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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 *)
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15 include "basic_2/notation/relations/predsn_5.ma".
16 include "basic_2/reduction/lpr.ma".
17 include "basic_2/reduction/cpx.ma".
19 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
21 definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
22 λh,g,G. lpx_sn (cpx h g G).
24 interpretation "extended parallel reduction (local environment, sn variant)"
25 'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
27 (* Basic inversion lemmas ***************************************************)
29 lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
30 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
32 lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
33 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
35 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
37 lemma lpx_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
38 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
40 lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
41 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
43 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
45 lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
46 ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
47 /2 width=1 by lpx_sn_inv_pair/ qed-.
49 (* Basic properties *********************************************************)
51 lemma lpx_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡[h, g] L.
52 /2 width=1 by lpx_sn_refl/ qed.
54 lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
55 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
56 /2 width=1 by lpx_sn_pair/ qed.
58 lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
59 #h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
62 (* Basic forward lemmas *****************************************************)
64 lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
65 /2 width=2 by lpx_sn_fwd_length/ qed-.