(* *)
(**************************************************************************)
+include "static_2/static/rdeq_drops.ma".
include "static_2/static/rdeq_fqup.ma".
include "static_2/static/rdeq_rdeq.ma".
include "basic_2/rt_transition/rpx_fsle.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
-lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
+lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀I,T.
+ ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[V] L2.
/3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
-lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
+lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀I,V.
+ ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
-lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
- ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
+lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀p.
+ ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
-lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
+lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀p,I,V.
+ ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
-lemma rpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (rpx h G).
+lemma rpx_tdeq_conf: ∀h,G. s_r_confluent1 … cdeq (rpx h G).
/2 width=5 by tdeq_rex_conf/ qed-.
-lemma rpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+lemma rpx_tdeq_div: ∀h,T1,T2. T1 ≛ T2 →
∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
/2 width=5 by tdeq_rex_div/ qed-.
-lemma cpx_tdeq_conf_sex: ∀h,o,G. R_confluent2_rex … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
-#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
+lemma cpx_tdeq_conf_sex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq.
+#h #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
- /4 width=3 by tdeq_sort, deg_next, ex2_intro/
+ elim (tdeq_inv_sort1 … H0) -H0 #s1 #H destruct
+ /3 width=3 by tdeq_sort, ex2_intro/
| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
>(tdeq_inv_lref1 … H0) -H0
elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
/3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
+| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
elim (rpx_inv_bind … H1) -H1 #HL01 #H1
elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (rdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
+ lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
+ lapply (rdeq_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
+ elim (tdeq_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
/3 width=5 by cpx_zeta, ex2_intro/
| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
]
qed-.
-lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∀T2. T0 ≛[h, o] T2 →
- ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
-#h #o #G #L #T0 #T1 #HT01 #T2 #HT02
+lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+ ∀T2. T0 ≛ T2 →
+ ∃∃T. T1 ≛ T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
+#h #G #L #T0 #T1 #HT01 #T2 #HT02
elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02
/2 width=3 by rex_refl, ex2_intro/
qed-.
-lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
+lemma tdeq_cpx_trans: ∀h,G,L,T2. ∀T0:term. T2 ≛ T0 →
∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L #T2 #T0 #HT20 #T1 #HT01
+ ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛ T1.
+#h #G #L #T2 #T0 #HT20 #T1 #HT01
elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
qed-.
(* Basic_2A1: uses: cpx_lleq_conf *)
-lemma cpx_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
-#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
+lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. L0 ≛[T0] L2 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T.
+#h #G #L0 #T0 #T1 #HT01 #L2 #HL02
elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02
/2 width=3 by rex_refl, ex2_intro/
qed-.
(* Basic_2A1: uses: lleq_cpx_trans *)
-lemma rdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
+lemma rdeq_cpx_trans: ∀h,G,L2,L0,T0. L2 ≛[T0] L0 →
∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_rdeq_conf … o … HT01 L2) -HT01
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1.
+#h #G #L2 #L0 #T0 #HL20 #T1 #HT01
+elim (cpx_rdeq_conf … HT01 L2) -HT01
/3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
qed-.
-lemma rpx_rdeq_conf: ∀h,o,G,T. confluent2 … (rpx h G T) (rdeq h o T).
+lemma rpx_rdeq_conf: ∀h,G,T. confluent2 … (rpx h G T) (rdeq T).
/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-.
-lemma rdeq_rpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (rpx_rdeq_conf … o … HLK2 L1)
+lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[T] K2.
+#h #G #T #L2 #K2 #HLK2 #L1 #HL12
+elim (rpx_rdeq_conf … HLK2 L1)
/3 width=3 by rdeq_sym, ex2_intro/
qed-.
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