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notational change for lexs
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2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
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10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
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14
15 include "ground_2/relocation/rtmap_sand.ma".
16 include "basic_2/relocation/drops.ma".
17
18 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
19
20 (* Main properties **********************************************************)
21
22 theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
23                        lexs_transitive RN1 RN2 RN RN1 RP1 →
24                        lexs_transitive RP1 RP2 RP RN1 RP1 →
25                        ∀L1,L0. L1 ⪤*[RN1, RP1, f] L0 →
26                        ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
27                        L1 ⪤*[RN, RP, f] L2.
28 #RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
29 [ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
30 | #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
31   #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
32 | #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
33   #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
34 ]
35 qed-.
36
37 (* Basic_2A1: includes: lpx_sn_trans *)
38 theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP →
39                                   lexs_transitive RP RP RP RN RP →
40                                   Transitive … (lexs RN RP f).
41 /2 width=9 by lexs_trans_gen/ qed-.
42
43 (* Basic_2A1: includes: lpx_sn_conf *)
44 theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
45                   ∀L,f.
46                   (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K V) →
47                   (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K V) →
48                   pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
49 #RN1 #RP1 #RN2 #RP2 #L elim L -L
50 [ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
51   /2 width=3 by lexs_atom, ex2_intro/
52 | #L #I #V #IH #f elim (pn_split f) * #g #H destruct
53   #HN #HP #Y1 #H1 #Y2 #H2
54   [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
55     elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
56     elim (HP … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
57     elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
58   | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
59     elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
60     elim (HN … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
61     elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
62   ]
63 ]
64 qed-.
65
66 theorem lexs_canc_sn: ∀RN,RP,f. Transitive … (lexs RN RP f) →
67                                 symmetric … (lexs RN RP f) →
68                                 left_cancellable … (lexs RN RP f).
69 /3 width=3 by/ qed-.
70
71 theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
72                                 symmetric … (lexs RN RP f) →
73                                 right_cancellable … (lexs RN RP f).
74 /3 width=3 by/ qed-.
75
76 lemma lexs_meet: ∀RN,RP,L1,L2.
77                  ∀f1. L1 ⪤*[RN, RP, f1] L2 →
78                  ∀f2. L1 ⪤*[RN, RP, f2] L2 →
79                  ∀f. f1 ⋒ f2 ≡ f → L1 ⪤*[RN, RP, f] L2.
80 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
81 #f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
82 elim (pn_split f2) * #g2 #H2 destruct
83 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
84 [ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
85 | elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
86 ] -Hf /3 width=5 by lexs_next, lexs_push/
87 qed-.
88
89 lemma lexs_join: ∀RN,RP,L1,L2.
90                  ∀f1. L1 ⪤*[RN, RP, f1] L2 →
91                  ∀f2. L1 ⪤*[RN, RP, f2] L2 →
92                  ∀f. f1 ⋓ f2 ≡ f → L1 ⪤*[RN, RP, f] L2.
93 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
94 #f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
95 elim (pn_split f2) * #g2 #H2 destruct
96 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
97 [ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
98 | elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
99 ] -Hf /3 width=5 by lexs_next, lexs_push/
100 qed-.