(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predtysnstar_4.ma".
-include "basic_2/relocation/lex.ma".
+include "basic_2/notation/relations/predtysnstar_3.ma".
+include "static_2/relocation/lex.ma".
include "basic_2/rt_computation/cpxs_ext.ma".
-(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
+(* EXTENDED PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS *************)
-definition lpxs (h) (G): relation lenv ≝
- lex (cpxs h G).
+definition lpxs (G): relation lenv ≝
+ lex (cpxs G).
interpretation
- "unbound parallel rt-computation on all entries (local environment)"
- 'PRedTySnStar h G L1 L2 = (lpxs h G L1 L2).
+ "extended parallel rt-computation on all entries (local environment)"
+ 'PRedTySnStar G L1 L2 = (lpxs G L1 L2).
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lpxs_pair_refl *)
-lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
+lemma lpxs_bind_refl_dx (G):
+ ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 →
+ ∀I. ❨G,L1.ⓘ[I]❩ ⊢ ⬈* L2.ⓘ[I].
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
+lemma lpxs_pair (G):
+ ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 →
+ ∀V1,V2. ❨G,L1❩ ⊢ V1 ⬈* V2 →
+ ∀I. ❨G,L1.ⓑ[I]V1❩ ⊢ ⬈* L2.ⓑ[I]V2.
/2 width=1 by lex_pair/ qed.
-lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
+lemma lpxs_refl (G):
+ reflexive … (lpxs G).
/2 width=1 by lex_refl/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpxs_inv_atom1 *)
-lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
+lemma lpxs_inv_atom_sn (G):
+ ∀L2. ❨G,⋆❩ ⊢ ⬈* L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
+lemma lpxs_inv_bind_sn (G):
+ ∀I1,L2,K1. ❨G,K1.ⓘ[I1]❩ ⊢ ⬈* L2 →
+ ∃∃I2,K2. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ I1 ⬈* I2 & L2 = K2.ⓘ[I2].
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_pair1 *)
-lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair_sn (G):
+ ∀I,L2,K1,V1. ❨G,K1.ⓑ[I]V1❩ ⊢ ⬈* L2 →
+ ∃∃K2,V2. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ V1 ⬈* V2 & L2 = K2.ⓑ[I]V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_atom2 *)
-lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+lemma lpxs_inv_atom_dx (G):
+ ∀L1. ❨G,L1❩ ⊢ ⬈* ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lpxs_inv_pair2 *)
-lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair_dx (G):
+ ∀I,L1,K2,V2. ❨G,L1❩ ⊢ ⬈* K2.ⓑ[I]V2 →
+ ∃∃K1,V1. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ V1 ⬈* V2 & L1 = K1.ⓑ[I]V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
(* Basic_2A1: was: lpxs_ind_alt *)
-lemma lpxs_ind (h) (G): ∀Q:relation lenv.
- Q (⋆) (⋆) → (
- ∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
- Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
- ) → (
- ∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 →
- Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
+lemma lpxs_ind (G) (Q:relation …):
+ Q (⋆) (⋆) → (
+ ∀I,K1,K2.
+ ❨G,K1❩ ⊢ ⬈* K2 →
+ Q K1 K2 → Q (K1.ⓘ[I]) (K2.ⓘ[I])
+ ) → (
+ ∀I,K1,K2,V1,V2.
+ ❨G,K1❩ ⊢ ⬈* K2 → ❨G,K1❩ ⊢ V1 ⬈* V2 →
+ Q K1 K2 → Q (K1.ⓑ[I]V1) (K2.ⓑ[I]V2)
+ ) →
+ ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.