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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/rajuststar_5.ma".
16 include "basic_2/substitution/fpa.ma".
18 (* MULTIPLE AJUSTMENT *******************************************************)
20 definition fpas: bool → bi_relation lenv term ≝ λs. bi_TC … (fpa s).
23 "multiple ajustment (restricted closure)"
24 'RAjustStar L1 T1 s L2 T2 = (fpas s L1 T1 L2 T2).
26 (* Basic eliminators ********************************************************)
28 lemma fpas_ind: ∀L1,T1,s. ∀R:relation2 …. R L1 T1 →
29 (∀L,L2,T,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ → R L T → R L2 T2) →
30 ∀L2,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L2 T2.
31 #L1 #T1 #s #R #IH1 #IH2 #L2 #T2 #H
32 @(bi_TC_star_ind … IH1 IH2 L2 T2 H) //
35 lemma fpas_ind_dx: ∀L2,T2,s. ∀R:relation2 …. R L2 T2 →
36 (∀L1,L,T1,T. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → R L T → R L1 T1) →
37 ∀L1,T1. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L1 T1.
38 #L2 #T2 #s #R #IH1 #IH2 #L1 #T1 #H
39 @(bi_TC_star_ind_dx … IH1 IH2 L1 T1 H) //
42 (* Basic properties *********************************************************)
44 lemma fpas_refl: ∀s. bi_reflexive … (fpas s).
45 /2 width=1 by bi_inj/ qed.
47 lemma fpas_sym: ∀s. bi_symmetric … (fpas s).
48 /3 width=1 by fpa_sym, bi_TC_symmetric/ qed-.
50 lemma fpa_fpas: ∀L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
51 /2 width=1 by bi_inj/ qed.
53 lemma fpas_strap1: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ →
54 ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
55 /2 width=4 by bi_step/ qed-.
57 lemma fpas_strap2: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ →
58 ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
59 /2 width=4 by bi_TC_strap/ qed-.