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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/btpredalt_8.ma".
16 include "basic_2/reduction/fpb_fleq.ma".
17 include "basic_2/reduction/fpbq.ma".
19 (* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
21 (* Basic properties *********************************************************)
23 lemma fleq_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2.
24 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
25 #h #o #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fpbq_lleq/
28 lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
29 ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
30 #h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
31 /3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
34 (* Advanced eliminators *****************************************************)
36 lemma fpbq_ind_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ∀R:Prop.
37 (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → R) →
38 (⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R) →
39 ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → R.
40 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #R #HI1 #HI2 #H elim (fpbq_fpbqa … H) /2 width=1 by/
43 (* aternative definition of fpb *********************************************)
45 lemma fpb_fpbq_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
46 ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ ∧ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥).
47 /3 width=10 by fpb_fpbq, fpb_inv_fleq, conj/ qed.
49 lemma fpbq_inv_fpb_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
50 (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
51 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 @(fpbq_ind_alt … H) -H //