a,I,L,V,W,T. L ⊢ 𝐍⦃W⦄ → L. ⓑ{I} V ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄.
+#a #I #L #V #W #T #HW #HT #X #H
+elim (cpr_fwd_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct
+>(HW … HW0) -W0 >(HT … HT0) -T0 //
+qed.
+
+(* Basic_1: was only: nf2_appl_lref *)
+lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄.
+#L #V #T #HV #HT #HS #X #H
+elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct
+>(HV … HV0) -V0 >(HT … HT0) -T0 //
+qed.
+
+(* Basic_1: was: nf2_dec *)
+axiom cnf_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨
+ ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
+
+(* Basic_1: removed theorems 1: nf2_abst_shift *)
--- /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/reduction/cpr_cif.ma".
+include "basic_2/reduction/cnf_crf.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Main properties on context-sensitive irreducible terms *******************)
+
+theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+/2 width=3 by cpr_fwd_cif/ qed.
+
+(* Main inversion lemmas on context-sensitive irreducible terms *************)
+
+theorem cnf_inv_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+/2 width=4 by cnf_inv_crf/ 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/reduction/crf.ma".
+include "basic_2/reduction/cnf.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Advanced inversion lemmas on context-sensitive reducible terms ***********)
+
+(* Note: this property is unusual *)
+lemma cnf_inv_crf: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
+#L #T #H elim H -L -T
+[ #L #K #V #i #HLK #H
+ elim (cnf_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #L #V #T #_ #IHT #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #I #L #V #T * #H1 #H2 destruct
+ [ elim (cnf_inv_zeta … H2)
+ | elim (cnf_inv_tau … H2)
+ ]
+|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+ [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
+ |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
+ ]
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+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/reduction/cpr_lift.ma".
+include "basic_2/reduction/cnf.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: nf2_csort_lref *)
+lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍⦃#i⦄.
+#L #i #HL #X #H
+elim (cpr_inv_lref1 … H) -H // *
+#K #V1 #V2 #HLK #_ #_
+lapply (ldrop_mono … HL … HLK) -L #H destruct
+qed.
+
+(* Basic_1: was: nf2_lref_abst *)
+lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍⦃#i⦄.
+#L #K #V #i #HLK #X #H
+elim (cpr_inv_lref1 … H) -H // *
+#K0 #V1 #V2 #HLK0 #_ #_
+lapply (ldrop_mono … HLK … HLK0) -L #H destruct
+qed.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: nf2_lift *)
+lemma cnf_lift: ∀L0,L,T,T0,d,e.
+ L ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄.
+#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
+elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
+<(HLT … HT1) in HT0; -L #HT0
+>(lift_mono … HT10 … HT0) -T1 -X //
+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/cpqs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx pr2_head_1 *)
+(* Note: cpr_flat: does not hold in basic_1 *)
+inductive cpr: lenv → relation term ≝
+| cpr_atom : ∀I,L. cpr L (⓪{I}) (⓪{I})
+| cpr_delta: ∀L,K,V,V2,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV → cpr K V V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → cpr L (#i) W2
+| cpr_bind : ∀a,I,L,V1,V2,T1,T2.
+ cpr L V1 V2 → cpr (L. ⓑ{I} V1) T1 T2 →
+ cpr L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| cpr_flat : ∀I,L,V1,V2,T1,T2.
+ cpr L V1 V2 → cpr L T1 T2 →
+ cpr L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| cpr_zeta : ∀L,V,T1,T,T2. cpr (L.ⓓV) T1 T →
+ ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV. T1) T2
+| cpr_tau : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV. T1) T2
+| cpr_beta : ∀a,L,V1,V2,W,T1,T2.
+ cpr L V1 V2 → cpr (L.ⓛW) T1 T2 →
+ cpr L (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
+| cpr_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
+ cpr L V1 V → ⇧[0, 1] V ≡ V2 → cpr L W1 W2 → cpr (L.ⓓW1) T1 T2 →
+ cpr L (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV2. T2)
+.
