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 â\80¦ (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ TC â\80¦ (lexs RN RP (â\86\91f)) (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. (LTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC â\80¦ (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
+ TC â\80¦ (lexs RN RP (â\86\91f)) (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 â\80¦ (lexs RN RP (â\86\91f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ TC â\80¦ (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. (LTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC â\80¦ (lexs RN RP (â\86\91f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
+ TC â\80¦ (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 ⪤*[LTC … RN, RP, f] L2.
+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, LTC … RP, f] L2.
+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. (LTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC â\80¦ (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
+ TC â\80¦ (lexs RN RP (â\86\91f)) (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. (LTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC â\80¦ (lexs RN RP (â\86\91f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
+ TC â\80¦ (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, LTC … RP, f] L2 → L1⪤* [RN, LTC … RP, f] L2.
+ ∀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
(* 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, LTC … RP, f] L2.
+ ∀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 ⪤*[LTC … RN, RP, f] L2 → TC … (lexs RN RP f) L1 L2.
+ ∀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, LTC … RP, f] L2 → TC … (lexs RN RP f) L1 L2.
+ ∀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-.