/3 width=1 by lfxs_refl, inj/ qed.
(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
-lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
+lemma tc_lfxs_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
-/3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair_refl, inj/
qed.
(* Advanced eliminators *****************************************************)
lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
-/4 width=5 by lfpx_cpx_trans, cpxs_bind_dx, lfpx_pair/ qed.
+/4 width=5 by lfpx_cpx_trans, cpxs_bind_dx, lfpx_pair_refl/ qed.
lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
-/4 width=5 by lfpx_cpxs_trans, cpxs_bind_dx, lfpx_pair/ qed.
+/4 width=5 by lfpx_cpxs_trans, cpxs_bind_dx, lfpx_pair_refl/ qed.
elim (tdneq_inv_pair … H2) -H2
[ #H elim H -H //
| -IHT #H lapply (csx_cpx_trans … o … HLT0) // -HT
- #HT0 lapply (csx_lfpx_conf … HT0 … (L.ⓛW0)) -HT0 /4 width=1 by lfpx_pair/
+ #HT0 lapply (csx_lfpx_conf … HT0 … (L.ⓛW0)) -HT0 /4 width=1 by lfpx_pair_refl/
| -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
]
qed.
[ #V1 #T1 #HLV1 #HLT1 #H destruct
elim (tdneq_inv_pair … H2) -H2
[ #H elim H -H //
- | /4 width=3 by csx_cpx_trans, csx_lfpx_conf, lfpx_pair/
+ | /4 width=3 by csx_cpx_trans, csx_lfpx_conf, lfpx_pair_refl/
| -IHV /4 width=3 by csx_cpx_trans, cpx_cpxs, cpx_pair_sn/
]
| -IHV -IHT -H2
| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
lapply (cpx_lifts_bi … HLV10 (Ⓣ) … (L.ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /3 width=1 by drops_refl, drops_drop/ #HLV23
@csx_abbr /2 width=3 by csx_cpx_trans/ -HV
- @(csx_lfpx_conf … (L.ⓓW0)) /2 width=1 by lfpx_pair/ -W1
+ @(csx_lfpx_conf … (L.ⓓW0)) /2 width=1 by lfpx_pair_refl/ -W1
/4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
]
qed-.
(* Properties with uncounted context-sensitive rt-computation for terms *****)
(* Basic_2A1: uses: lpxs_pair lpxs_pair_refl *)
-lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
-/2 width=1 by tc_lfxs_pair/ qed.
+lemma lfpxs_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
+/2 width=1 by tc_lfxs_pair_refl/ qed.
(* Basic_2A1: uses: lpxs_cpx_trans *)
lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
lemma cpxs_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
-/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair, cpxs_bind/ qed.
+/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair_refl, cpxs_bind/ qed.
(* Advanced inversion lemmas on uncounted rt-computation for terms **********)
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
+/3 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro/
qed-.
lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
elim (cpx_inv_abbr1 … HU02) -HU02 *
[ #V2 #T2 #HV02 #HT02 #H destruct
lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+ /4 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
| #T2 #HT02 #HUT2
lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by lfpxs_pair, cpxs_trans, ex3_intro, or_intror/
+ /4 width=3 by lfpxs_pair_refl, cpxs_trans, ex3_intro, or_intror/
]
| #U1 #HTU1 #HU01
elim (cpx_lifts_sn … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
(**************************************************************************)
include "basic_2/rt_computation/lfpxs.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
include "basic_2/rt_computation/csx_lsubr.ma".
include "basic_2/rt_computation/lfsx_drops.ma".
include "basic_2/rt_computation/lfsx_lfpxs.ma".
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lsx_lref_be_lpxs *)
-axiom lfsx_pair_lpxs: ∀h,o,G,K1,V. ⦃G, K1⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
- ∀K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h, V] K2 →
- ∀I. G ⊢ ⬈*[h, o, #0] 𝐒⦃K2.ⓑ{I}V⦄.
-(*
-#h #o #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
-#V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
-#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
-#L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
-#Y #H #HLK2 elim (lpx_inv_pair1 … H) -H
-#K2 #V2 #HK02 #HV02 #H destruct
-elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
-[ /4 width=8 by lpxs_strap1, lleq_lref/
-| @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
+lemma lfsx_pair_lfpxs: ∀h,o,G,K1,V. ⦃G, K1⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
+ ∀K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h, V] K2 →
+ ∀I. G ⊢ ⬈*[h, o, #0] 𝐒⦃K2.ⓑ{I}V⦄.
+#h #o #G #K1 #V #H
+@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
+@(lfsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
+@lfsx_intro #Y #HY #HnY
+elim (lfpx_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
+elim (tdeq_dec h o V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
+[ /5 width=5 by lfsx_lfdeq_trans, lfpxs_step_dx, lfdeq_pair/
+| @lfsx_lfpx_trans
+ [2: @(IHV0 … HnV02 K0 … I) -IHV0 -HnV02
+ [ /2 width=3 by lfpxs_cpx_trans/
+ |
+ |
+ ]
+ |1: skip
+ |3: @lfpx_pair /2 width=3 by lfpx_cpx_conf/
+ ]
/3 width=4 by lsx_cpx_trans_O, lsx_lpx_trans, lpxs_cpx_trans, lpxs_strap1/ (**) (* full auto too slow *)
]
qed.
-*)
+
(* Basic_2A1: uses: lsx_lref_be *)
lemma lfsx_lref_pair: ∀h,o,G,K,V. ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄ →
∀I,L,i. ⬇*[i] L ≡ K.ⓑ{I}V → G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄.
/2 width=1 by lfxs_refl/ qed.
(* Basic_2A1: uses: lpr_pair *)
-lemma lfpr_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡[h, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair/ qed.
+lemma lfpr_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡[h, T] L.ⓑ{I}V2.
+/2 width=1 by lfxs_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
(* Advanced properties ******************************************************)
-(* Basic_2A1: uses: lpx_refl lpx_pair *)
+(* Basic_2A1: uses: lpx_refl *)
lemma lfpx_refl: ∀h,G,T. reflexive … (lfpx h G T).
/2 width=1 by lfxs_refl/ qed.
-(* Basic_2A1: uses: lpx_refl lpx_pair *)
-lemma lfpx_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair/ qed.
+(* Basic_2A1: uses: lpx_pair *)
+lemma lfpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
+/2 width=1 by lfxs_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
lemma lfdeq_refl: ∀h,o,T. reflexive … (lfdeq h o T).
/2 width=1 by lfxs_refl/ qed.
-lemma lfdeq_pair: ∀h,o,V1,V2. V1 ≡[h, o] V2 →
- ∀I,L. ∀T:term. L.ⓑ{I}V1 ≡[h, o, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair/ qed.
+lemma lfdeq_pair_refl: ∀h,o,V1,V2. V1 ≡[h, o] V2 →
+ ∀I,L. ∀T:term. L.ⓑ{I}V1 ≡[h, o, T] L.ⓑ{I}V2.
+/2 width=1 by lfxs_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
/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/
/4 width=3 by lexs_refl, ext2_refl, ex2_intro/
qed.
-lemma lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
+lemma lfxs_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #HV12 #I #T
elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
elim (pn_split f) * #g #H destruct