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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/notation/relations/predtysnstar_4.ma".
16 include "static_2/relocation/lex.ma".
17 include "basic_2/rt_computation/cpxs_ext.ma".
19 (* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
21 definition lpxs (h) (G): relation lenv ≝
25 "unbound parallel rt-computation on all entries (local environment)"
26 'PRedTySnStar h G L1 L2 = (lpxs h G L1 L2).
28 (* Basic properties *********************************************************)
30 (* Basic_2A1: uses: lpxs_pair_refl *)
31 lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
32 ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
33 /2 width=1 by lex_bind_refl_dx/ qed.
35 lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
36 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ⬈*[h] V2 →
37 ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
38 /2 width=1 by lex_pair/ qed.
40 lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
41 /2 width=1 by lex_refl/ qed.
43 (* Basic inversion lemmas ***************************************************)
45 (* Basic_2A1: was: lpxs_inv_atom1 *)
46 lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
47 /2 width=2 by lex_inv_atom_sn/ qed-.
49 lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
50 ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
51 /2 width=1 by lex_inv_bind_sn/ qed-.
53 (* Basic_2A1: was: lpxs_inv_pair1 *)
54 lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
55 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
56 /2 width=1 by lex_inv_pair_sn/ qed-.
58 (* Basic_2A1: was: lpxs_inv_atom2 *)
59 lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
60 /2 width=2 by lex_inv_atom_dx/ qed-.
62 (* Basic_2A1: was: lpxs_inv_pair2 *)
63 lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
64 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
65 /2 width=1 by lex_inv_pair_dx/ qed-.
67 (* Basic eliminators ********************************************************)
69 (* Basic_2A1: was: lpxs_ind_alt *)
70 lemma lpxs_ind (h) (G): ∀Q:relation lenv.
74 Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
77 ⦃G, K1⦄ ⊢ ⬈*[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 →
78 Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
80 ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
81 /3 width=4 by lex_ind/ qed-.