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 *)
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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):
24 (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) →
25 (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) →
26 ∀L0. L1 ⪤*[RN1, RP1, f] L0 →
27 ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
29 #RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
30 [ #f #_ #_ #L0 #H1 #L2 #H2
31 lapply (lexs_inv_atom1 … H1) -H1 #H destruct
32 lapply (lexs_inv_atom1 … H2) -H2 #H destruct
33 /2 width=1 by lexs_atom/
34 | #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
35 #HN #HP #L0 #H1 #L2 #H2
36 [ elim (lexs_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
37 elim (lexs_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
38 lapply (HP … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
39 lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_push, drops_drop/
40 | elim (lexs_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
41 elim (lexs_inv_next1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
42 lapply (HN … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
43 lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_next, drops_drop/
48 theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) →
49 (∀g,I,K. lexs_transitive RP RP RP RN RP g K I) →
50 Transitive … (lexs RN RP f).
51 /2 width=9 by lexs_trans_gen/ qed-.
53 theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ →
54 ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2.
55 #R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
56 [ #f #Hf #L2 #H >(lexs_inv_atom1 … H) -L2 // ]
57 #f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
58 [ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
59 elim (lexs_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
60 /3 width=1 by lexs_push/
63 theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
65 (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
66 (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
67 pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
68 #RN1 #RP1 #RN2 #RP2 #L elim L -L
69 [ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
70 /2 width=3 by lexs_atom, ex2_intro/
71 | #L #I0 #IH #f elim (pn_split f) * #g #H destruct
72 #HN #HP #Y1 #H1 #Y2 #H2
73 [ elim (lexs_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
74 elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
75 elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
76 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
77 | elim (lexs_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
78 elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
79 elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
80 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
85 theorem lexs_canc_sn: ∀RN,RP,f. Transitive … (lexs RN RP f) →
86 symmetric … (lexs RN RP f) →
87 left_cancellable … (lexs RN RP f).
90 theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
91 symmetric … (lexs RN RP f) →
92 right_cancellable … (lexs RN RP f).
95 lemma lexs_meet: ∀RN,RP,L1,L2.
96 ∀f1. L1 ⪤*[RN, RP, f1] L2 →
97 ∀f2. L1 ⪤*[RN, RP, f2] L2 →
98 ∀f. f1 ⋒ f2 ≘ f → L1 ⪤*[RN, RP, f] L2.
99 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
100 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
101 elim (pn_split f2) * #g2 #H2 destruct
102 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
103 [ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
104 | elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
105 ] -Hf /3 width=5 by lexs_next, lexs_push/
108 lemma lexs_join: ∀RN,RP,L1,L2.
109 ∀f1. L1 ⪤*[RN, RP, f1] L2 →
110 ∀f2. L1 ⪤*[RN, RP, f2] L2 →
111 ∀f. f1 ⋓ f2 ≘ f → L1 ⪤*[RN, RP, f] L2.
112 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
113 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
114 elim (pn_split f2) * #g2 #H2 destruct
115 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
116 [ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
117 | elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
118 ] -Hf /3 width=5 by lexs_next, lexs_push/