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15 include "basic_2/rt_computation/rsx_csx.ma".
16 include "basic_2/rt_computation/fpbs_cpx.ma".
17 include "basic_2/rt_computation/fpbs_csx.ma".
18 include "basic_2/rt_computation/fsb_fpbg.ma".
20 (* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
22 (* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
24 lemma fsb_inv_csx (h):
25 ∀G,L,T. ≥𝐒[h] ❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
26 #h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
29 (* Propreties with context-sensitive stringly rt-normalizing terms **********)
31 lemma csx_fsb_fpbs (h):
32 ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
33 ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫.
34 #h #G1 #L1 #T1 #H @(csx_ind … H) -T1
35 #T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
37 lapply (fpbs_csx_conf … H10) // -HT1 #HT0
38 generalize in match IHu; -IHu generalize in match H10; -H10
39 @(rsx_ind … (csx_rsx … HT0)) -L0
40 #L0 #_ #IHd #H10 #IHu @fsb_intro
41 #G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
42 [ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
44 elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0
46 | #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
47 [ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
49 elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
50 elim (teqx_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
51 [ elim (teqx_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
52 /4 width=7 by fsb_fpbs_trans, teqx_reqx_lpx_fpbs, fpbs_fqup_trans/
53 | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
54 elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
55 /3 width=14 by fpbs_cpxs_teqx_fqup_lpx_trans/
62 ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ≥𝐒[h] ❪G,L,T❫.
63 /2 width=5 by csx_fsb_fpbs/ qed.
65 (* Advanced eliminators *****************************************************)
67 lemma csx_ind_fpb (h) (Q:relation3 …):
70 (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
73 ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
74 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
76 lemma csx_ind_fpbg (h) (Q:relation3 …):
79 (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
82 ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
83 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.