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/grammar/lenv_weight.ma".
18 include "basic_2/relocation/lexs.ma".
20 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
22 (* Main properties **********************************************************)
24 (* Basic_2A1: includes: lpx_sn_trans *)
25 theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RP →
26 lexs_transitive RP RP RN RP →
27 Transitive … (lexs RN RP f).
28 #RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
29 [ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
30 | #I #K1 #K #V1 #V #f #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 | #I #K1 #K #V1 #V #f #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_conf *)
38 theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
39 lpx_sn_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
40 lpx_sn_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
41 ∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f).
42 #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 generalize in match f; -f
43 @(f_ind … lw … L0) -L0 #x #IH *
44 [ #_ #f #X1 #H1 #X2 #H2 -x
45 >(lexs_inv_atom1 … H1) -X1
46 >(lexs_inv_atom1 … H2) -X2 /2 width=3 by lexs_atom, ex2_intro/
47 | #L0 #I #V0 #Hx #f elim (pn_split f) *
48 #g #H #X1 #H1 #X2 #H2 destruct
49 [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
50 elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
51 elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
52 elim (HRP … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_push, ex2_intro/
53 | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
54 elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
55 elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
56 elim (HRN … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_next, ex2_intro/
61 theorem lexs_canc_sn: ∀RN,RP,f. Transitive … (lexs RN RP f) →
62 symmetric … (lexs RN RP f) →
63 left_cancellable … (lexs RN RP f).
66 theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
67 symmetric … (lexs RN RP f) →
68 right_cancellable … (lexs RN RP f).
71 theorem lexs_meet: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
72 ∀f2. L1 ⦻*[RN, RP, f2] L2 →
73 ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
74 #RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
75 #I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
76 #g2 #H #H2 #f #Hf destruct
77 [1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
79 [ elim (sand_inv_npx … Hf) | elim (sand_inv_ppx … Hf) | elim (sand_inv_nnx … Hf) | elim (sand_inv_pnx … Hf) ] -Hf
80 /3 width=5 by lexs_next, lexs_push/
83 theorem lexs_join: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
84 ∀f2. L1 ⦻*[RN, RP, f2] L2 →
85 ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
86 #RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
87 #I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
88 #g2 #H #H2 #f #Hf destruct
89 [1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
91 [ elim (sor_inv_npx … Hf) | elim (sor_inv_ppx … Hf) | elim (sor_inv_nnx … Hf) | elim (sor_inv_pnx … Hf) ] -Hf
92 /3 width=5 by lexs_next, lexs_push/