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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 "static_2/relocation/lex.ma".
16 include "basic_2/notation/relations/pconveta_4.ma".
17 include "basic_2/rt_conversion/cpce.ma".
19 (* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
21 definition lpce (h) (G): relation lenv ≝ lex (cpce h G).
24 "parallel eta-conversion on all entries (local environment)"
25 'PConvEta h G L1 L2 = (lpce h G L1 L2).
27 (* Advanced properties ******************************************************)
29 lemma lpce_pair (h) (G):
30 ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 →
31 ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2.
32 /2 width=1 by lex_pair/ qed.
34 (* Basic inversion lemmas ***************************************************)
36 lemma lpce_inv_atom_sn (h) (G):
37 ∀L2. ⦃G,⋆⦄ ⊢ ⬌η[h] L2 → L2 = ⋆.
38 /2 width=2 by lex_inv_atom_sn/ qed-.
40 lemma lpce_inv_atom_dx (h) (G):
41 ∀L1. ⦃G,L1⦄ ⊢ ⬌η[h] ⋆ → L1 = ⋆.
42 /2 width=2 by lex_inv_atom_dx/ qed-.
44 (* Advanced inversion lemmas ************************************************)
46 lemma lpce_inv_unit_sn (h) (G):
47 ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
48 ∃∃K2. ⦃G, K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
49 /2 width=1 by lex_inv_unit_sn/ qed-.
51 lemma lpce_inv_pair_sn (h) (G):
52 ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] L2 →
53 ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L2 = K2.ⓑ{I}V2.
54 /2 width=1 by lex_inv_pair_sn/ qed-.
56 lemma lpce_inv_unit_dx (h) (G):
57 ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓤ{I} →
58 ∃∃K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L1 = K1.ⓤ{I}.
59 /2 width=1 by lex_inv_unit_dx/ qed-.
61 lemma lpce_inv_pair_dx (h) (G):
62 ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2 →
63 ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L1 = K1.ⓑ{I}V1.
64 /2 width=1 by lex_inv_pair_dx/ qed-.
66 lemma lpce_inv_pair (h) (G):
67 ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬌η[h] L2.ⓑ{I2}V2 →
68 ∧∧ ⦃G,L1⦄ ⊢ ⬌η[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬌η[h] V2 & I1 = I2.
69 /2 width=1 by lex_inv_pair/ qed-.