+
+interpretation "context-sensitive parallel reduction (term)"
+ 'PRed L T1 T2 = (cpr L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
+#L1 #T1 #T2 #H elim H -L1 -T1 -T2
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ lapply (ldrop_fwd_ldrop2_length … HLK1) #Hi
+ lapply (ldrop_fwd_O1_length … HLK1) #H2i
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // -Hi
+ <H2i -H2i <minus_plus_m_m /3 width=6/
+|3,7: /4 width=1/
+|4,6: /3 width=1/
+|5,8: /4 width=3/
+]
+qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
+#T1 #T2 #HT12 #L
+lapply (cpr_lsubr_trans … HT12 L ?) //
+qed.
+
+lemma cpqs_cpr: ∀L,T1,T2. L ⊢ T1 ➤* T2 → L ⊢ T1 ➡ T2.
+#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
+qed.
+
+lemma cpss_cpr: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡ T2.
+/3 width=1/ qed.
+
+(* Basic_1: includes by definition: pr0_refl *)
+lemma cpr_refl: ∀T,L. L ⊢ T ➡ T.
+/2 width=1/ qed.
+
+lemma cpr_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
+#L #K #V #T1 #d #HLK
+elim (cpqs_delift … T1 … HLK) -HLK /3 width=4/
+qed-.
+
+lemma cpr_append: l_appendable_sn … cpr.
+#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
+#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_ldrop2_length … HK0) #H
+@(cpr_delta … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
+qed.
+
+lemma cpr_ext_bind: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
+ ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
+#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I
+lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➡ V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 * -L -T1 -T2
+[ #I #L #J #H destruct /2 width=1/
+| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #L #V #T1 #T2 #_ #J #H destruct
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #J #H destruct
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpr_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➡ T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➡ V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+/2 width=3 by cpr_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
+lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+#L #T2 #k #H
+elim (cpr_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+ T2 = #i ∨
+ ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➡ V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #H
+elim (cpr_inv_atom1 … H) -H /2 width=1/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma cpr_inv_gref1: ∀L,T2,p. L ⊢ §p ➡ T2 → T2 = §p.
+#L #T2 #p #H
+elim (cpr_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+ L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+#L #U1 #U2 * -L -U1 -U2
+[ #I #L #b #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
+| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
+| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+ L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+/2 width=3 by cpr_inv_bind1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+ L. ⓓ V1 ⊢ T1 ➡ T2 &
+ U2 = ⓓ{a} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
+#a #L #V1 #T1 #U2 #H
+elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
+qed-.
+
+(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
+lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➡ U2 →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓛ V1 ⊢ T1 ➡ T2 &
+ U2 = ⓛ{a} V2. T2.
+#a #L #V1 #T1 #U2 #H
+elim (cpr_inv_bind1 … H) -H *
+[ /3 width=5/
+| #T #_ #_ #_ #H destruct
+]
+qed-.
+
+fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
+ ∀I,V1,U1. U = ⓕ{I} V1. U1 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | (L ⊢ U1 ➡ U2 ∧ I = Cast)
+ | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
+ U1 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+ L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+ U1 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
+#L #U #U2 * -L -U -U2
+[ #I #L #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
+| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
+| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
+| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
+| #a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=9/
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W1 #U1 #H destruct /3 width=13/
+]
+qed-.
+
+lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I} V1. U1 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | (L ⊢ U1 ➡ U2 ∧ I = Cast)
+ | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
+ U1 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+ L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+ U1 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
+/2 width=3 by cpr_inv_flat1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐ V1. U1 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+ U2 = ⓐ V2. T2
+ | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
+ U1 = ⓛ{a}W. T1 & U2 = ⓓ{a}V2. T2
+ | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+ L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+ U1 = ⓓ{a}W1. T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
+#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+[ /3 width=5/
+| #_ #H destruct
+| /3 width=9/
+| /3 width=13/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #a #V2 #W #U1 #U2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
+lemma cpr_inv_cast1: ∀L,V1,U1,U2. L ⊢ ⓝ V1. U1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+ U2 = ⓝ V2. T2
+ ) ∨
+ L ⊢ U1 ➡ U2.
+#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+[ /3 width=5/
+| /2 width=1/
+| #a #V2 #W #T1 #T2 #_ #_ #_ #_ #H destruct
+| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 &
+ U2 = ⓛ{a} V2. T2.
