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/lib/star.ma".
16 include "basic_2/relocation/lexs.ma".
18 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
20 definition s_rs_transitive_isid: relation (relation3 lenv bind bind) ≝ λRN,RP.
21 ∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.lexs RN RP f).
23 (* Properties with transitive closure ***************************************)
25 lemma lexs_tc_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
26 ∀f. reflexive … (TC … (lexs RN RP f)).
27 /3 width=1 by lexs_refl, TC_reflexive/ qed.
29 lemma lexs_tc_next_sn: ∀RN,RP. c_reflexive … RN →
30 ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RN L1 I1 I2 →
31 TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
32 #RN #RP #HRN #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
33 /3 width=3 by lexs_next, TC_strap, inj/
36 lemma lexs_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
37 ∀f,I1,I2,L1. (LTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
38 TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
39 #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
40 /4 width=5 by lexs_refl, lexs_next, step, inj/
43 lemma lexs_tc_push_sn: ∀RN,RP. c_reflexive … RP →
44 ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RP L1 I1 I2 →
45 TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
46 #RN #RP #HRP #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
47 /3 width=3 by lexs_push, TC_strap, inj/
50 lemma lexs_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
51 ∀f,I1,I2,L1. (LTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
52 TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
53 #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
54 /4 width=5 by lexs_refl, lexs_push, step, inj/
57 lemma lexs_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[LTC … RN, RP, f] L2.
58 #RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
59 /3 width=1 by lexs_push, lexs_next, inj/
62 lemma lexs_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[RN, LTC … RP, f] L2.
63 #RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
64 /3 width=1 by lexs_push, lexs_next, inj/
67 (* Main properties with transitive closure **********************************)
69 theorem lexs_tc_next: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
70 ∀f,I1,I2,L1. (LTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
71 TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
72 #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
73 /4 width=5 by lexs_tc_next_sn, lexs_tc_refl, trans_TC/
76 theorem lexs_tc_push: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
77 ∀f,I1,I2,L1. (LTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
78 TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
79 #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
80 /4 width=5 by lexs_tc_push_sn, lexs_tc_refl, trans_TC/
83 (* Basic_2A1: uses: TC_lpx_sn_ind *)
84 theorem lexs_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
85 ∀f,L1,L. L1 ⪤*[RN, RP, f] L → 𝐈⦃f⦄ →
86 ∀L2. L ⪤*[RN, LTC … RP, f] L2 → L1⪤* [RN, LTC … RP, f] L2.
87 #RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
88 [ #f #_ #Y #H -HRP >(lexs_inv_atom1 … H) -Y // ]
89 #f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
90 [ elim (isid_inv_next … Hf) -Hf //
91 | lapply (isid_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
92 elim (lexs_inv_push1 … H) -H #I2 #L2 #HL2 #HI2 #H destruct
93 @lexs_push [ /2 width=1 by/ ] -L2 -IH
94 @(TC_strap … HI1) -HI1
95 @(HRP … HL1) // (**) (* auto fails *)
99 (* Advanced properties ******************************************************)
101 (* Basic_2A1: uses: TC_lpx_sn_inv_lpx_sn_LTC *)
102 lemma lexs_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
103 ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (lexs RN RP f) L1 L2 → L1 ⪤*[RN, LTC … RP, f] L2.
104 #RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
105 /3 width=3 by lexs_tc_step_dx, lexs_tc_inj_dx/
108 (* Advanced inversion lemmas ************************************************)
110 lemma lexs_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
111 ∀f,L1,L2. L1 ⪤*[LTC … RN, RP, f] L2 → TC … (lexs RN RP f) L1 L2.
112 #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
113 /2 width=1 by lexs_tc_next, lexs_tc_push_sn, lexs_atom, inj/
116 (* Basic_2A1: uses: lpx_sn_LTC_TC_lpx_sn *)
117 lemma lexs_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
118 ∀f,L1,L2. L1 ⪤*[RN, LTC … RP, f] L2 → TC … (lexs RN RP f) L1 L2.
119 #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
120 /2 width=1 by lexs_tc_push, lexs_tc_next_sn, lexs_atom, inj/