(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsnstar_2.ma".
-include "basic_2/grammar/lpx_sn_tc.ma".
+include "basic_2/notation/relations/predsnstar_3.ma".
+include "basic_2/relocation/lpx_sn_tc.ma".
include "basic_2/reduction/lpr.ma".
(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-definition lprs: relation lenv ≝ TC … lpr.
+definition lprs: relation3 genv lenv lenv ≝
+ λG. TC … (lpr G).
interpretation "parallel computation (local environment, sn variant)"
- 'PRedSnStar L1 L2 = (lprs L1 L2).
+ 'PRedSnStar G L1 L2 = (lprs G L1 L2).
(* Basic eliminators ********************************************************)
-lemma lprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
- ∀L2. L1 ⊢ ➡* L2 → R L2.
-#L1 #R #HL1 #IHL1 #L2 #HL12
+lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
+#G #L1 #R #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) //
qed-.
-lemma lprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. L1 ⊢ ➡ L → L ⊢ ➡* L2 → R L → R L1) →
- ∀L1. L1 ⊢ ➡* L2 → R L1.
-#L2 #R #HL2 #IHL2 #L1 #HL12
+lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
+#G #L2 #R #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) //
qed-.
(* Basic properties *********************************************************)
-lemma lpr_lprs: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ➡* L2.
+lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
/2 width=1/ qed.
-lemma lprs_refl: ∀L. L ⊢ ➡* L.
+lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
/2 width=1/ qed.
-lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
+lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
/2 width=3/ qed.
-lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
+lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
/2 width=3/ qed.
-lemma lprs_pair_refl: ∀L1,L2. L1 ⊢ ➡* L2 → ∀I,V. L1. ⓑ{I} V ⊢ ➡* L2. ⓑ{I} V.
+lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
/2 width=1 by TC_lpx_sn_pair_refl/ qed.
(* Basic inversion lemmas ***************************************************)
-lemma lprs_inv_atom1: ∀L2. ⋆ ⊢ ➡* L2 → L2 = ⋆.
+lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
-lemma lprs_inv_atom2: ∀L1. L1 ⊢ ➡* ⋆ → L1 = ⋆.
+lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lprs_fwd_length: ∀L1,L2. L1 ⊢ ➡* L2 → |L1| = |L2|.
+lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
/2 width=2 by TC_lpx_sn_fwd_length/ qed-.