1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/notation/relations/predtysn_4.ma".
16 include "basic_2/relocation/lex.ma".
17 include "basic_2/rt_transition/cpx_ext.ma".
19 (* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENVIRONMENTS ******************)
21 definition lpx: sh → genv → relation lenv ≝
25 "uncounted parallel rt-transition (local environment)"
26 'PRedTySn h G L1 L2 = (lpx h G L1 L2).
28 (* Basic properties *********************************************************)
30 lemma lpx_bind: ∀h,G,K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
31 ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
32 /2 width=1 by lex_bind/ qed.
34 lemma lpx_refl: ∀h,G. reflexive … (lpx h G).
35 /2 width=1 by lex_refl/ qed.
37 (* Advanced properties ******************************************************)
39 lemma lpx_bind_refl_dx: ∀h,G,K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
40 ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
41 /2 width=1 by lex_bind_refl_dx/ qed.
43 lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
44 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
45 /2 width=1 by lpx_sn_pair/ qed.
47 (* Basic inversion lemmas ***************************************************)
49 (* Basic_2A1: was: lpx_inv_atom1 *)
50 lemma lpx_inv_atom_sn: ∀h,G,L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
51 /2 width=2 by lex_inv_atom_sn/ qed-.
53 lemma lpx_inv_bind_sn: ∀h,I1,G,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
54 ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
56 /2 width=1 by lex_inv_bind_sn/ qed-.
58 (* Basic_2A1: was: lpx_inv_atom2 *)
59 lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
60 /2 width=2 by lex_inv_atom_dx/ qed-.
62 lemma lpx_inv_bind_dx: ∀h,I2,G,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
63 ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
65 /2 width=1 by lex_inv_bind_dx/ qed-.
67 (* Advanced inversion lemmas ************************************************)
69 (* Basic_2A1: was: lpx_inv_pair1 *)
70 lemma lpx_inv_pair_sn: ∀h,I,G,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
71 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
73 /2 width=1 by lex_inv_pair_sn/ qed-.
75 (* Basic_2A1: was: lpx_inv_pair2 *)
76 lemma lpx_inv_pair_dx: ∀h,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
77 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
79 /2 width=1 by lex_inv_pair_dx/ qed-.
81 lemma lpx_inv_pair: ∀h,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
82 ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
83 /2 width=1 by lex_inv_pair/ qed-.