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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/rt_transition/fpb_fdeq.ma".
16 include "basic_2/rt_transition/fpbq.ma".
18 (* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
20 (* Properties with proper parallel rst-transition for closures **************)
22 lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
23 ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
24 #h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
25 /3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
28 (* Basic_2A1: fpb_fpbq_alt *)
29 lemma fpb_fpbq_ffdneq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
30 ∧∧ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥).
31 /3 width=10 by fpb_fpbq, fpb_inv_fdeq, conj/ qed-.
33 (* Inversrion lemmas with proper parallel rst-transition for closures *******)
35 (* Basic_2A1: uses: fpbq_ind_alt *)
36 lemma fpbq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
37 ∨∨ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄
38 | ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
39 #h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
40 [ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
41 /3 width=1 by fpb_fqu, fdeq_intro_sn, or_intror, or_introl/
42 | #T2 #H elim (tdeq_dec h o T1 T2)
43 /4 width=1 by fpb_cpx, fdeq_intro_sn, or_intror, or_introl/
44 | #L2 elim (rdeq_dec h o L1 L2 T1)
45 /4 width=1 by fpb_lpx, fdeq_intro_sn, or_intror, or_introl/
46 | /2 width=1 by or_introl/
50 (* Basic_2A1: fpbq_inv_fpb_alt *)
51 lemma fpbq_ffdneq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
52 (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
53 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
54 elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 //