(* *)
(**************************************************************************)
-include "basic_2/computation/lprs_cprs.ma".
-include "basic_2/conversion/cpc_cpc.ma".
-include "basic_2/equivalence/cpcs_cprs.ma".
+include "basic_2/rt_equivalence/cpcs_lprs.ma".
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
-#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
-[ /3 width=3 by ex2_intro/
-| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
- [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
- | /3 width=5 by cprs_strap2, ex2_intro/
- ]
-]
-qed-.
-
-(* Basic_1: was: pc3_gen_sort *)
-lemma cpcs_inv_sort: ∀G,L,s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌* ⋆s2 → s1 = s2.
-#G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H
-#T #H1 >(cprs_inv_sort1 … H1) -T #H2
-lapply (cprs_inv_sort1 … H2) -L #H destruct //
-qed-.
-
-lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-#a #G #L #W1 #T1 #T #H
-elim (cpcs_inv_cprs … H) -H #X #H1 #H2
-elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
-/3 width=6 by cprs_bind, ex2_2_intro/
-qed-.
-
-lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
-
-(* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌* ⓛ{a}W.T → ⊥.
-#a #G #L #W #T #s #H
-elim (cpcs_inv_cprs … H) -H #X #H1
->(cprs_inv_sort1 … H1) -X #H2
-elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: pc3_gen_lift *)
-lemma cpcs_inv_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K →
- ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 →
- ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
-#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
-elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
-elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
-elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
->(lift_inj … HXU … HTU) -X -U -l -k /2 width=3 by cprs_div/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lpr_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
-/4 width=5 by cprs_div, lpr_cprs_trans/
-qed-.
-
-lemma lprs_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
-/4 width=5 by cprs_div, lprs_cprs_trans/
-qed-.
-
-lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cpr_cprs_div/
-qed-.
-
-lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cprs_cpr_div/
-qed-.
-
-lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cprs_div/
-qed-.
-
-lemma lprs_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
-/2 width=3 by cprs_div/
-qed-.
-
-(* Basic_1: was: pc3_wcpr0_t *)
-(* Basic_1: note: pc3_wcpr0_t should be renamed *)
-(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
-lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
-/2 width=3 by cprs_div/
-qed-.
-
-(* Basic_1: was only: pc3_pr0_pr2_t *)
-(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
-lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-/3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
-
-(* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
-#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
-elim (cpcs_inv_cprs … HV12) -HV12
-elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_flat, cprs_div/
-qed.
-
-lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
-/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
-
-lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
-#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, cprs_bind/
-qed.
-
-lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
-#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
-/3 width=5 by cprs_div, cprs_bind/
-qed.
-
-lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
- ∀L2. L2 ⫃ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, lsubr_cprs_trans/
-qed-.
-
-(* Basic_1: was: pc3_lift *)
-lemma cpcs_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K →
- ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 →
- ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
-#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
-elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
-elim (lift_total T l k) /3 width=12 by cprs_div, cprs_lift/
-qed.
-
-lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
- ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
-#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
-
-(* More inversion lemmas ****************************************************)
-
-(* Note: there must be a proof suitable for llpr *)
-lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
-elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
-lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
-lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
-/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
-qed-.
-
-lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
-#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
-qed-.
+(* CONTEXT-SENSITIVE PARALLEL R-EQUIVALENCE FOR TERMS ***********************)
(* Main properties **********************************************************)
(* Basic_1: was pc3_t *)
-theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
+theorem cpcs_trans (h) (G) (L): Transitive … (cpcs h G L).
+#h #G #L #T1 #T #HT1 #T2 @(trans_TC … HT1)
+qed-.
-theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+theorem cpcs_canc_sn (h) (G) (L): left_cancellable … (cpcs h G L).
/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
-theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+theorem cpcs_canc_dx (h) (G) (L): right_cancellable … (cpcs h G L).
/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
-lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
- ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+(* Advanced properties ******************************************************)
+
+lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
-lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
- ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+(* Advanced properties with r-transition for full local environments ********)
+
(* Basic_1: was: pc3_wcpr0 *)
-lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
qed-.
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