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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/reducibility/cpr.ma".
16 include "basic_2/reducibility/fpr.ma".
18 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
20 definition cfpr: lenv → bi_relation lenv term ≝
21 λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
24 "context-sensitive parallel reduction (closure)"
25 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
27 (* Basic properties *********************************************************)
29 lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
32 lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
33 #L1 #L2 #T1 #T2 * /3 width=1/
36 (* Basic inversion lemmas ***************************************************)
38 lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
39 #L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
42 (* Advanced inversion lemmas ************************************************)
44 lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
46 ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
48 #I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
49 elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
50 >shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
51 elim (tpr_inv_bind1 … H) -H *
52 [ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
53 | #T0 #_ #_ #H destruct