+#a #L #V1 #T1 #U2 #H #I #W
+elim (cpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}W) ?) -HT12 /2 width=1/ /2 width=5/
+qed-.
+
+
+lemma cpr_fwd_ext_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
+ U2 = ⓛ{a}V2.T2.
+#a #L #V1 #T1 #U2 #H #b #I #W
+elim (cpr_fwd_abst1 … H I W) -H /3 width=5/
+qed-.
+
+lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #L #V1 #T1 #T #H #b
+elim (cpr_inv_bind1 … H) -H *
+[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+| #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpr_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X
+ >shift_append_assoc normalize #H
+ elim (cpr_inv_bind1 … H) -H *
+ [ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
+
+(* Basic_1: removed theorems 12:
+ pr0_subst0_back pr0_subst0_fwd pr0_subst0
+ pr2_head_2 pr2_cflat clear_pr2_trans
+ pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
+ pr2_subst1
+ pr2_gen_ctail pr2_ctail
+*)
+(* Basic_1: removed local theorems 4:
+ pr0_delta_tau pr0_cong_delta
+ pr2_free_free pr2_free_delta
+*)
--- /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/reduction/cif.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Advanced forward lemmas on context-sensitive irreducible terms ***********)
+
+lemma cpr_fwd_cif: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T2 = T1.
+#L #T1 #T2 #H elim H -L -T1 -T2
+[ //
+| #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
+ elim (cif_inv_delta … HLK ?) //
+| #a * #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+ [ elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+ lapply (IHT1 … HT1) -IHT1 #H destruct //
+ | elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
+ ]
+| * #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+ [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ | elim (cif_inv_ri2 … H) /2 width=1/
+ ]
+| #L #V1 #T1 #T #T2 #_ #_ #_ #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+| #L #V1 #T1 #T2 #_ #_ #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+qed-.
+
+lemma cpss_fwd_cif_eq: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ 𝐈⦃T1⦄ → T2 = T1.
+/3 width=3 by cpr_fwd_cif, cpss_cpr/ 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/relocation/ldrop_ldrop.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: includes: pr0_lift pr2_lift *)
+lemma cpr_lift: l_liftable cpr.
+#K #T1 #T2 #H elim H -K -T1 -T2
+[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
+ | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
+ ]
+| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
+| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+| #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
+ elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+ elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
+| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
+| #a #K #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /4 width=5/
+| #a #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+ elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+ elim (lift_trans_ge … HV2 … HV3 ?) -V2 // /4 width=5/
+]
+qed.
+
+(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
+lemma cpr_inv_lift1: l_deliftable_sn cpr.
+#L #U1 #U2 #H elim H -L -U1 -U2
+[ * #L #i #K #d #e #_ #T1 #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+ elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+ elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
+ elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
+ ]
+| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
+| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -V1
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
+| #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
+ elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
+| #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #K #d #e #HLK #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HLK … HV01) -V1
+ elim (IHT12 (K.ⓛW0) … HT01) -T1 /2 width=1/ /3 width=5/
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #d #e #HLK #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
+ elim (IHT12 (K.ⓓW0) … HT01) -T1 /2 width=1/ #T3 #HT32 #HT03
+ elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
+ elim (lift_trans_le … HV3 … HV2 ?) -V // #V #HV3 #HV2
+ @ex2_intro [2: /3 width=2/ | skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
+]
+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/grammar/tshf.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Forward lemmas on same head forms for terms ******************************)
+
+lemma cpr_fwd_tshf1: ∀L,T1,T2. L ⊢ T1 ➡ T2 → T1 ≈ T1 →
+ T2 ≈ T1 ∨ (L = ⋆ → ⊥).
+#L #T1 #T2 #H elim H -L -T1 -T2
+[ /2 width=1/
+| #L #K #V1 #V2 #W2 #i #HLK #_ #_ #_ #_
+ @or_intror #H destruct
+ lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct /2 width=1/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
+ lapply (IHT12 HT1U2) -IHT12 -HT1U2 * #HUT2 /3 width=1/
+ lapply (simple_tshf_repl_sn … HUT2 HT1) /3 width=1/
+| #L #V #T #T1 #T2 #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct
+| #L #V #T1 #T2 #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
+ elim (simple_inv_bind … H)
+]
+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/relocation/ldrop.ma".
