(* Properties with generic extension of a context sensitive relation ********)
lemma tc_lfxs_lex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ⪤[LTC … R] L2 → L1 ⪤**[R, T] L2.
+ ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤**[R, T] L2.
#R #HR #L1 #L2 #T *
/5 width=7 by tc_lfxs_tc, lexs_inv_tc_dx, lexs_co, ext2_inv_tc, ext2_refl/
qed.
lemma tc_lfxs_lex_lfeq: ∀R. c_reflexive … R →
- ∀L1,L. L1 ⪤[LTC … R] L → ∀L2,T. L ≐[T] L2 →
+ ∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
L1 ⪤**[R, T] L2.
/3 width=3 by tc_lfxs_lex, tc_lfxs_step_dx, lfeq_fwd_lfxs/ qed.
s_rs_transitive … R (λ_.lex R) →
lfeq_transitive R →
∀L1,L2,T. L1 ⪤**[R, T] L2 →
- ∃∃L. L1 ⪤[LTC … R] L & L ≐[T] L2.
+ ∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
@(tc_lfxs_ind_sn … H1R … H) -H -L2
lapply (lexs_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
lapply (sle_lexs_trans … HL02 … Hf21) -f1 // #HL02
- lapply (lexs_co ?? cfull (LTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
+ lapply (lexs_co ?? cfull (CTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
/8 width=11 by lexs_inv_tc_dx, lexs_tc_dx, lexs_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)
]
qed-.