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 "ground_2/relocation/rtmap_sor.ma".
17 include "basic_2/relocation/lexs.ma".
18 include "basic_2/relocation/drops.ma".
20 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
22 (* Main properties **********************************************************)
24 theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
25 lexs_transitive RN1 RN2 RN RN1 RP1 →
26 lexs_transitive RP1 RP2 RP RN1 RP1 →
27 ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
28 ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
30 #RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
31 [ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
32 | #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
33 #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
34 | #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
35 #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
39 (* Basic_2A1: includes: lpx_sn_trans *)
40 theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP →
41 lexs_transitive RP RP RP RN RP →
42 Transitive … (lexs RN RP f).
43 /2 width=9 by lexs_trans_gen/ qed-.
45 (* Basic_2A1: includes: lpx_sn_conf *)
46 theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
48 (∀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) →
49 (∀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) →
50 pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
51 #RN1 #RP1 #RN2 #RP2 #L elim L -L
52 [ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
53 /2 width=3 by lexs_atom, ex2_intro/
54 | #L #I #V #IH #f elim (pn_split f) * #g #H destruct
55 #HN #HP #Y1 #H1 #Y2 #H2
56 [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
57 elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
58 elim (HP … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
59 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
60 | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
61 elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
62 elim (HN … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
63 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
68 theorem lexs_canc_sn: ∀RN,RP,f. Transitive … (lexs RN RP f) →
69 symmetric … (lexs RN RP f) →
70 left_cancellable … (lexs RN RP f).
73 theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
74 symmetric … (lexs RN RP f) →
75 right_cancellable … (lexs RN RP f).
78 lemma lexs_meet: ∀RN,RP,L1,L2.
79 ∀f1. L1 ⦻*[RN, RP, f1] L2 →
80 ∀f2. L1 ⦻*[RN, RP, f2] L2 →
81 ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
82 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
83 #f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
84 elim (pn_split f2) * #g2 #H2 destruct
85 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
86 [ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
87 | elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
88 ] -Hf /3 width=5 by lexs_next, lexs_push/
91 lemma lexs_join: ∀RN,RP,L1,L2.
92 ∀f1. L1 ⦻*[RN, RP, f1] L2 →
93 ∀f2. L1 ⦻*[RN, RP, f2] L2 →
94 ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
95 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
96 #f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
97 elim (pn_split f2) * #g2 #H2 destruct
98 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
99 [ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
100 | elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
101 ] -Hf /3 width=5 by lexs_next, lexs_push/