+
+(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
+
+(* reducible binary items *)
+definition ri2: predicate item2 ≝
+ λI. I = Bind2 true Abbr ∨ I = Flat2 Cast.
+
+(* irreducible binary binders *)
+definition ib2: relation2 bool bind2 ≝
+ λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr.
+
+(* reducible terms *)
+inductive crf: lenv → predicate term ≝
+| crf_delta : ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → crf L (#i)
+| crf_appl_sn: ∀L,V,T. crf L V → crf L (ⓐV. T)
+| crf_appl_dx: ∀L,V,T. crf L T → crf L (ⓐV. T)
+| crf_ri2 : ∀I,L,V,T. ri2 I → crf L (②{I}V. T)
+| crf_ib2_sn : ∀a,I,L,V,T. ib2 a I → crf L V → crf L (ⓑ{a,I}V. T)
+| crf_ib2_dx : ∀a,I,L,V,T. ib2 a I → crf (L.ⓑ{I}V) T → crf L (ⓑ{a,I}V. T)
+| crf_beta : ∀a,L,V,W,T. crf L (ⓐV. ⓛ{a}W. T)
+| crf_theta : ∀a,L,V,W,T. crf L (ⓐV. ⓓ{a}W. T)
+.
+
+interpretation
+ "context-sensitive reducibility (term)"
+ 'Reducible L T = (crf L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact trf_inv_atom_aux: ∀I,L,T. L ⊢ 𝐑⦃T⦄ → L = ⋆ → T = ⓪{I} → ⊥.
+#I #L #T * -L -T
+[ #L #K #V #i #HLK #H1 #H2 destruct
+ lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
+| #L #V #T #_ #_ #H destruct
+| #L #V #T #_ #_ #H destruct
+| #J #L #V #T #_ #_ #H destruct
+| #a #J #L #V #T #_ #_ #_ #H destruct
+| #a #J #L #V #T #_ #_ #_ #H destruct
+| #a #L #V #W #T #_ #H destruct
+| #a #L #V #W #T #_ #H destruct
+]
+qed-.
+
+lemma trf_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥.
+/2 width=6 by trf_inv_atom_aux/ qed-.
+
+fact crf_inv_lref_aux: ∀L,T. L ⊢ 𝐑⦃T⦄ → ∀i. T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
+#L #T * -L -T
+[ #L #K #V #j #HLK #i #H destruct /2 width=3/
+| #L #V #T #_ #i #H destruct
+| #L #V #T #_ #i #H destruct
+| #J #L #V #T #_ #i #H destruct
+| #a #J #L #V #T #_ #_ #i #H destruct
+| #a #J #L #V #T #_ #_ #i #H destruct
+| #a #L #V #W #T #i #H destruct
+| #a #L #V #W #T #i #H destruct
+]
+qed-.
+
+lemma crf_inv_lref: ∀L,i. L ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
+/2 width=3 by crf_inv_lref_aux/ qed-.
+
+fact crf_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W. U →
+ L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄.
+#a #I #L #W #U #T #HI * -T
+[ #L #K #V #i #_ #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #J #L #V #T #H1 #H2 destruct
+ elim H1 -H1 #H destruct
+ elim HI -HI #H destruct
+| #b #J #L #V #T #_ #HV #H destruct /2 width=1/
+| #b #J #L #V #T #_ #HT #H destruct /2 width=1/
+| #b #L #V #W #T #H destruct
+| #b #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crf_inv_ib2: ∀a,I,L,W,T. ib2 a I → L ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ →
+ L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄.
+/2 width=5 by crf_inv_ib2_aux/ qed-.
+
+fact crf_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW. U →
+ ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+#L #W #U #T * -T
+[ #L #K #V #i #_ #H destruct
+| #L #V #T #HV #H destruct /2 width=1/
+| #L #V #T #HT #H destruct /2 width=1/
+| #J #L #V #T #H1 #H2 destruct
+ elim H1 -H1 #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W0 #T #H destruct
+ @or3_intro2 #H elim (simple_inv_bind … H)
+| #a #L #V #W0 #T #H destruct
+ @or3_intro2 #H elim (simple_inv_bind … H)
+]
+qed-.
