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 "basic_2/relocation/lexs_lexs.ma".
16 include "basic_2/static/frees_fqup.ma".
17 include "basic_2/static/frees_frees.ma".
18 include "basic_2/static/lfxs.ma".
20 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
22 (* Advanced properties ******************************************************)
24 lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
25 lexs_frees_confluent … R1 cfull →
26 ∀L1,L2,V. L1 ⦻*[R1, V] L2 → ∀I,T.
27 ∃∃L. L1 ⦻*[R1, ②{I}V.T] L & L ⦻*[R2, V] L2.
28 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
29 [ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
30 elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
31 | elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
32 elim (frees_inv_flat … Hg) #y1 #y2 #H #_ #Hy
34 lapply(frees_mono … H … Hf) -H #H1
35 lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy
36 lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
37 elim (lexs_sle_split … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2
38 lapply (sle_lexs_trans … HL1 … Hfg) // #H
39 elim (HR … Hf … H) -HR -Hf -H
40 /4 width=7 by sle_lexs_trans, ex2_intro/
43 lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
44 lexs_frees_confluent … R1 cfull →
45 ∀L1,L2,T. L1 ⦻*[R1, T] L2 → ∀I,V.
46 ∃∃L. L1 ⦻*[R1, ⓕ{I}V.T] L & L ⦻*[R2, T] L2.
47 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
48 elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
49 elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
50 lapply(frees_mono … H … Hf) -H #H2
51 lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
52 lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
53 elim (lexs_sle_split … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2
54 lapply (sle_lexs_trans … HL1 … Hfg) // #H
55 elim (HR … Hf … H) -HR -Hf -H
56 /4 width=7 by sle_lexs_trans, ex2_intro/
59 (* Main properties **********************************************************)
61 theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
62 L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
63 L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
64 #R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
65 elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2))
66 /3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
69 theorem lfxs_flat: ∀R,I,L1,L2,V,T.
70 L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 →
72 #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
73 /3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
76 theorem lfxs_conf: ∀R1,R2.
77 lexs_frees_confluent R1 cfull →
78 lexs_frees_confluent R2 cfull →
79 R_confluent2_lfxs R1 R2 R1 R2 →
80 ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
81 #R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
82 lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
83 lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
84 elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
86 elim (HR1 … Hf … HL01) -HL01 #f1 #Hf1 #H1
87 elim (HR2 … Hf … HL02) -HL02 #f2 #Hf2 #H2
88 lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
89 lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
90 /3 width=5 by ex2_intro/
91 | #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02
92 elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
93 lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01
94 lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02
95 elim (HR12 … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/