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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2A/notation/relations/predsn_3.ma".
16 include "basic_2A/substitution/lpx_sn.ma".
17 include "basic_2A/reduction/cpr.ma".
19 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
21 definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (cpr G).
23 interpretation "parallel reduction (local environment, sn variant)"
24 'PRedSn G L1 L2 = (lpr G L1 L2).
26 (* Basic inversion lemmas ***************************************************)
28 lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
29 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
31 lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
32 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
33 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
35 lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
36 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
38 lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
39 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
40 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
42 (* Basic properties *********************************************************)
45 lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
46 /2 width=1 by lpx_sn_refl/ qed.
48 lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
49 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
50 /2 width=1 by lpx_sn_pair/ qed.
52 (* Basic forward lemmas *****************************************************)
54 lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
55 /2 width=2 by lpx_sn_fwd_length/ qed-.