+
+lemma crf_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+/2 width=3 by crf_inv_appl_aux/ 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/relocation/ldrop_append.ma".
+include "basic_2/reduction/crf.ma".
+
+(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
+
+(* Advanved properties ******************************************************)
+
+lemma crf_labst_last: ∀L,T,W. L ⊢ 𝐑⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄.
+#L #T #W #H elim H -L -T /2 width=1/
+#L #K #V #i #HLK
+lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+lapply (ldrop_O1_append_sn_le … HLK … (⋆.ⓛW)) -HLK /2 width=2/ -Hi /2 width=3/
+qed.
+
+lemma crf_trf: ∀T,W. ⋆ ⊢ 𝐑⦃T⦄ → ⋆.ⓛW ⊢ 𝐑⦃T⦄.
+#T #W #H lapply (crf_labst_last … W H) //
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact crf_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄ →
+ ∀L2. L1 = ⋆.ⓛW @@ L2 → L2 ⊢ 𝐑⦃T⦄.
+#L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
+[ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
+ lapply (ldrop_fwd_ldrop2_length … HLK1)
+ >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
+ elim (le_to_or_lt_eq i (|L2|) ?) /2 width=1/ -H #Hi destruct
+ [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
+ lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
+ normalize #H destruct /2 width=3/
+ | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
+ lapply (ldrop_inv_refl … H) -H #H destruct
+ ]
+| #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
+]
+qed.
+
+lemma crf_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐑⦃T⦄.
+/2 width=4/ qed-.
+
+lemma crf_inv_trf: ∀T,W. ⋆.ⓛW ⊢ 𝐑⦃T⦄ → ⋆ ⊢ 𝐑⦃T⦄.
+/2 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/unfold/lpqs.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+definition lpr: relation lenv ≝ lpx_sn cpr.
+
+interpretation "parallel reduction (local environment, sn variant)"
+ 'PRedSn L1 L2 = (lpr L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: includes: wcpr0_gen_sort *)
+lemma lpr_inv_atom1: ∀L2. ⋆ ⊢ ➡ L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: wcpr0_gen_head *)
+lemma lpr_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ➡ L2 →
+ ∃∃K2,V2. K1 ⊢ ➡ K2 & K1 ⊢ V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpr_inv_atom2: ∀L1. L1 ⊢ ➡ ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpr_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ➡ K2 & K1 ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lpr_refl: ∀L. L ⊢ ➡ L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpr_append: ∀K1,K2. K1 ⊢ ➡ K2 → ∀L1,L2. L1 ⊢ ➡ L2 →
+ L1 @@ K1 ⊢ ➡ L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpr_append/ qed.
+
+lemma lpqs_lpr: ∀L1,L2. L1 ⊢ ➤* L2 → L1 ⊢ ➡ L2.
+#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+lemma lpss_lpr: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➡ L2.
+/3 width=1/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpr_fwd_length: ∀L1,L2. L1 ⊢ ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpr_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➡ L →
+ ∃∃K2,L2. K1 ⊢ ➡ K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpr_fwd_append2: ∀L,K2,L2. L ⊢ ➡ K2 @@ L2 →
+ ∃∃K1,L1. K1 ⊢ ➡ K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
+
+(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /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/aaa_lift.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reduction/lpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_cpr_lpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ➡ T2 →
+ ∀L2. L1 ⊢ ➡ L2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #X #H
+ >(cpr_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+ elim (cpr_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
+ | * #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (lpr_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpr_inv_abbr1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
+ | #T2 #HT12 #HT2 #H destruct -IHV1
+ @(aaa_inv_lift (L2.ⓓV1) … HT2) -HT2 /2 width=1/ /3 width=1/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpr_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
+ | #b #V2 #W1 #U1 #U2 #HV12 #HU12 #H1 #H2 destruct
+ lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
+ lapply (IHT1 (ⓛ{b}W1.U2) … HL12) -IHT1 /2 width=1/ -L1 #H
+ elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+ lapply (lsuba_aaa_trans … HU2 (L2.ⓓV2) ?) -HU2 /2 width=2/ /2 width=3/
+ | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+ lapply (aaa_lift L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=1/ #HV2
+ lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1/ -L1 #H
+ elim (aaa_inv_abbr … H) -H /3 width=3/
+ ]
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpr_inv_cast1 … H) -H
+ [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
+ | -IHV1 /2 width=1/
+ ]
+]
+qed-.
