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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_3.ma".
16 include "basic_2/relocation/lpx_sn_tc.ma".
17 include "basic_2/reduction/lpr.ma".
19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
21 definition lprs: relation3 genv lenv lenv ≝
24 interpretation "parallel computation (local environment, sn variant)"
25 'PRedSnStar G L1 L2 = (lprs G L1 L2).
27 (* Basic eliminators ********************************************************)
29 lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
30 (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
31 ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
32 #G #L1 #R #HL1 #IHL1 #L2 #HL12
33 @(TC_star_ind … HL1 IHL1 … HL12) //
36 lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
37 (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
38 ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
39 #G #L2 #R #HL2 #IHL2 #L1 #HL12
40 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
43 (* Basic properties *********************************************************)
45 lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
48 lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
51 lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
54 lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
57 lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
58 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
60 (* Basic inversion lemmas ***************************************************)
62 lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
63 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
65 lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
66 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
68 (* Basic forward lemmas *****************************************************)
70 lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
71 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.