include "basic_2/s_computation/fqus_fqup.ma".
include "basic_2/static/lfdeq_drops.ma".
+include "basic_2/static/lfdeq_fqup.ma".
include "basic_2/static/lfdeq_lfdeq.ma".
(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-(* Properties with supclosure ***********************************************)
+(* Properties with extended structural successor for closures ***************)
-lemma fqu_tdeq_conf: ∀h,o,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, T1⦄ →
- â\88\80U2. U1 â\89¡[h, o] U2 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐ ⦃G2, L, T2⦄ & L2 ≡[h, o, T1] L & T1 ≡[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
-[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -H
+lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+ â\88\80U2. U1 â\89\9b[h, o] U2 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[h, o, T1] L & T1 ≛[h, o] T2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
+[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
/2 width=5 by fqu_lref_O, ex3_2_intro/
| #I #G #L #W1 #U1 #X #H
elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #_ #H destruct
/2 width=5 by fqu_pair_sn, ex3_2_intro/
| #p #I #G #L #W1 #U1 #X #H
elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
- /3 width=5 by lfdeq_pair, fqu_bind_dx, ex3_2_intro/
+ /3 width=5 by lfdeq_pair_refl, fqu_bind_dx, ex3_2_intro/
+| #p #I #G #L #W1 #U1 #Hb #X #H
+ elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
+ /3 width=5 by fqu_clear, ex3_2_intro/
| #I #G #L #W1 #U1 #X #H
elim (tdeq_inv_pair1 … H) -H #W2 #U2 #_ #HU12 #H destruct
/2 width=5 by fqu_flat_dx, ex3_2_intro/
-| #I #G #L #W #T1 #U1 #HTU1 #U2 #HU12
- elim (tdeq_inv_lifts_sn … HU12 … HTU1) -U1 #T2 #HTU2 #HT12
+| #I #G #L #T1 #U1 #HTU1 #U2 #HU12
+ elim (tdeq_inv_lifts_sn … HU12 … HTU1) -U1
/3 width=5 by fqu_drop, ex3_2_intro/
]
qed-.
-lemma tdeq_fqu_trans: ∀h,o,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, T1⦄ →
- â\88\80U2. U2 â\89¡[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐ ⦃G2, L, T2⦄ & T2 ≡[h, o] T1 & L ≡[h, o, T1] L2.
-#h #o #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
+lemma tdeq_fqu_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+ â\88\80U2. U2 â\89\9b[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
elim (fqu_tdeq_conf … o … H12 U2) -H12
/3 width=5 by lfdeq_sym, tdeq_sym, ex3_2_intro/
qed-.
-(* Basic_2A1: was just: lleq_fqu_trans *)
-lemma lfdeq_fqu_trans: ∀h,o,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U0⦄ & U0 ≡[h, o] U & K1 ≡[h, o, U] K2.
-#h #o #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+(* Basic_2A1: uses: lleq_fqu_trans *)
+lemma lfdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
[ #I #G #L2 #V2 #L1 #H elim (lfdeq_inv_zero_pair_dx … H) -H
#K1 #V1 #HV1 #HV12 #H destruct
- /3 width=7 by tdeq_lfdeq_conf_sn, fqu_lref_O, ex3_2_intro/
+ /3 width=7 by tdeq_lfdeq_conf, fqu_lref_O, ex3_2_intro/
| * [ #p ] #I #G #L2 #V #T #L1 #H
[ elim (lfdeq_inv_bind … H)
| elim (lfdeq_inv_flat … H)
] -H
/2 width=5 by fqu_pair_sn, ex3_2_intro/
-| #a #I #G #L2 #V #T #L1 #H elim (lfdeq_inv_bind … H) -H
+| #p #I #G #L2 #V #T #L1 #H elim (lfdeq_inv_bind … H) -H
/2 width=5 by fqu_bind_dx, ex3_2_intro/
+| #p #I #G #L2 #V #T #Hb #L1 #H elim (lfdeq_inv_bind_void … H) -H
+ /3 width=5 by fqu_clear, ex3_2_intro/
| #I #G #L2 #V #T #L1 #H elim (lfdeq_inv_flat … H) -H
/2 width=5 by fqu_flat_dx, ex3_2_intro/
-| #I #G #L2 #V2 #T #U #HTU #Y #HU
+| #I #G #L2 #T #U #HTU #Y #HU
elim (lfdeq_fwd_dx … HU) #L1 #V1 #H destruct
- /5 width=12 by lfdeq_inv_lifts_bi, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+ /5 width=14 by lfdeq_inv_lifts_bi, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with optional structural successor for closures ***************)
+
+lemma tdeq_fquq_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
+[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_2A1: was just: lleq_fquq_trans *)
-lemma lfdeq_fquq_trans: ∀h,o,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U0⦄ & U0 ≡[h, o] U & K1 ≡[h, o, U] K2.
-#h #o #G1 #G2 #L2 #K2 #T #U #H elim H -H
+lemma lfdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -H
[ #H #L1 #HL12 elim (lfdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
qed-.
+(* Properties with plus-iterated structural successor for closures **********)
+
(* Basic_2A1: was just: lleq_fqup_trans *)
-lemma lfdeq_fqup_trans: ∀h,o,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U0⦄ & U0 ≡[h, o] U & K1 ≡[h, o, U] K2.
-#h #o #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+lemma lfdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
[ #G2 #K2 #U #HTU #L1 #HL12 elim (lfdeq_fqu_trans … HTU … HL12) -L2
/3 width=5 by fqu_fqup, ex3_2_intro/
| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12
elim (lfdeq_fqu_trans … HU2 … HK0) -K #K1 #U1 #HU1 #HU12 #HK12
elim (tdeq_fqu_trans … HU1 … HU0) -U #K3 #U3 #HU03 #HU31 #HK31
@(ex3_2_intro … K3 U3) (**) (* full auto too slow *)
- /3 width=5 by lfdeq_trans, tdeq_lfdeq_conf_sn, fqup_strap1, tdeq_trans/
+ /3 width=5 by lfdeq_trans, tdeq_lfdeq_conf, fqup_strap1, tdeq_trans/
+]
+qed-.
+
+lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
+[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fqup, ex3_2_intro/
+| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
+ elim (tdeq_fqu_trans … H … HU21) -U1 #L0 #T #H1 #HTU #HL0
+ lapply (tdeq_lfdeq_div … HTU … HL0) -HL0 #HL0
+ elim (IH … HTU) -U #K2 #U1 #H2 #HUT1 #HKL2
+ elim (lfdeq_fqup_trans … H2 … HL0) -L #K #U #H2 #HU1 #HK2
+ lapply (tdeq_lfdeq_conf … HUT1 … HK2) -HK2 #HK2
+ /3 width=7 by lfdeq_trans, fqup_strap2, tdeq_trans, ex3_2_intro/
+]
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
+[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_2A1: was just: lleq_fqus_trans *)
-lemma lfdeq_fqus_trans: ∀h,o,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U0⦄ & U0 ≡[h, o] U & K1 ≡[h, o, U] K2.
-#h #o #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
+lemma lfdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
[ #H elim (lfdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
projects / helm.git / blobdiff