+
+lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
+/2 width=5 by aaa_cpr_lpr_conf/ qed-.
+
+lemma aaa_lpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ➡ L2 → L2 ⊢ T ⁝ A.
+/2 width=5 by aaa_cpr_lpr_conf/ 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/relocation/fsup.ma".
+include "basic_2/reduction/lpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties on context-sensitive parallel reduction for terms ********)
+
+fact cpr_conf_lpr_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T.
+/2 width=3/ qed-.
+
+fact cpr_conf_lpr_atom_delta:
+ ∀L0,i. (
+ ∀L,T.♯{L, T} < ♯{L0, #i} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ #i ➡ T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_lpr_delta_delta:
+ ∀L0,i. (
+ ∀L,T.♯{L, T} < ♯{L0, #i} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpr_conf_lpr_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➡ T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpr_conf_lpr_bind_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 →
+ ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ +ⓓV1.T1 ➡ T & L2 ⊢ X2 ➡ T.
+#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
+qed-.
+
+fact cpr_conf_lpr_zeta_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ X1 ➡ T & L2 ⊢ X2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
+qed-.
+
+fact cpr_conf_lpr_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➡ T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+fact cpr_conf_lpr_flat_tau:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓝV1.T1 ➡ T & L2 ⊢ T2 ➡ T.
+#L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
+qed-.
+
+fact cpr_conf_lpr_tau_tau:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
+qed-.
+
+fact cpr_conf_lpr_flat_beta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
+qed-.
+
+(* Basic-1: includes:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_lpr_flat_theta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2
+elim (cpr_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 /4 width=7/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY
+ @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *)
+]
+qed-.
+
+fact cpr_conf_lpr_beta_beta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
+lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/
+lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
+qed-.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact cpr_conf_lpr_theta_theta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+ ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ /4 width=7/
+qed-.
+
+theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
+#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
+[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpr_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_lpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_lpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_lpr_delta_delta/
+ ]
+| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
+ | /3 width=11 by cpr_conf_lpr_bind_zeta/
+ | /3 width=12 by cpr_conf_lpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #a2 #V2 #Y2 #Z2 #T2 #HV02 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
+ | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
+ | /3 width=8 by cpr_conf_lpr_flat_tau/
+ | /3 width=7 by cpr_conf_lpr_tau_tau/
+ | /3 width=11 by cpr_conf_lpr_flat_beta/
+ | /3 width=11 by cpr_conf_lpr_beta_beta/
+ | /3 width=14 by cpr_conf_lpr_flat_theta/
+ | /3 width=17 by cpr_conf_lpr_theta_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf: ∀L. confluent … (cpr L).
+/2 width=6 by cpr_conf_lpr/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma lpr_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
+ ∃∃T. L1 ⊢ T0 ➡ T & L1 ⊢ T1 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
+qed-.
+
+lemma lpr_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
+ ∃∃T. L1 ⊢ T0 ➡ T & L0 ⊢ T1 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+lemma fsup_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
+ ∃∃L,U1. L1 ⊢ ➡ L & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+elim (ldrop_lpr_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
+lapply (cpr_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+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/substitution/lpss_ldrop.ma".
+include "basic_2/reduction/lpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+fact cpr_cpss_conf_lpr_lpss_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
+/2 width=3/ qed-.
+
+fact cpr_cpss_conf_lpr_lpss_atom_delta:
+ ∀L0,i. (
+ ∀L,T.♯{L, T} < ♯{L0, #i} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_delta_atom:
+ ∀L0,i. (
+ ∀L,T.♯{L, T} < ♯{L0, #i} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_delta_delta:
+ ∀L0,i. (
+ ∀L,T.♯{L, T} < ♯{L0, #i} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_bind_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_tau:
+ ∀L0,V0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01
+#V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_beta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
+elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_theta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
+elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
+/4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
+qed-.
+
+lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
+#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
+[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpss_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
+ ]
+| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ] destruct
+ [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
+ | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
+ ]
+| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ] destruct
+ [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
+ | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
+ | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
+ | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_subst1 *)
+lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
+/2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
+
+lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
+qed-.
