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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/relocation/llpx_sn_ldrop.ma".
16 include "basic_2/relocation/fquq_alt.ma".
17 include "basic_2/reduction/cpr_lift.ma".
18 include "basic_2/reduction/llpr.ma".
20 (* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
22 (* Advanced inversion lemmas ************************************************)
24 lemma llpr_inv_lref_ge_dx: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
25 ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
26 ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
27 ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
28 /2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
30 lemma llpr_inv_lref_ge_sn: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
31 ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
32 ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
33 ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
34 /2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
36 lemma llpr_inv_lref_ge_bi: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
38 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
39 ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
40 /2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
42 lemma llpr_inv_bind_O: ∀a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [ⓑ{a,I}V.T, 0] L2 →
43 ⦃G, L1⦄ ⊢ ➡[V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V.
44 /2 width=2 by llpx_sn_inv_bind_O/ qed-.
46 lemma llpr_bind_repl_O: ∀I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V2 →
47 ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡ W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[T, 0] L2.ⓑ{J}W2.
48 /2 width=4 by llpx_sn_bind_repl_O/ qed-.
50 (* Properties on context-sensitive parallel reduction for terms *************)
52 lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
53 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
54 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
55 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
56 /3 width=10 by llpr_lref, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
57 #G #L #K #U #T #e #HLK #HUT #U2 #HU2
58 elim (lift_total U2 0 (e+1)) #T2 #HUT2
59 lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
62 lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
63 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
64 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
65 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
66 [ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
67 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
71 lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
72 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
73 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
74 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
75 /3 width=10 by llpr_lref, cpr_pair_sn, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
76 #G #L #K #U #T #e #HLK #HUT #U2 #HU2
77 elim (lift_total U2 0 (e+1)) #T2 #HUT2
78 lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
81 lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
82 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
83 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
84 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
85 [ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
86 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/