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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/predsnstar_2.ma".
16 include "basic_2/grammar/lpx_sn_tc.ma".
17 include "basic_2/reduction/lpr.ma".
19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
21 definition lprs: relation lenv ≝ TC … lpr.
23 interpretation "parallel computation (local environment, sn variant)"
24 'PRedSnStar L1 L2 = (lprs L1 L2).
26 (* Basic eliminators ********************************************************)
28 lemma lprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
29 (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
30 ∀L2. L1 ⊢ ➡* L2 → R L2.
31 #L1 #R #HL1 #IHL1 #L2 #HL12
32 @(TC_star_ind … HL1 IHL1 … HL12) //
35 lemma lprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
36 (∀L1,L. L1 ⊢ ➡ L → L ⊢ ➡* L2 → R L → R L1) →
37 ∀L1. L1 ⊢ ➡* L2 → R L1.
38 #L2 #R #HL2 #IHL2 #L1 #HL12
39 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
42 (* Basic properties *********************************************************)
44 lemma lpr_lprs: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ➡* L2.
47 lemma lprs_refl: ∀L. L ⊢ ➡* L.
50 lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
53 lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
56 lemma lprs_pair_refl: ∀L1,L2. L1 ⊢ ➡* L2 → ∀I,V. L1. ⓑ{I} V ⊢ ➡* L2. ⓑ{I} V.
57 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
59 (* Basic inversion lemmas ***************************************************)
61 lemma lprs_inv_atom1: ∀L2. ⋆ ⊢ ➡* L2 → L2 = ⋆.
62 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
64 lemma lprs_inv_atom2: ∀L1. L1 ⊢ ➡* ⋆ → L1 = ⋆.
65 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
67 (* Basic forward lemmas *****************************************************)
69 lemma lprs_fwd_length: ∀L1,L2. L1 ⊢ ➡* L2 → |L1| = |L2|.
70 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.