+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/star.ma".
-include "basic_2/relocation/lexs.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-definition s_rs_transitive_isid: relation (relation3 lenv bind bind) ≝ λRN,RP.
- ∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.lexs RN RP f).
-
-(* Properties with transitive closure ***************************************)
-
-lemma lexs_tc_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f. reflexive … (TC … (lexs RN RP f)).
-/3 width=1 by lexs_refl, TC_reflexive/ qed.
-
-lemma lexs_tc_next_sn: ∀RN,RP. c_reflexive … RN →
- ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RN L1 I1 I2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_next, TC_strap, inj/
-qed.
-
-lemma lexs_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_refl, lexs_next, step, inj/
-qed.
-
-lemma lexs_tc_push_sn: ∀RN,RP. c_reflexive … RP →
- ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RP L1 I1 I2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRP #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_push, TC_strap, inj/
-qed.
-
-lemma lexs_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_refl, lexs_push, step, inj/
-qed.
-
-lemma lexs_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[CTC … RN, RP, f] L2.
-#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by lexs_push, lexs_next, inj/
-qed.
-
-lemma lexs_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[RN, CTC … RP, f] L2.
-#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by lexs_push, lexs_next, inj/
-qed.
-
-(* Main properties with transitive closure **********************************)
-
-theorem lexs_tc_next: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_tc_next_sn, lexs_tc_refl, trans_TC/
-qed.
-
-theorem lexs_tc_push: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_tc_push_sn, lexs_tc_refl, trans_TC/
-qed.
-
-(* Basic_2A1: uses: TC_lpx_sn_ind *)
-theorem lexs_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f,L1,L. L1 ⪤*[RN, RP, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤*[RN, CTC … RP, f] L2 → L1⪤* [RN, CTC … RP, f] L2.
-#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
-[ #f #_ #Y #H -HRP >(lexs_inv_atom1 … H) -Y // ]
-#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
-[ elim (isid_inv_next … Hf) -Hf //
-| lapply (isid_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
- elim (lexs_inv_push1 … H) -H #I2 #L2 #HL2 #HI2 #H destruct
- @lexs_push [ /2 width=1 by/ ] -L2 -IH
- @(TC_strap … HI1) -HI1
- @(HRP … HL1) // (**) (* auto fails *)
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lexs_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (lexs RN RP f) L1 L2 → L1 ⪤*[RN, CTC … RP, f] L2.
-#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_tc_step_dx, lexs_tc_inj_dx/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lexs_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤*[CTC … RN, RP, f] L2 → TC … (lexs RN RP f) L1 L2.
-#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
-/2 width=1 by lexs_tc_next, lexs_tc_push_sn, lexs_atom, inj/
-qed-.
-
-lemma lexs_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤*[RN, CTC … RP, f] L2 → TC … (lexs RN RP f) L1 L2.
-#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
-/2 width=1 by lexs_tc_push, lexs_tc_next_sn, lexs_atom, inj/
-qed-.