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15 include "basic_2/substitution/lpx_sn.ma".
17 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
19 (* Properties on transitive_closure *****************************************)
21 lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
22 ∀L1,L2. TC … (lpx_sn R) L1 L2 →
23 ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
24 #R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
25 [ /2 width=1 by lpx_sn_refl/
26 | /3 width=1 by TC_reflexive, lpx_sn_refl/
27 | /3 width=5 by lpx_sn_pair, step/
31 lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
32 ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
33 ∀V1,V2. LTC … R L1 V1 V2 →
34 TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
35 #R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
36 [ /2 width=1 by TC_lpx_sn_pair_refl/
37 | /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
41 lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
42 ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
43 TC … (lpx_sn R) L1 L2.
44 #R #HR #L1 #L2 #H elim H -L1 -L2
45 /2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
48 (* Inversion lemmas on transitive closure ***********************************)
50 lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
51 #R #L1 #H @(TC_ind_dx … L1 H) -L1
52 [ /2 width=2 by lpx_sn_inv_atom2/
53 | #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
57 lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
58 ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
59 ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
60 #R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
61 [ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
62 | #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
63 elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
64 lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
68 lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
72 TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
73 S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
75 ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
76 #R #HR #S #IH1 #IH2 #L2 elim L2 -L2
77 [ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
78 | #L2 #I #V2 #IHL2 #X #H
79 elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
80 #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
84 lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
86 [ /2 width=2 by lpx_sn_inv_atom1/
87 | #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
91 fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
92 ∀L1,L2. TC … (lpx_sn R) L1 L2 →
93 ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
94 ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
95 #R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
96 [ #J #K #W #H destruct
97 | #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
101 lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
102 ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
103 ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
104 /2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
106 lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
107 ∀L1,L2. TC … (lpx_sn R) L1 L2 →
108 lpx_sn (LTC … R) L1 L2.
109 /3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
111 (* Forward lemmas on transitive closure *************************************)
113 lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
114 #R #L1 #L2 #H elim H -L2
115 [ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
116 | #L #L2 #_ #HL2 #IHL1
117 >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //