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14
15 include "basic_2/reducibility/cfpr_cpr.ma".
16
17 (* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
18
19 (* Advanced propertis *******************************************************)
20
21 lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
22                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
23 #L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
24 elim (fpr_inv_all … H) /3 width=4/
25 qed.
26
27 (* Advanced forward lemmas **************************************************)
28
29 lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
30                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
31                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
32 * #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
33 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
34 [ elim (cpr_inv_abbr1 … H) -H *
35   [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
36   | #T #_ #_ #H destruct
37   ]
38 | elim (cpr_inv_abst1 … H Abst V2) -H
39   #V #T #HV1 #_ #H destruct /3 width=4/
40 ]
41 qed-.
42
43 (* Advanced inversion lemmas ************************************************)
44
45 lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
46                      ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
47                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
48                               L2 = K2.ⓑ{I}V2.
49 #I1 #K1 #X #V1 #T1 #T2 #H
50 elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
51 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
52 qed-.
53
54 lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
55                      ∃∃K1,V1. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
56                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
57                               L1 = K1.ⓑ{I}V1.
58 #I2 #X #K2 #V2 #T1 #T2 #H
59 elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
60 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
61 qed-.