(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
lemma rexs_pair_refl: ∀R. c_reflexive … R →
- ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
+ ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R,T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
/3 width=3 by rexs_step_dx, rex_pair_refl, inj/
qed.
lemma rexs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (sex cfull (cext2 R) f) L1 L2 →
- L1 ⪤*[R, T] L2.
+ L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #f #Hf #H elim H -L2
[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
/5 width=7 by sex_sdj, rexs_step_dx, sdj_isid_sn, inj, ex2_intro/
lemma rexs_ind_sn: ∀R. c_reflexive … R →
∀L1,T. ∀Q:predicate …. Q L1 →
- (∀L,L2. L1 ⪤*[R, T] L → L ⪤[R, T] L2 → Q L → Q L2) →
- ∀L2. L1 ⪤*[R, T] L2 → Q L2.
+ (∀L,L2. L1 ⪤*[R,T] L → L ⪤[R,T] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤*[R,T] L2 → Q L2.
#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by rex_refl/
qed-.
lemma rexs_ind_dx: ∀R. c_reflexive … R →
∀L2,T. ∀Q:predicate …. Q L2 →
- (∀L1,L. L1 ⪤[R, T] L → L ⪤*[R, T] L2 → Q L → Q L1) →
- ∀L1. L1 ⪤*[R, T] L2 → Q L1.
+ (∀L1,L. L1 ⪤[R,T] L → L ⪤*[R,T] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤*[R,T] L2 → Q L1.
#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by rex_refl/
qed-.
(* Advanced inversion lemmas ************************************************)
lemma rexs_inv_bind_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H @(rexs_ind_sn … HR … H) -L2
[ /3 width=1 by rexs_refl, conj/
| #L #L2 #_ #H * elim (rex_inv_bind_void … H) -H /3 width=3 by rexs_step_dx, conj/
(* Advanced forward lemmas **************************************************)
lemma rexs_fwd_bind_dx_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind_void … H) -H //
qed-.