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. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- 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 #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. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- 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 #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by lexs_refl, lexs_push, step, inj/
qed.
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 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 #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 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 #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by lexs_tc_push_sn, lexs_tc_refl, trans_TC/
qed.