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15 include "basic_2/grammar/lpx_sn_tc.ma".
16 include "basic_2/reduction/lpr.ma".
18 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
20 definition lprs: relation lenv ≝ TC … lpr.
22 interpretation "parallel computation (local environment, sn variant)"
23 'PRedSnStar L1 L2 = (lprs L1 L2).
25 (* Basic eliminators ********************************************************)
27 lemma lprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
28 (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
29 ∀L2. L1 ⊢ ➡* L2 → R L2.
30 #L1 #R #HL1 #IHL1 #L2 #HL12
31 @(TC_star_ind … HL1 IHL1 … HL12) //
34 lemma lprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
35 (∀L1,L. L1 ⊢ ➡ L → L ⊢ ➡* L2 → R L → R L1) →
36 ∀L1. L1 ⊢ ➡* L2 → R L1.
37 #L2 #R #HL2 #IHL2 #L1 #HL12
38 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
41 (* Basic properties *********************************************************)
43 lemma lpr_lprs: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ➡* L2.
46 lemma lprs_refl: ∀L. L ⊢ ➡* L.
49 lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
52 lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
55 lemma lprs_pair_refl: ∀L1,L2. L1 ⊢ ➡* L2 → ∀I,V. L1. ⓑ{I} V ⊢ ➡* L2. ⓑ{I} V.
56 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
58 (* Basic inversion lemmas ***************************************************)
60 lemma lprs_inv_atom1: ∀L2. ⋆ ⊢ ➡* L2 → L2 = ⋆.
61 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
63 lemma lprs_inv_atom2: ∀L1. L1 ⊢ ➡* ⋆ → L1 = ⋆.
64 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
66 (* Basic forward lemmas *****************************************************)
68 lemma lprs_fwd_length: ∀L1,L2. L1 ⊢ ➡* L2 → |L1| = |L2|.
69 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.