include "basic_2/static/lsubf_frees.ma".
include "basic_2/static/lfxs.ma".
include "basic_2/rt_transition/cpx_drops.ma".
+include "basic_2/rt_transition/cpx_ext.ma".
(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
(* Properties with context-sensitive free variables *************************)
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
+(* check sle_refl *)
lemma cpx_frees_conf_lfpx: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
- ∀L2. L1 ⪤*[cpx h G, cfull, f1] L2 →
+ ∀L2. L1 ⪤*[cpx_ext h G, cfull, f1] L2 →
∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
-#h #G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1
+#h #G #L1 #T1 @(fqup_wf_ind_eq (Ⓣ) … G L1 T1) -G -L1 -T1
#G0 #L0 #U0 #IH #G #L1 * *
[ -IH #s #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
lapply (frees_inv_sort … H1) -H1 #Hg1
elim (cpx_inv_sort1 … H0) -H0 #H destruct
- /3 width=3 by frees_sort_gen, sle_refl, ex2_intro/
+ /3 width=3 by frees_sort, sle_refl, ex2_intro/
| #i #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
elim (frees_inv_lref_drops … H1) -H1 *
- [ -IH #HL1 #Hg1
+ [ -IH #f1 #HL1 #Hf1 #H destruct
elim (cpx_inv_lref1_drops … H0) -H0
[ #H destruct
- /5 width=9 by frees_lref_atom, drops_atom2_lexs_conf, coafter_isuni_isid, sle_refl, ex2_intro/
- | * -H2 -Hg1 #I #K1 #V1 #V2 #HLK1
+ /4 width=9 by frees_atom_drops, drops_atom2_lexs_conf, sle_refl, ex2_intro/
+ | * -H2 -Hf1 #I #K1 #V1 #V2 #HLK1
lapply (drops_TF … HLK1) -HLK1 #HLK1
lapply (drops_mono … HLK1 … HL1) -L1 #H destruct
]
| #f1 #I #K1 #V1 #Hf1 #HLK1 #H destruct
elim (cpx_inv_lref1_drops … H0) -H0
[ #H destruct
- elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #K2 #V2 #HLK2 #HK12 #HV12
+ elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #Z #K2 #HLK2 #HK12 #H
+ elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -L1 -K1 -V1 #f2 #Hf2 #Hf21
- /4 width=7 by frees_lref_pushs, frees_lref_pair, drops_refl, sle_next, ex2_intro, sle_pushs/
- | * #J #Y #X #V2 #H #HV12 #HVU2
+ /4 width=7 by frees_lref_pushs, frees_pair_drops, drops_refl, sle_pushs, sle_next, ex2_intro/
+ | * #Z #Y #X #V2 #H #HV12 #HVU2
lapply (drops_mono … H … HLK1) -H #H destruct
- elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #K2 #V0 #HLK2 #HK12 #_
+ elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #I2 #K2 #HLK2 #HK12 #H
+ elim (ext2_inv_pair_sn … H) -H #V0 #_ #H destruct
lapply (drops_isuni_fwd_drop2 … HLK2) // -V0 #HLK2
elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -I -L1 -K1 -V1 #f2 #Hf2 #Hf21
lapply (frees_lifts … Hf2 … HLK2 … HVU2 ??) /4 width=7 by sle_weak, ex2_intro, sle_pushs/
]
+ | #f1 #I #K1 #HLK1 #Hf1 #H destruct
+ elim (cpx_inv_lref1_drops … H0) -H0
+ [ -IH #H destruct
+ elim (lexs_drops_conf_next … H2 … HLK1) -H2 -HLK1 [ |*: // ] #Z #K2 #HLK2 #_ #H
+ lapply (ext2_inv_unit_sn … H) -H #H destruct
+ /3 width=3 by frees_unit_drops, sle_refl, ex2_intro/
+ | * -H2 -Hf1 #Z #Y1 #X1 #X2 #HLY1
+ lapply (drops_mono … HLK1 … HLY1) -L1 #H destruct
+ ]
]
| -IH #l #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
lapply (frees_inv_gref … H1) -H1 #Hg1
lapply (cpx_inv_gref1 … H0) -H0 #H destruct
- /3 width=3 by frees_gref_gen, sle_refl, ex2_intro/
+ /3 width=3 by frees_gref, sle_refl, ex2_intro/
| #p #I #V1 #T1 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
elim (frees_inv_bind … H1) -H1 #gV1 #gT1 #HgV1 #HgT1 #Hg1
elim (cpx_inv_bind1 … H0) -H0 *
[ #V2 #T2 #HV12 #HT12 #H destruct
lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
- lapply (lexs_inv_tl … I … HL12T … HV12 ?) // -HL12T #HL12T
+ lapply (lexs_inv_tl … (BPair I V1) (BPair I V2) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
elim (sor_isfin_ex gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/
/4 width=10 by frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
| #T2 #HT12 #HUT2 #H0 #H1 destruct -HgV1
lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
