1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground_2/relocation/rtmap_sand.ma".
16 include "basic_2/relocation/drops.ma".
18 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
20 (* Main properties **********************************************************)
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 →
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/
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-.
43 (* Basic_2A1: includes: lpx_sn_conf *)
44 theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
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/
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).
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).
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/
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/