+
+lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Basic_1: includes: pr0_subst1_fwd *)
+lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /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/relocation/ldrop_lpx_sn.ma".
+include "basic_2/reduction/cpr_lift.ma".
+include "basic_2/reduction/lpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properies on local environment slicing ***********************************)
+
+(* Basic_1: includes: wcpr0_drop *)
+lemma lpr_ldrop_conf: dropable_sn lpr.
+/3 width=5 by lpx_sn_deliftable_dropable, cpr_inv_lift1/ qed-.
+
+(* Basic_1: includes: wcpr0_drop_back *)
+lemma ldrop_lpr_trans: dedropable_sn lpr.
+/3 width=9 by lpx_sn_liftable_dedropable, cpr_lift/ qed-.
+
+lemma lpr_ldrop_trans_O1: dropable_dx lpr.
+/2 width=3 by lpx_sn_dropable/ 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/reduction/lpr_cpr.ma".
+
+(* SN PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ******************************)
+
+(* Main properties **********************************************************)
+
+theorem lpr_conf: confluent … lpr.
+/3 width=6 by lpx_sn_conf, cpr_conf_lpr/
+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/reduction/lpr_cpss.ma".
+
+(* SN PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ******************************)
+
+(* Properties on sn parallel substitution on local environments *************)
+
+lemma lpr_lpss_conf: confluent2 … lpr lpss.
+/3 width=6 by lpx_sn_conf, cpr_cpss_conf_lpr_lpss/
+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/static/ssta.ma".
+include "basic_2/reduction/cpr.ma".
+
+lemma arith1: ∀x,y,z,w. z < y → x + (y + w) - z = x + y - z + w.
+#x #y #z #w #H <le_plus_minus_comm // /3 width=1 by lt_to_le, le_plus_a/
+qed-.
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+notation "hvbox( ⦃term 46 h, break term 46 L⦄ ⊢ break term 46 T1 ➡ break [ term 46 g ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'XPRed $h $g $L $T1 $T2 }.
+
+inductive xpr (h) (g): lenv → relation term ≝
+| xpr_cpr : ∀I,L,U2. L ⊢ ⓪{I} ➡ U2 → xpr h g L (⓪{I}) U2
+| xpr_ssta: ∀I,L,U2,l. ⦃h, L⦄ ⊢ ⓪{I} •[g] ⦃l+1, U2⦄ → xpr h g L (⓪{I}) U2
+| xpr_bind: ∀a,I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g (L.ⓑ{I}V1) T1 T2 →
+ L ⊢ ⓑ{a,I}V2.T2 ➡ U2 → xpr h g L (ⓑ{a,I}V1.T1) U2
+| xpr_flat: ∀I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g L T1 T2 →
+ L ⊢ ⓕ{I}V2.T2 ➡ U2 → xpr h g L (ⓕ{I}V1.T1) U2
+.
+
+interpretation
+ "context-sensitive extended parallel reduction (term)"
+ 'XPRed h g L T1 T2 = (xpr h g L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Note: this is "∀h,g,L. reflexive … (xpr h g L)" *)
+lemma xpr_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
+#h #g #T elim T -T /2 width=1/ * /2 width=5/
+qed.
+
+lemma cpr_xpr: ∀h,g,T1,L,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+#h #g #T1 elim T1 -T1 /2 width=1/ * /2 width=5/
+qed.
+
+lemma ssta_xpr: ∀h,g,T1,L,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+#h #g #T1 elim T1 -T1 /2 width=2/ * [|*]
+[ #a #I #V1 #T1 #_ #IHT1 #L #X #l #H
+ elim (ssta_inv_bind1 … H) -H #T2 #HT12 #H destruct /3 width=5/
+| #V1 #T1 #_ #IHT1 #L #X #l #H
+ elim (ssta_inv_appl1 … H) -H #T2 #HT12 #H destruct /3 width=5/
+| #V1 #T1 #_ #IHT1 #L #X #l #H
+ lapply (ssta_inv_cast1 … H) -H /3 width=5/
+]
+qed.
+
+include "basic_2/relocation/lift_lift.ma".
+include "basic_2/relocation/fsup.ma".
