(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
theorem lifts_div4: ∀f2,Tf,T. ⇧*[f2] Tf ≘ T → ∀g2,Tg. ⇧*[g2] Tg ≘ T →
(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
theorem lifts_div4: ∀f2,Tf,T. ⇧*[f2] Tf ≘ T → ∀g2,Tg. ⇧*[g2] Tg ≘ T →
∃∃T0. ⇧*[f1] T0 ≘ Tf & ⇧*[g1] T0 ≘ Tg.
#f2 #Tf #T #H elim H -f2 -Tf -T
[ #f2 #s #g2 #Tg #H #f1 #g1 #_
∃∃T0. ⇧*[f1] T0 ≘ Tf & ⇧*[g1] T0 ≘ Tg.
#f2 #Tf #T #H elim H -f2 -Tf -T
[ #f2 #s #g2 #Tg #H #f1 #g1 #_
| #f2 #p #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
elim (lifts_inv_bind2 … H) -H #Vg #Tg #HVg #HTg #H destruct
elim (IHV … HVg … H0) -IHV -HVg
| #f2 #p #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
elim (lifts_inv_bind2 … H) -H #Vg #Tg #HVg #HTg #H destruct
elim (IHV … HVg … H0) -IHV -HVg
/3 width=5 by lifts_bind, ex2_intro/
| #f2 #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
elim (lifts_inv_flat2 … H) -H #Vg #Tg #HVg #HTg #H destruct
/3 width=5 by lifts_bind, ex2_intro/
| #f2 #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
elim (lifts_inv_flat2 … H) -H #Vg #Tg #HVg #HTg #H destruct
- ∀T1. ⇧*[1] T1 ≘ T →
- ∃∃T0. ⇧*[1] T0 ≘ Tf & ⇧*[f] T0 ≘ T1.
-/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
+ ∀T1. ⇧[1] T1 ≘ T →
+ ∃∃T0. ⇧[1] T0 ≘ Tf & ⇧*[f] T0 ≘ T1.
+/4 width=6 by lifts_div4, pr_pat_div_id_dx, pr_pat_div_push_next/ qed-.
theorem lifts_div3: ∀f2,T,T2. ⇧*[f2] T2 ≘ T → ∀f,T1. ⇧*[f] T1 ≘ T →
∀f1. f2 ⊚ f1 ≘ f → ⇧*[f1] T1 ≘ T2.
#f2 #T #T2 #H elim H -f2 -T -T2
[ #f2 #s #f #T1 #H >(lifts_inv_sort2 … H) -T1 //
| #f2 #i2 #i #Hi2 #f #T1 #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
theorem lifts_div3: ∀f2,T,T2. ⇧*[f2] T2 ≘ T → ∀f,T1. ⇧*[f] T1 ≘ T →
∀f1. f2 ⊚ f1 ≘ f → ⇧*[f1] T1 ≘ T2.
#f2 #T #T2 #H elim H -f2 -T -T2
[ #f2 #s #f #T1 #H >(lifts_inv_sort2 … H) -T1 //
| #f2 #i2 #i #Hi2 #f #T1 #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
| #f2 #l #f #T1 #H >(lifts_inv_gref2 … H) -T1 //
| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
| #f2 #l #f #T1 #H >(lifts_inv_gref2 … H) -T1 //
| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
#f1 #T1 #T #H elim H -f1 -T1 -T
[ #f1 #s #f2 #T2 #H >(lifts_inv_sort1 … H) -T2 //
| #f1 #i1 #i #Hi1 #f2 #T2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
#f1 #T1 #T #H elim H -f1 -T1 -T
[ #f1 #s #f2 #T2 #H >(lifts_inv_sort1 … H) -T2 //
| #f1 #i1 #i #Hi1 #f2 #T2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
| #f1 #l #f2 #T2 #H >(lifts_inv_gref1 … H) -T2 //
| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
| #f1 #l #f2 #T2 #H >(lifts_inv_gref1 … H) -T2 //
| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- ∀T. ⇧*[1]T1 ≘ T → ⇧*[⫯f]T ≘ T2 →
- ∃∃T0. ⇧*[f]T1 ≘ T0 & ⇧*[1]T0 ≘ T2.
-/4 width=6 by lifts_trans, lifts_split_trans, after_uni_one_dx/ qed-.
+ ∀T. ⇧[1]T1 ≘ T → ⇧*[⫯f]T ≘ T2 →
+ ∃∃T0. ⇧*[f]T1 ≘ T0 & ⇧[1]T0 ≘ T2.
+/4 width=6 by lifts_trans, lifts_split_trans, pr_after_push_unit/ qed-.
(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
theorem lifts_conf: ∀f1,T,T1. ⇧*[f1] T ≘ T1 → ∀f,T2. ⇧*[f] T ≘ T2 →
(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
theorem lifts_conf: ∀f1,T,T1. ⇧*[f1] T ≘ T1 → ∀f,T2. ⇧*[f] T ≘ T2 →
#f1 #T #T1 #H elim H -f1 -T -T1
[ #f1 #s #f #T2 #H >(lifts_inv_sort1 … H) -T2 //
| #f1 #i #i1 #Hi1 #f #T2 #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
#f1 #T #T1 #H elim H -f1 -T -T1
[ #f1 #s #f #T2 #H >(lifts_inv_sort1 … H) -T2 //
| #f1 #i #i1 #Hi1 #f #T2 #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
| #f1 #l #f #T2 #H >(lifts_inv_gref1 … H) -T2 //
| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
| #f1 #l #f #T2 #H >(lifts_inv_gref1 … H) -T2 //
| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/3 width=6 by lifts_div3, lifts_fwd_isid/
qed-.
(* Basic_2A1: includes: lift_mono *)
lemma lifts_mono: ∀f,T. is_mono … (lifts f T).
/3 width=6 by lifts_div3, lifts_fwd_isid/
qed-.
(* Basic_2A1: includes: lift_mono *)
lemma lifts_mono: ∀f,T. is_mono … (lifts f T).
- ∀l1,T1. ⇧*[l1] T1 ≘ T →
- ∀l2,T2. ⇧*[l2] T ≘ T2 → ⇧*[l1+l2] T1 ≘ T2.
+ ∀l1,T1. ⇧[l1] T1 ≘ T →
+ ∀l2,T2. ⇧[l2] T ≘ T2 → ⇧[l1+l2] T1 ≘ T2.