- lapply (lexs_inv_tl … Abbr … V1 V1 HL12T ??) // -HL12T #HL12T
+ lapply (lexs_inv_tl … (BPair Abbr V1) (BPair Abbr V1) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
elim (IH … HgT1 … HL12T … HT12) // -L1 -T1 #gT2 #HgT2 #HgT21
lapply (frees_inv_lifts_SO (Ⓣ) … HgT2 … L2 … HUT2) [ /3 width=1 by drops_refl, drops_drop/ ] -V1 -T2
/5 width=6 by sor_inv_sle_dx, sle_trans, sle_tl, ex2_intro/
lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
- lapply (lexs_inv_tl … Abst … HL12T … HW12 ?) // -HL12T #HL12T
+ lapply (lexs_inv_tl … (BPair Abst W1) (BPair Abst W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
- lapply (sor_trans2 … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
- lapply (sor_sym … Hg0) -Hg0 #Hg0
+ lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+ lapply (sor_comm … Hg0) -Hg0 #Hg0
elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
- elim (sor_isfin_ex gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
- elim (lsubf_frees_trans … HgT2 (⫯gVT2) … (L2.ⓓⓝW2.V2))
- [2: /4 width=4 by lsubf_refl, lsubf_beta, sor_inv_sle_dx, sor_inv_sle_sn, sle_inv_tl_sn/ ]
- -HgT2 #gT0 #HgT0 #HgT20
+ elim (sor_isfin_ex (⫱gT2) gV2) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
+ elim (lsubf_beta_tl_dx … HgV2 … HgVT2 … W2) [ |*: /1 width=3 by lsubf_refl/ ] #gT0 #HL2 #HgT02
+ lapply (lsubf_frees_trans … HgT2 … HL2) -HgT2 -HL2 #HgT0
+ lapply (sor_comm … HgVT2) -HgVT2 #HgVT2 (**) (* this should be removed *)
elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #gV0 #HgV0 #H
elim (sor_isfin_ex gV0 (⫱gT0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
- @(ex2_intro … g2)
- [ @(frees_bind … Hg2) /2 width=5 by frees_flat/
- | -L2 @(sor_inv_sle … Hg2) -Hg2
- [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
- | @sle_xn_tl [2:|*: // ] @(sle_trans … HgT20) -HgT20
- /4 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl, sle_next/
- ] (**) (* full auto too slow *)
- ]
+ @(ex2_intro … g2) /3 width=5 by frees_flat, frees_bind/ -h -p -G -L1 -L2 -V1 -V2 -W1 -W2 -T1 -T2
+ @(sor_inv_sle … Hg2) -Hg2
+ [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
+ | @(sle_trans … HgT02) -HgT02
+ /3 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl/
+ ] (**) (* full auto too slow *)
| #p #V2 #V #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #H0 #H1 #H destruct
elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
- lapply (lexs_inv_tl … Abbr … HL12T … HW12 ?) // -HL12T #HL12T
+ lapply (lexs_inv_tl … (BPair Abbr W1) (BPair Abbr W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
- lapply (sor_trans2 … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+ lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
elim (sor_isfin_ex (↑gV2) gT2) /3 width=3 by frees_fwd_isfin, isfin_push/ #gV0 #HgV0 #H
elim (sor_isfin_ex gW2 (⫱gV0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
- lapply (sor_trans2 … Hg2 … (⫱gT2) … Hg) /2 width=1 by sor_tl/ #Hg2
+ lapply (sor_assoc_sn … Hg2 … (⫱gT2) … Hg) /2 width=1 by sor_tl/ #Hg2
lapply (frees_lifts (Ⓣ) … HgV2 … (L2.ⓓW2) … HV2 ??)
- [4: lapply (sor_sym … Hg) |*: /3 width=3 by drops_refl, drops_drop/ ] -V2 (**) (* full auto too slow *)
+ [4: lapply (sor_comm … Hg) |*: /3 width=3 by drops_refl, drops_drop/ ] -V2 (**) (* full auto too slow *)
/4 width=10 by frees_flat, frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
]
]
(* Basic_2A1: uses: cpx_frees_trans *)
lemma cpx_frees_conf: ∀h,G. R_frees_confluent (cpx h G).
-/3 width=7 by cpx_frees_conf_lfpx, lexs_refl/ qed-.
+/4 width=7 by cpx_frees_conf_lfpx, lexs_refl, ext2_refl/ qed-.
(* Basic_2A1: uses: lpx_frees_trans *)
-lemma lfpx_frees_conf: ∀h,G. lexs_frees_confluent (cpx h G) cfull.
+lemma lfpx_frees_conf: ∀h,G. lexs_frees_confluent (cpx_ext h G) cfull.
/2 width=7 by cpx_frees_conf_lfpx/ qed-.
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