(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div4: â\88\80f2,Tf,T. â¬\86*[f2] Tf â\89¡ T â\86\92 â\88\80g2,Tg. â¬\86*[g2] Tg â\89¡ T →
+theorem lifts_div4: â\88\80f2,Tf,T. â¬\86*[f2] Tf â\89\98 T â\86\92 â\88\80g2,Tg. â¬\86*[g2] Tg â\89\98 T →
∀f1,g1. H_at_div f2 g2 f1 g1 →
- â\88\83â\88\83T0. â¬\86*[f1] T0 â\89¡ Tf & â¬\86*[g1] T0 â\89¡ Tg.
+ â\88\83â\88\83T0. â¬\86*[f1] T0 â\89\98 Tf & â¬\86*[g1] T0 â\89\98 Tg.
#f2 #Tf #T #H elim H -f2 -Tf -T
[ #f2 #s #g2 #Tg #H #f1 #g1 #_
lapply (lifts_inv_sort2 … H) -H #H destruct
]
qed-.
-lemma lifts_div4_one: â\88\80f,Tf,T. â¬\86*[â\86\91f] Tf â\89¡ T →
- â\88\80T1. â¬\86*[1] T1 â\89¡ T →
- â\88\83â\88\83T0. â¬\86*[1] T0 â\89¡ Tf & â¬\86*[f] T0 â\89¡ T1.
+lemma lifts_div4_one: â\88\80f,Tf,T. â¬\86*[⫯f] Tf â\89\98 T →
+ â\88\80T1. â¬\86*[1] T1 â\89\98 T →
+ â\88\83â\88\83T0. â¬\86*[1] T0 â\89\98 Tf & â¬\86*[f] T0 â\89\98 T1.
/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
-theorem lifts_div3: â\88\80f2,T,T2. â¬\86*[f2] T2 â\89¡ T â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89¡ T →
- â\88\80f1. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f1] T1 â\89¡ T2.
+theorem lifts_div3: â\88\80f2,T,T2. â¬\86*[f2] T2 â\89\98 T â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89\98 T →
+ â\88\80f1. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f1] T1 â\89\98 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
(* Basic_1: was: lift1_lift1 (left to right) *)
(* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
(* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
-theorem lifts_trans: â\88\80f1,T1,T. â¬\86*[f1] T1 â\89¡ T â\86\92 â\88\80f2,T2. â¬\86*[f2] T â\89¡ T2 →
- â\88\80f. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f] T1 â\89¡ T2.
+theorem lifts_trans: â\88\80f1,T1,T. â¬\86*[f1] T1 â\89\98 T â\86\92 â\88\80f2,T2. â¬\86*[f2] T â\89\98 T2 →
+ â\88\80f. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f] T1 â\89\98 T2.
#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
qed-.
(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-theorem lifts_conf: â\88\80f1,T,T1. â¬\86*[f1] T â\89¡ T1 â\86\92 â\88\80f,T2. â¬\86*[f] T â\89¡ T2 →
- â\88\80f2. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f2] T1 â\89¡ T2.
+theorem lifts_conf: â\88\80f1,T,T1. â¬\86*[f1] T â\89\98 T1 â\86\92 â\88\80f,T2. â¬\86*[f] T â\89\98 T2 →
+ â\88\80f2. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f2] T1 â\89\98 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
(* Advanced proprerties *****************************************************)
(* Basic_2A1: includes: lift_inj *)
-lemma lifts_inj: ∀f,T1,U. ⬆*[f] T1 ≡ U → ∀T2. ⬆*[f] T2 ≡ U → T1 = T2.
+lemma lifts_inj: ∀f. is_inj2 … (lifts f).
#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
/3 width=6 by lifts_div3, lifts_fwd_isid/
qed-.
(* Basic_2A1: includes: lift_mono *)
-lemma lifts_mono: ∀f,T,U1. ⬆*[f] T ≡ U1 → ∀U2. ⬆*[f] T ≡ U2 → U1 = U2.
+lemma lifts_mono: ∀f,T. is_mono … (lifts f T).
#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
/3 width=6 by lifts_conf, lifts_fwd_isid/
qed-.