(* *)
(**************************************************************************)
-include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/csx_alt.ma".
-include "basic_2/computation/lsx_lpxs.ma".
+include "basic_2/computation/csx_lift.ma".
+include "basic_2/computation/lcosx_cpxs.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-(* Advanced forward lemmas **************************************************)
-
-include "basic_2/computation/cpxs_lleq.ma".
-
-(*
-lemma lsx_cpxs_conf: ∀h,g,G,L1,T1,T2. G ⊢ ⋕⬊*[h, g, T1] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
- G ⊢ ⋕⬊*[h, g, T2] L2.
-#h #g #G #L1 #T1 #T2 #H @(lsx_ind … H) -L1
-#L1 #HL1 #IHL1 #L2 #HL12 #HT12 @lsx_intro
-#L3 #HL23 #HnL23 elim (lleq_dec T2 L1 L2 0) [| -HnL23 ]
-[ #HT2 @(IHL1 L3) /2 width=3 by lpxs_trans/
-
- @(lsx_lpxs_trans … HL23)
-| #HnT2 @(lsx_lpxs_trans … HL23) @(IHL1 … L2) //
- #HT1 @HnT2 @(lleq_cpxs_conf_dx … HT12) //
+(* Advanced properties ******************************************************)
+
+lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
+ ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
+ ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
+#h #g #I #G #K1 #V #i #d #Hdi #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 (lpxs_ldrop_conf … HLK0 … HL02) -HL02
+#Y #H #HLK2 elim (lpxs_inv_pair1 … H) -H
+#K2 #V2 #HK02 #HV02 #H destruct
+lapply (lpxs_trans … HK10 … HK02) #HK12
+elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 -HK10 | -IHK0 -HnL02 -HLK0 ]
+[ /4 width=8 by lleq_lref/
+| @(IHV0 … HnV02 … HK12 … HLK2) -IHV0 -HnV02 -HK12 -HLK2
+ /3 width=4 by lsx_cpxs_trans_O, lsx_lpxs_trans, lpxs_cpxs_trans/ (**) (* full auto too slow *)
]
-*)
-
-lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V1. ⦃G, L1⦄ ⊢ ⬊*[h, g] V1 →
- ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V1.T] L2 →
- ∀V2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 →
- G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V2.
-#h #g #a #I #G #L1 #V1 #H @(csx_ind_alt … H) -V1
-#V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
-#L2 #HL2 #IHL2 #V2 #HL12 #HV12 @lsx_intro
-#Y #H #HnT elim (lpxs_inv_pair1 … H) -H
-#L3 #V3 #HL23 #HV23 #H destruct
-lapply (lpxs_trans … HL12 … HL23) #HL13
-elim (eq_term_dec V2 V3)
-[ (* -HV13 -HL2 -IHV1 -HL12 *) #H destruct
- @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
-| -IHL2 -HnT #HnV23 elim (eq_term_dec V1 V2)
- [ #H -HV12 destruct
- (* @(lsx_lpxs_trans … (L2.ⓑ{I}V2)) /2 width=1 by lpxs_pair/ *)
- @(IHV1 … HnV23) -IHV1 -HnV23 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
- @(lsx_lpxs_trans … HL23)
-
-
-lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V. ⦃G, L1⦄ ⊢ ⬊*[h, g] V →
- ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V.
-#h #g #a #I #G #L1 #V #H @(csx_ind_alt … H) -V
-#V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro
-#Y #H #HnT elim (lpxs_inv_pair1 … H) -H
-#L3 #V3 #HL23 #HV13 #H destruct
-lapply (lpxs_trans … HL12 … HL23) #HL13
-elim (eq_term_dec V1 V3)
-[ -HV13 -HL2 -IHV1 -HL12 #H destruct
- @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
-| -IHL2 -HnT #HnV13
- @(IHV1 … HnV13) -IHV1 -HnV13 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
- @(lsx_lpxs_trans … HL23)
-
-
-(* Advanced inversion lemmas ************************************************)
-
+qed.
+lemma lsx_lref_be: ∀h,g,I,G,K,V,i,d. d ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
+ G ⊢ ⋕⬊*[h, g, V, 0] K →
+ ∀L. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L.
+/2 width=8 by lsx_lref_be_lpxs/ qed.
(* Main properties **********************************************************)
-axiom csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
-(*
+theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕⬊*[h, g, T, d] L.
#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
-#G1 #L1 #T1 #IH #G2 #L2 * *
-[ #k #HG #HL #HT destruct //
-| #i #HG #HL #HT destruct
- #H
-| #p #HG #HL #HT destruct //
-| #a #I #V2 #T2 #HG #HL #HT #H destruct
- elim (csx_fwd_bind … H) -H
- #HV2 #HT2
-| #I #V2 #T2 #HG #HL #HT #H destruct
+#Z #Y #X #IH #G #L * * //
+[ #i #HG #HL #HT #H #d destruct
+ elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
+ elim (ylt_split i d) /2 width=1 by lsx_lref_skip/
+ #Hdi #Hi elim (ldrop_O1_lt … Hi) -Hi
+ #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
+ /4 width=6 by lsx_lref_be, fqup_lref/
+| #a #I #V #T #HG #HL #HT #H #d destruct
+ elim (csx_fwd_bind … H) -H /3 width=1 by lsx_bind/
+| #I #V #T #HG #HL #HT #H #d destruct
elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
-*)
+]
+qed.
projects / helm.git / blobdiff