1 lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
2 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
3 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
5 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
6 [ /2 width=1 by or_introl/
7 | #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
8 elim (fpb_fpbu … H1) -H1 #H1
11 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
12 #L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
13 [ -HT1 elim (fqus_inv_gen … HT2) -HT2
14 [ #H elim (fqup_inv_step_sn … H) -H
15 /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
16 | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
17 [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
18 | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
19 /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
22 | elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
23 /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
27 (* alternative proof that needs decidability of bteq to go in fpbs.ma
28 * or lpx_fpbc_trans to go in fpbs_lift.ma (possibly)
31 axiom lpx_bteq_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1⦄ ⊢➡ [h, g] L →
32 ⦃G1, L, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄.
34 lemma fpbs_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
35 ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ ∨
36 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
37 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1 [ /2 width=1 by or_introl/ ] (**) (* auto fails without brackets *)
38 #G1 #G #L1 #L #T1 #T *
39 [ #G0 #L0 #T0 #H #H02 #IH1 elim (fquq_inv_gen … H) -H
40 [ -IH1 /4 width=5 by fpbc_fqu, ex2_3_intro, or_intror/
41 | -H02 * #HG #HL #HT destruct /2 width=1 by/
43 | #T0 #HT10 #H02 #IH02 elim (eq_term_dec T1 T0) #H destruct
45 | -IH02 /5 width=5 by fpbc_cpx, ex2_3_intro, or_intror/
47 | #L0 #HL10 #_ * [ /3 width=3 by or_introl, lpx_bteq_trans/ ]
48 * #G3 #L3 #T3 #H13 #H32