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15 include "basic_2/reduction/lpx.ma".
16 include "basic_2/computation/lprs.ma".
18 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
20 definition lpxs: ∀h. sd h → relation lenv ≝ λh,g. TC … (lpx h g).
22 interpretation "extended parallel computation (local environment, sn variant)"
23 'PRedSnStar h g L1 L2 = (lpxs h g L1 L2).
25 (* Basic eliminators ********************************************************)
27 lemma lpxs_ind: ∀h,g,L1. ∀R:predicate lenv. R L1 →
28 (∀L,L2. ⦃h, L1⦄ ⊢ ➡*[g] L → ⦃h, L⦄ ⊢ ➡[g] L2 → R L → R L2) →
29 ∀L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L2.
30 #h #g #L1 #R #HL1 #IHL1 #L2 #HL12
31 @(TC_star_ind … HL1 IHL1 … HL12) //
34 lemma lpxs_ind_dx: ∀h,g,L2. ∀R:predicate lenv. R L2 →
35 (∀L1,L. ⦃h, L1⦄ ⊢ ➡[g] L → ⦃h, L⦄ ⊢ ➡*[g] L2 → R L → R L1) →
36 ∀L1. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L1.
37 #h #g #L2 #R #HL2 #IHL2 #L1 #HL12
38 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
41 (* Basic properties *********************************************************)
43 axiom lprs_lpxs: ∀h,g,L1,L2. L1 ⊢ ➡* L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
47 lemma lpx_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
50 lemma lpxs_refl: ∀h,g,L. ⦃h, L⦄ ⊢ ➡*[g] L.
53 lemma lpxs_strap1: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡*[g] L → ⦃h, L⦄ ⊢ ➡[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
56 lemma lpxs_strap2: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡[g] L → ⦃h, L⦄ ⊢ ➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
59 lemma lpxs_pair_refl: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ∀I,V. ⦃h, L1. ⓑ{I}V⦄ ⊢ ➡*[g] L2. ⓑ{I}V.
60 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
62 (* Basic inversion lemmas ***************************************************)
64 lemma lpxs_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡*[g] L2 → L2 = ⋆.
65 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
67 lemma lpxs_inv_atom2: ∀h,g,L1. ⦃h, L1⦄ ⊢ ➡*[g] ⋆ → L1 = ⋆.
68 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
70 (* Basic forward lemmas *****************************************************)
72 lemma lpxs_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → |L1| = |L2|.
73 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.