(* *)
(**************************************************************************)
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+(* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
-include "basic_2/relocation/lifts_tdeq.ma".
-include "basic_2/s_computation/fqus_fqup.ma".
+include "static_2/relocation/lifts_teqx.ma".
+include "static_2/s_computation/fqus_fqup.ma".
include "basic_2/rt_transition/cpx_drops.ma".
+include "basic_2/rt_transition/cpx_lsubr.ma".
(* Properties on supclosure *************************************************)
-lemma fqu_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
+lemma fqu_cpx_trans (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/
[ #I #G #L2 #V2 #X2 #HVX2
- elim (lifts_total X2 (ð\9d\90\94â\9d´1â\9dµ))
+ elim (lifts_total X2 (ð\9d\90\94â\9d¨1â\9d©))
/3 width=3 by fqu_drop, cpx_delta, ex2_intro/
-| #I #G #L2 #V #T2 #X2 #HTX2 #U2 #HTU2
- elim (cpx_lifts … HTU2 (Ⓣ) … (L2.ⓑ{I}V) … HTX2)
+| /5 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit, fqu_clear, ex2_intro/
+| #I #G #L2 #T2 #X2 #HTX2 #U2 #HTU2
+ elim (cpx_lifts_sn … HTU2 (Ⓣ) … (L2.ⓘ[I]) … HTX2)
/3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/
]
qed-.
-lemma fquq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
+lemma fquq_cpx_trans (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂⸮[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
[ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqup_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma fqup_cpx_trans (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂+[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
/3 width=3 by fqu_fqup, ex2_intro/
| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
]
qed-.
-lemma fqus_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
+lemma fqus_cpx_trans (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂*[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
[ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqu_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
+lemma fqu_cpx_trans_tneqx (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❨1❩)
#U2 #HVU2 @(ex3_intro … U2)
[1,3: /3 width=7 by cpx_delta, fqu_drop/
- | #H lapply (tdeq_inv_lref1 … H) -H
+ | #H lapply (teqx_inv_lref1 … H) -H
#H destruct /2 width=5 by lifts_inv_lref2_uni_lt/
]
-| #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②{I}V2.T))
+| #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②[I]V2.T))
[1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
- | #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
]
-| #p #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
[1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
- | #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
]
-| #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓕ{I}V.T2))
- [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
- | #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
+ [1,3: /4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit, fqu_clear/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
]
-| #I #G #L #V #T1 #U1 #HTU1 #T2 #HT12 #H0
- elim (cpx_lifts … HT12 (Ⓣ) … (L.ⓑ{I}V) … HTU1) -HT12 /3 width=1 by drops_refl, drops_drop/
- #U2 #HTU2 #HU12 @(ex3_intro … U2)
- [1,3: /3 width=1 by fqu_drop/
- | #H elim (tdeq_inv_lifts … H … HTU1) -U1
- #X2 #H <(lifts_inj … HTU2 … H) -U2 /2 width=1 by/
+| #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓕ[I]V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
]
+| #I #G #L #T1 #U1 #HTU1 #T2 #HT12 #H0
+ elim (cpx_lifts_sn … HT12 (Ⓣ) … (L.ⓘ[I]) … HTU1) -HT12
+ /4 width=6 by fqu_drop, drops_refl, drops_drop, teqx_inv_lifts_bi, ex3_intro/
]
qed-.
-lemma fquq_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
-[ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_ntdeq … H12 … HTU2 H) -T2
+lemma fquq_cpx_trans_tneqx (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂⸮[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
+[ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tneqx … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]
qed-.
-lemma fqup_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_ntdeq … H12 … HTU2 H) -T2
+lemma fqup_cpx_trans_tneqx (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂+[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tneqx … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
- #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_ntdeq … H1 … HTU1 H) -T1
+ #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_tneqx … H1 … HTU1 H) -T1
/3 width=8 by fqup_strap2, ex3_intro/
]
qed-.
-lemma fqus_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
-[ #H12 elim (fqup_cpx_trans_ntdeq … H12 … HTU2 H) -T2
+lemma fqus_cpx_trans_tneqx (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂*[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
+[ #H12 elim (fqup_cpx_trans_tneqx … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]
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