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/relocation/rtmap_sand.ma".
16 include "static_2/relocation/drops.ma".
18 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
20 (* Main properties **********************************************************)
22 theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
24 (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_transitive_sex RN1 RN2 RN RN1 RP1 g K I) →
25 (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_transitive_sex 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 (sex_inv_atom1 … H1) -H1 #H destruct
32 lapply (sex_inv_atom1 … H2) -H2 #H destruct
33 /2 width=1 by sex_atom/
34 | #K1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
35 #HN #HP #L0 #H1 #L2 #H2
36 [ elim (sex_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
37 elim (sex_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 sex_push, drops_drop/
40 | elim (sex_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
41 elim (sex_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 sex_next, drops_drop/
48 theorem sex_trans (RN) (RP) (f):
49 (∀g,I,K. R_pw_transitive_sex RN RN RN RN RP g K I) →
50 (∀g,I,K. R_pw_transitive_sex RP RP RP RN RP g K I) →
51 Transitive … (sex RN RP f).
52 /2 width=9 by sex_trans_gen/ qed-.
54 theorem sex_trans_id_cfull (R1) (R2) (R3):
55 ∀L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈❨f❩ →
56 ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2.
57 #R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
58 [ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ]
59 #f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
60 [ elim (pr_isi_inv_next … Hf) | lapply (pr_isi_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
61 elim (sex_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
62 /3 width=1 by sex_push/
65 theorem sex_conf (RN1) (RP1) (RN2) (RP2):
67 (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
68 (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
69 pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L.
70 #RN1 #RP1 #RN2 #RP2 #L elim L -L
71 [ #f #_ #_ #L1 #H1 #L2 #H2 >(sex_inv_atom1 … H1) >(sex_inv_atom1 … H2) -H2 -H1
72 /2 width=3 by sex_atom, ex2_intro/
73 | #L #I0 #IH #f elim (pr_map_split_tl f) * #g #H destruct
74 #HN #HP #Y1 #H1 #Y2 #H2
75 [ elim (sex_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
76 elim (sex_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
77 elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
78 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, sex_push, ex2_intro/
79 | elim (sex_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
80 elim (sex_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
81 elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
82 elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, sex_next, ex2_intro/
87 lemma sex_repl (RN) (RP) (SN) (SP) (L1) (f):
88 (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_replace3_sex … RN SN RN RP SN SP g K1 I) →
89 (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_replace3_sex … RP SP RN RP SN SP g K1 I) →
90 ∀L2. L1 ⪤[RN,RP,f] L2 → ∀K1. L1 ⪤[SN,SP,f] K1 →
91 ∀K2. L2 ⪤[SN,SP,f] K2 → K1 ⪤[RN,RP,f] K2.
92 #RN #RP #SN #SP #L1 elim L1 -L1
93 [ #f #_ #_ #Y #HY #Y1 #HY1 #Y2 #HY2
94 lapply (sex_inv_atom1 … HY) -HY #H destruct
95 lapply (sex_inv_atom1 … HY1) -HY1 #H destruct
96 lapply (sex_inv_atom1 … HY2) -HY2 #H destruct //
97 | #L1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
98 #HN #HP #Y #HY #Y1 #HY1 #Y2 #HY2
99 [ elim (sex_inv_push1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct
100 elim (sex_inv_push1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct
101 elim (sex_inv_push1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct
102 /5 width=13 by sex_push, drops_refl, drops_drop/
103 | elim (sex_inv_next1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct
104 elim (sex_inv_next1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct
105 elim (sex_inv_next1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct
106 /5 width=13 by sex_next, drops_refl, drops_drop/
111 theorem sex_canc_sn (RN) (RP):
112 ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) →
113 left_cancellable … (sex RN RP f).
116 theorem sex_canc_dx (RN) (RP):
117 ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) →
118 right_cancellable … (sex RN RP f).
121 lemma sex_meet (RN) (RP) (L1) (L2):
122 ∀f1. L1 ⪤[RN,RP,f1] L2 →
123 ∀f2. L1 ⪤[RN,RP,f2] L2 →
124 ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
125 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
126 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
127 elim (pr_map_split_tl f2) * #g2 #H2 destruct
128 try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
129 [ elim (pr_sand_inv_next_push … Hf) | elim (pr_sand_inv_next_bi … Hf)
130 | elim (pr_sand_inv_push_bi … Hf) | elim (pr_sand_inv_push_next … Hf)
131 ] -Hf /3 width=5 by sex_next, sex_push/
134 lemma sex_join (RN) (RP) (L1) (L2):
135 ∀f1. L1 ⪤[RN,RP,f1] L2 →
136 ∀f2. L1 ⪤[RN,RP,f2] L2 →
137 ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
138 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
139 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
140 elim (pr_map_split_tl f2) * #g2 #H2 destruct
141 try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
142 [ elim (pr_sor_inv_next_push … Hf) | elim (pr_sor_inv_next_bi … Hf)
143 | elim (pr_sor_inv_push_bi … Hf) | elim (pr_sor_inv_push_next … Hf)
144 ] -Hf /3 width=5 by sex_next, sex_push/