+include "basic_2/substitution/lpss_ldrop.ma".
+
}
]
class "water"
- [ { "reducibility" * } {
- [ { "context-sensitive focalized reduction" * } {
- [ "cfpr ( ? ⊢ ⦃?,?⦄ ➡ ⦃?,?⦄ )" "cnfpr_ltpss" + "cfpr_aaa" + "cfpr_cpr" + "cfpr_cfpr" * ]
- }
- ]
- [ { "context-free focalized reduction" * } {
- [ "lfpr ( ⦃?⦄ ➡ ⦃?⦄ )" "lfpr_alt ( ⦃?⦄ ➡➡ ⦃?⦄ )" "lfpr_aaa" + "lfpr_cpr" + "lfpr_fpr" + "lfpr_lfpr" * ]
- [ "fpr ( ⦃?,?⦄ ➡ ⦃?,?⦄ )" "fpr_cpr" + "fpr_fpr" * ]
- }
- ]
+ [ { "reduction" * } {
[ { "context-sensitive normal forms" * } {
- [ "cnf ( ? ⊢ 𝐍⦃?⦄ )" "cnf_lift" + "cnf_cif" * ]
- }
- ]
- [ { "revised context-sensitive reduction" * } {
- [ "lpr ( ? ⊢ ➡ ? )" "lpr_ldrop" + "lpr_cpr" + "lpr_lpr" * ]
- [ "cpr ( ? ⊢ ? ➡ ? )" "cpr_lift" * ]
+ [ "cnf ( ? ⊢ 𝐍⦃?⦄ )" "cnf_liftt" + "cnf_crf" + "cnf_cif" * ]
+ [ "chnf ( ? ⊢ 𝐇𝐍⦃?⦄ )" * ]
}
]
[ { "context-sensitive reduction" * } {
- [ "cpr ( ? ⊢ ? ➡ ? )" "cpr_lift" + "cpr_tpss" + "cpr_ltpss_dx" + "cpr_ltpss_sn" + "cpr_delift" + "cpr_aaa" + "cpr_ltpr" + "cpr_cpr" * ]
+ [ "lpr ( ? ⊢ ➡ ? )" "lpr_ldrop" + "lpr_cpss" + "lpr_lpss" + "lpr_aaa" + "lpr_cpr" + "lpr_lpr" * ]
+ [ "cpr ( ? ⊢ ? ➡ ? )" "lpr_tshf" + "cpr_lift" + "cpr_cif" * ]
}
]
[ { "context-sensitive reducible forms" * } {
[ "crf ( ? ⊢ 𝐑⦃?⦄ )" "crf_append" "cif ( ? ⊢ 𝐈⦃?⦄ )" "cif_append" * ]
}
]
- [ { "context-free normal forms" * } {
- [ "thnf ( 𝐇𝐍⦃?⦄ )" * ]
- }
- ]
- [ { "context-free reduction" * } {
- [ "ltpr ( ? ➡ ? )" "ltpr_ldrop" + "ltpr_tps" + "ltpr_tpss" + "ltpr_ltpss_dx" + "ltpr_ltpss_sn" + "ltpr_aaa" + "ltpr_ltpr" * ]
- [ "tpr ( ? ➡ ? )" "tpr_lift" + "tpr_delift" + "tpr_tpr" * ]
- }
- ]
}
]
class "green"
]
class "yellow"
[ { "substitution" * } {
-(*
- [ { "basic local env. thinning" * } {
- [ "thin ( ? ▼*[?,?] ≡ ? )" "thin_ldrop" + "thin_delift" * ]
- }
- ]
- [ { "inverse basic term relocation" * } {
- [ "delift ( ? ⊢ ? ▼*[?,?] ≡ ? )" "delift_alt ( ? ⊢ ? ▼▼*[?,?] ≡ ? )" "delift_lift" + "delift_tpss" + "delift_ltpss" + "delift_delift" * ]
- }
- ]
-*)
[ { "parallel substitution" * } {
[ "lpss ( ? ⊢ ▶* ? )" "lpss_ldrop" + "lpss_cpss" + "lpss_lpss" * ]
[ "cpss ( ? ⊢ ? ▶* ? )" "cpss_lift" * ]
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