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15 include "basic_2/rt_computation/rdsx_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,o,G,L,T. ≥[h, o] 𝐒⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
25 #h #o #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
28 (* Propreties with context-sensitive stringly rt-normalizing terms **********)
30 lemma csx_fsb_fpbs: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
31 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
32 #h #o #G1 #L1 #T1 #H @(csx_ind … H) -T1
33 #T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
35 lapply (fpbs_csx_conf … H10) // -HT1 #HT0
36 generalize in match IHu; -IHu generalize in match H10; -H10
37 @(rdsx_ind … (csx_rdsx … HT0)) -L0
38 #L0 #_ #IHd #H10 #IHu @fsb_intro
39 #G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
40 [ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
42 elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0
44 | #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
45 [ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
47 elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
48 elim (tdeq_dec h o T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
49 [ elim (tdeq_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
50 /4 width=7 by fsb_fpbs_trans, tdeq_rdeq_lpx_fpbs, fpbs_fqup_trans/
51 | elim (cpxs_tdneq_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
52 elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
53 /3 width=14 by fpbs_cpxs_tdeq_fqup_lpx_trans/
59 lemma csx_fsb: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ≥[h, o] 𝐒⦃G, L, T⦄.
60 /2 width=5 by csx_fsb_fpbs/ qed.
62 (* Advanced eliminators *****************************************************)
64 lemma csx_ind_fpb: ∀h,o. ∀Q:relation3 genv lenv term.
65 (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
66 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
69 ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q G L T.
70 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
72 lemma csx_ind_fpbg: ∀h,o. ∀Q:relation3 genv lenv term.
73 (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
74 (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
77 ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q G L T.
78 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.