(* *)
(**************************************************************************)
-include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/lpxs_lpxs.ma".
-include "basic_2/computation/lsx.ma".
+include "basic_2/computation/lsx_alt.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
(* Advanced properties ******************************************************)
-lemma lsx_leqy_conf: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → G ⊢ ⋕⬊*[h, g, T, d] L2.
-#h #g #G #L1 #T #d #H @(lsx_ind … H) -L1
-#L1 #_ #IHL1 #L2 #H1L12 #H2L12 @lsx_intro
-#L3 #H1L23 #HnL23 lapply (lpxs_fwd_length … H1L23)
-#H2L23 elim (lsuby_lpxs_trans_lleq … H1L12 … H1L23) -H1L12 -H1L23
-#L0 #H1L03 #H1L10 #H lapply (lpxs_fwd_length … H1L10)
-#H2L10 elim (H T) -H //
-#_ #H @(IHL1 … H1L10) -IHL1 -H1L10 /3 width=1 by/
-qed-.
-
-lemma lsx_ge: ∀h,g,G,L,T,d1,d2. d1 ≤ d2 →
- G ⊢ ⋕⬊*[h, g, T, d1] L → G ⊢ ⋕⬊*[h, g, T, d2] L.
-#h #g #G #L #T #d1 #d2 #Hd12 #H @(lsx_ind … H) -L
-/5 width=7 by lsx_intro, lleq_ge/
-qed-.
-
-lemma lsx_lleq_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. L1 ⋕[T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsx_ind … H) -L1
-#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
-#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
-/5 width=4 by lleq_canc_sn, lleq_trans/
-qed-.
-
-lemma lsx_lpxs_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 d) /3 width=4 by lsx_lleq_trans/
-qed-.
-
-fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
- ∀Y,T. G ⊢ ⋕⬊*[h, g, T, ⫯d] Y →
+fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
+ ∀Y,T. G ⊢ ⬊*[h, g, T, ⫯l] Y →
∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- G â\8a¢ â\8b\95â¬\8a*[h, g, â\93\91{a,I}V.T, d] L2.
-#h #g #a #I #G #L1 #V #d #H @(lsx_ind … H) -L1
-#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind … H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro
+ G â\8a¢ â¬\8a*[h, g, â\93\91{a,I}V.T, l] L2.
+#h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind_alt … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro_alt
#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
#HL10 #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
+[ #HnV elim (lleq_dec V L1 L2 l)
[ #HV @(IHL1 … L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
| -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
]
]
qed-.
-lemma lsx_bind: ∀h,g,a,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
- â\88\80T. G â\8a¢ â\8b\95â¬\8a*[h, g, T, ⫯d] L.ⓑ{I}V →
- G â\8a¢ â\8b\95â¬\8a*[h, g, â\93\91{a,I}V.T, d] L.
+lemma lsx_bind: ∀h,g,a,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
+ â\88\80T. G â\8a¢ â¬\8a*[h, g, T, ⫯l] L.ⓑ{I}V →
+ G â\8a¢ â¬\8a*[h, g, â\93\91{a,I}V.T, l] L.
/2 width=3 by lsx_bind_lpxs_aux/ qed.
-lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
- â\88\80L2,T. G â\8a¢ â\8b\95â¬\8a*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- G â\8a¢ â\8b\95â¬\8a*[h, g, â\93\95{I}V.T, d] L2.
-#h #g #I #G #L1 #V #d #H @(lsx_ind … H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind … H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro
+lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
+ â\88\80L2,T. G â\8a¢ â¬\8a*[h, g, T, l] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ G â\8a¢ â¬\8a*[h, g, â\93\95{I}V.T, l] L2.
+#h #g #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_alt … H) -L2
+#L2 #HL2 #IHL2 #HL12 @lsx_intro_alt
#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
+[ #HnV elim (lleq_dec V L1 L2 l)
[ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
| -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
]
]
qed-.
-lemma lsx_flat: ∀h,g,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
- â\88\80T. G â\8a¢ â\8b\95â¬\8a*[h, g, T, d] L â\86\92 G â\8a¢ â\8b\95â¬\8a*[h, g, â\93\95{I}V.T, d] L.
+lemma lsx_flat: ∀h,g,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
+ â\88\80T. G â\8a¢ â¬\8a*[h, g, T, l] L â\86\92 G â\8a¢ â¬\8a*[h, g, â\93\95{I}V.T, l] L.
/2 width=3 by lsx_flat_lpxs/ qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lsx_fwd_lref_be: ∀h,g,I,G,L,d,i. d ≤ yinj i → G ⊢ ⋕⬊*[h, g, #i, d] L →
- ∀K,V. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, V, 0] K.
-#h #g #I #G #L #d #i #Hdi #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
-#K2 #HK12 #HnK12 lapply (ldrop_fwd_drop2 … HLK1)
-#H2LK1 elim (ldrop_lpxs_trans … H2LK1 … HK12) -H2LK1 -HK12
-#L2 #HL12 #H2LK2 #H elim (leq_ldrop_conf_be … H … HLK1) -H /2 width=1 by ylt_inj/
-#Y #_ #HLK2 lapply (ldrop_fwd_drop2 … HLK2)
-#HY lapply (ldrop_mono … HY … H2LK2) -HY -H2LK2 #H destruct
-/4 width=10 by lleq_inv_lref_ge/
-qed-.
-
-lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L →
- G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
-#h #g #a #I #G #L #V1 #T #d #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 @lsx_intro
-#Y #H #HT elim (lpxs_inv_pair1 … H) -H
-#L2 #V2 #HL12 #_ #H destruct
-@(lsx_leqy_conf … (L2.ⓑ{I}V1)) /2 width=1 by lsuby_succ/
-@IHL1 // #H @HT -IHL1 -HL12 -HT
-@(lleq_lsuby_trans … (L2.ⓑ{I}V1))
-/2 width=2 by lleq_fwd_bind_dx, lsuby_succ/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a, I}V.T, d] L →
- G ⊢ ⋕⬊*[h, g, V, d] L ∧ G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
-/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.
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