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14
15 include "basic_2/notation/relations/predtysnstar_3.ma".
16 include "static_2/relocation/lex.ma".
17 include "basic_2/rt_computation/cpxs_ext.ma".
18
19 (* EXTENDED PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS *************)
20
21 definition lpxs (G): relation lenv ≝
22            lex (cpxs G).
23
24 interpretation
25   "extended parallel rt-computation on all entries (local environment)"
26   'PRedTySnStar G L1 L2 = (lpxs G L1 L2).
27
28 (* Basic properties *********************************************************)
29
30 (* Basic_2A1: uses: lpxs_pair_refl *)
31 lemma lpxs_bind_refl_dx (G):
32       ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 →
33       ∀I. ❨G,L1.ⓘ[I]❩ ⊢ ⬈* L2.ⓘ[I].
34 /2 width=1 by lex_bind_refl_dx/ qed.
35
36 lemma lpxs_pair (G):
37       ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 →
38       ∀V1,V2. ❨G,L1❩ ⊢ V1 ⬈* V2 →
39       ∀I. ❨G,L1.ⓑ[I]V1❩ ⊢ ⬈* L2.ⓑ[I]V2.
40 /2 width=1 by lex_pair/ qed.
41
42 lemma lpxs_refl (G):
43       reflexive … (lpxs G).
44 /2 width=1 by lex_refl/ qed.
45
46 (* Basic inversion lemmas ***************************************************)
47
48 (* Basic_2A1: was: lpxs_inv_atom1 *)
49 lemma lpxs_inv_atom_sn (G):
50       ∀L2. ❨G,⋆❩ ⊢ ⬈* L2 → L2 = ⋆.
51 /2 width=2 by lex_inv_atom_sn/ qed-.
52
53 lemma lpxs_inv_bind_sn (G):
54       ∀I1,L2,K1. ❨G,K1.ⓘ[I1]❩ ⊢ ⬈* L2 →
55       ∃∃I2,K2. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ I1 ⬈* I2 & L2 = K2.ⓘ[I2].
56 /2 width=1 by lex_inv_bind_sn/ qed-.
57
58 (* Basic_2A1: was: lpxs_inv_pair1 *)
59 lemma lpxs_inv_pair_sn (G):
60       ∀I,L2,K1,V1. ❨G,K1.ⓑ[I]V1❩ ⊢ ⬈* L2 →
61       ∃∃K2,V2. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ V1 ⬈* V2 & L2 = K2.ⓑ[I]V2.
62 /2 width=1 by lex_inv_pair_sn/ qed-.
63
64 (* Basic_2A1: was: lpxs_inv_atom2 *)
65 lemma lpxs_inv_atom_dx (G):
66       ∀L1. ❨G,L1❩ ⊢ ⬈* ⋆ → L1 = ⋆.
67 /2 width=2 by lex_inv_atom_dx/ qed-.
68
69 (* Basic_2A1: was: lpxs_inv_pair2 *)
70 lemma lpxs_inv_pair_dx (G):
71       ∀I,L1,K2,V2. ❨G,L1❩ ⊢ ⬈* K2.ⓑ[I]V2 →
72       ∃∃K1,V1. ❨G,K1❩ ⊢ ⬈* K2 & ❨G,K1❩ ⊢ V1 ⬈* V2 & L1 = K1.ⓑ[I]V1.
73 /2 width=1 by lex_inv_pair_dx/ qed-.
74
75 (* Basic eliminators ********************************************************)
76
77 (* Basic_2A1: was: lpxs_ind_alt *)
78 lemma lpxs_ind (G) (Q:relation …):
79       Q (⋆) (⋆) → (
80         ∀I,K1,K2.
81         ❨G,K1❩ ⊢ ⬈* K2 →
82         Q K1 K2 → Q (K1.ⓘ[I]) (K2.ⓘ[I])
83       ) → (
84         ∀I,K1,K2,V1,V2.
85         ❨G,K1❩ ⊢ ⬈* K2 → ❨G,K1❩ ⊢ V1 ⬈* V2 →
86         Q K1 K2 → Q (K1.ⓑ[I]V1) (K2.ⓑ[I]V2)
87       ) →
88       ∀L1,L2. ❨G,L1❩ ⊢ ⬈* L2 → Q L1 L2.
89 /3 width=4 by lex_ind/ qed-.