(* *)
(**************************************************************************)
-include "basic_2/relocation/lifts_tdeq.ma".
-include "basic_2/static/lfxs_lfxs.ma".
include "basic_2/static/lfdeq_fqup.ma".
-include "basic_2/rt_transition/lfpx_frees.ma".
-include "basic_2/rt_transition/lfpx.ma". (**) (* should be in lfpx_frees.ma *)
+include "basic_2/static/lfdeq_lfdeq.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
(* Properties with degree-based equivalence for local environments **********)
+lemma lfpx_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.
+/3 width=5 by lfpx_fsge_comp, lfxs_pair_sn_split/ qed-.
+
+lemma lfpx_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.
+/3 width=5 by lfpx_fsge_comp, lfxs_flat_dx_split/ qed-.
+
+lemma lfpx_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.
+/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split/ qed-.
+
+lemma lfpx_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.
+/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split_void/ qed-.
+
+lemma lfpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (lfpx h G).
+/2 width=5 by tdeq_lfxs_conf/ qed-.
+
+lemma lfpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
+/2 width=5 by tdeq_lfxs_div/ qed-.
+
lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (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/
[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
>(tdeq_inv_lref1 … H0) -H0
- elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
- elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
+ elim (lfpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
+ elim (lfdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
+ elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
+ elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
+ lapply (lfdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
elim (IHT … HT02 … H1 … H2) -L0 -T0
/3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
+ lapply (lfdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- elim (tdeq_inv_lifts … HT1 … HUT1) -T1
+ 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
elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
+ lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
+ lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- elim (tdeq_lifts … HV1 … HVU1) -V1
+ elim (tdeq_lifts_sn … HV1 … HVU1) -V1
/4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
]
qed-.
-lemma cpx_tdeq_conf: ∀h,o,G,L,T0,T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- â\88\80T2. T0 â\89¡[h, o] T2 →
- â\88\83â\88\83T. T1 â\89¡[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
+lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+ â\88\80T2. T0 â\89\9b[h, o] T2 →
+ â\88\83â\88\83T. T1 â\89\9b[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
#h #o #G #L #T0 #T1 #HT01 #T2 #HT02
elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
/2 width=3 by lfxs_refl, ex2_intro/
qed-.
-lemma tdeq_cpx_trans: ∀h,o,G,L,T2,T0. T2 ≡[h, o] T0 →
+lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T2 â¬\88[h] T & T â\89¡[h, o] T1.
+ â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T2 â¬\88[h] T & T â\89\9b[h, o] T1.
#h #o #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: was just: cpx_lleq_conf *)
+(* Basic_2A1: uses: cpx_lleq_conf *)
lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- â\88\80L2. L0 â\89¡[h, o, T0] L2 →
- â\88\83â\88\83T. â¦\83G, L2â¦\84 â\8a¢ T0 â¬\88[h] T & T1 â\89¡[h, o] T.
+ â\88\80L2. L0 â\89\9b[h, o, T0] L2 →
+ â\88\83â\88\83T. â¦\83G, L2â¦\84 â\8a¢ T0 â¬\88[h] T & T1 â\89\9b[h, o] T.
#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
/2 width=3 by lfxs_refl, ex2_intro/
qed-.
-(* Basic_2A1: was just: lleq_cpx_trans *)
-lemma lfdeq_cpx_trans: â\88\80h,o,G,L2,L0,T0. L2 â\89¡[h, o, T0] L0 →
+(* Basic_2A1: uses: lleq_cpx_trans *)
+lemma lfdeq_cpx_trans: â\88\80h,o,G,L2,L0,T0. L2 â\89\9b[h, o, T0] L0 →
∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- â\88\83â\88\83T. â¦\83G, L2â¦\84 â\8a¢ T0 â¬\88[h] T & T â\89¡[h, o] T1.
+ â\88\83â\88\83T. â¦\83G, L2â¦\84 â\8a¢ T0 â¬\88[h] T & T â\89\9b[h, o] T1.
#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
/3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
qed-.
lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
-/3 width=6 by lfpx_frees_conf, cpx_tdeq_conf_lexs, frees_lfdeq_conf_lexs, lfxs_conf/ qed-.
+/3 width=6 by lfpx_fsge_comp, lfdeq_fsge_comp, cpx_tdeq_conf_lexs, lfxs_conf/ qed-.
-(* Basic_2A1: was just: lleq_lpx_trans *)
+(* Basic_2A1: uses: lleq_lpx_trans *)
lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- â\88\83â\88\83K1. â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, T] K1 & K1 â\89¡[h, o, T] K2.
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ â\88\83â\88\83K1. â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, T] K1 & K1 â\89\9b[h, o, T] K2.
#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
elim (lfpx_lfdeq_conf … o … HLK2 L1)
/3 width=3 by lfdeq_sym, ex2_intro/
qed-.
-(*
-(* Properties with supclosure ***********************************************)
-
-lemma lpx_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
- #K0 #V0 #H1KL1 #_ #H destruct
- elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
- #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
- /2 width=4 by fqu_lref_O, ex3_intro/
-| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
- [ elim (lleq_inv_bind … H)
- | elim (lleq_inv_flat … H)
- ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
-| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
- /3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
-| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
- /2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (drop_O1_le (Ⓕ) (k+1) K1)
- [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
- #H2KL elim (lpx_drop_trans_O1 … H1KL1 … HL1) -L1
- #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
- /3 width=4 by fqu_drop, ex3_intro/
- | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
- lapply (lleq_fwd_length … H2KL1) //
- ]
-]
-qed-.
-
-lemma lpx_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
-elim (fquq_inv_gen … H) -H
-[ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqu_fquq, ex3_intro/
-| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
-]
-qed-.
-
-lemma lpx_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqu_fqup, ex3_intro/
-| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
- #K #HT1 #H1KL #H2KL elim (lpx_lleq_fqu_trans … HT2 … H1KL H2KL) -L
- /3 width=5 by fqup_strap1, ex3_intro/
-]
-qed-.
-
-lemma lpx_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
-elim (fqus_inv_gen … H) -H
-[ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqup_fqus, ex3_intro/
-| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
-]
-qed-.
-*)
projects / helm.git / blobdiff