(* *)
(**************************************************************************)
+include "basic_2/rt_transition/fpbc_fqup.ma".
+include "basic_2/rt_transition/fpbc_lpx.ma".
include "basic_2/rt_computation/rsx_csx.ma".
include "basic_2/rt_computation/fpbs_cpx.ma".
include "basic_2/rt_computation/fpbs_csx.ma".
(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx (h):
- ∀G,L,T. ≥𝐒[h] ❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
-#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
+lemma fsb_inv_csx:
+ ∀G,L,T. ≥𝐒 ❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*𝐒 T.
+#G #L #T #H @(fsb_ind_alt … H) -G -L -T
+/5 width=1 by csx_intro, cpx_fpbc/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs (h):
- ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
- ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫.
-#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
+lemma csx_fsb_fpbs:
+ ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*𝐒 T1 →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → ≥𝐒 ❪G2,L2,T2❫.
+#G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
#G0 #L0 #T0 #IHu #H10
lapply (fpbs_csx_conf … H10) // -HT1 #HT0
generalize in match IHu; -IHu generalize in match H10; -H10
-@(rsx_ind … (csx_rsx … HT0)) -L0
-#L0 #_ #IHd #H10 #IHu @fsb_intro
-#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
-[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
-| #T2 #HT02 #HnT02
- elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0
- /3 width=4 by/
-| #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
+@(rsx_ind … (csx_rsx … HT0)) -L0 #L0 #_ #IHd #H10 #IHu
+@fsb_intro #G2 #L2 #T2 #H
+elim (fpbc_fwd_lpx … H) -H * [ -IHd -IHc | -IHu -IHd |]
+[ /5 width=5 by fsb_fpb_trans, fpbs_fqup_trans, fqu_fqup/
+| #T3 #HT03 #HnT03 #H32
+ elim (fpbs_cpx_tneqg_trans … H10 … HT03 HnT03) -T0
+ /4 width=5 by fsb_fpb_trans, sfull_dec/
+| #L3 #HL03 #HnL03 #HL32
+ @(fsb_fpb_trans … HL32) -L2
+ @(IHd … HL03 HnL03) -IHd -HnL03 [ -IHu -IHc |]
[ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
- | #G3 #L3 #T3 #H03 #_
- elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
- elim (teqx_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
- [ elim (teqx_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
- /4 width=7 by fsb_fpbs_trans, teqx_reqx_lpx_fpbs, fpbs_fqup_trans/
- | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
- elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
- /3 width=14 by fpbs_cpxs_teqx_fqup_lpx_trans/
+ | #G4 #L4 #T4 #H04 #_
+ elim (lpx_fqup_trans … H04 … HL03) -L3 #L3 #T3 #HT03 #H34 #HL34
+ elim (teqx_dec T0 T3) [ -IHc -HT03 #HT03 | -IHu #HnT03 ]
+ [ elim (teqg_fqup_trans … H34 … HT03) -T3 // #L2 #T3 #H03 #HT34 #HL23
+ /4 width=10 by fsb_fpbs_trans, teqg_reqg_lpx_fpbs, fpbs_fqup_trans/
+ | elim (cpxs_tneqg_fwd_step_sn … HT03 HnT03) -HT03 -HnT03 /2 width=1 by sfull_dec/ #T2 #HT02 #HnT02 #HT23
+ elim (fpbs_cpx_tneqg_trans … H10 … HT02 HnT02) -T0 /2 width=1 by sfull_dec/ #T0 #HT10 #HnT10 #H02
+ /3 width=17 by fpbs_cpxs_teqg_fqup_lpx_trans/
]
]
]
qed.
-lemma csx_fsb (h):
- â\88\80G,L,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] T â\86\92 â\89¥ð\9d\90\92[h] ❪G,L,T❫.
+lemma csx_fsb (G) (L) (T):
+ â\9dªG,Lâ\9d« â\8a¢ â¬\88*ð\9d\90\92 T â\86\92 â\89¥ð\9d\90\92 ❪G,L,T❫.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
-lemma csx_ind_fpb (h) (Q:relation3 …):
+lemma csx_ind_fpbc (Q:relation3 …):
(∀G1,L1,T1.
- ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
- (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ ❪G1,L1❫ ⊢ ⬈*𝐒 T1 →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒 T → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
-lemma csx_ind_fpbg (h) (Q:relation3 …):
+lemma csx_ind_fpbg (Q:relation3 …):
(∀G1,L1,T1.
- ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
- (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ ❪G1,L1❫ ⊢ ⬈*𝐒 T1 →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒 T → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.