(* 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 →
- ∀f1,g1. H_at_div f2 g2 f1 g1 →
+ ∀f1,g1. H_pr_pat_div f2 g2 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
- elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by at_div_pp/ ]
+ elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by pr_pat_div_push_bi/ ]
/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
qed-.
lemma lifts_div4_one: ∀f,Tf,T. ⇧*[⫯f] Tf ≘ T →
- ∀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
- #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
+ #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat_dx/
| #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
- #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
+ #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat/
| #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
qed-.
lemma lifts_trans4_one (f) (T1) (T2):
- ∀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 →
#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
- #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
+ #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat_sn/
| #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
(* Basic_2A1: includes: lift_inj *)
lemma lifts_inj: ∀f. is_inj2 … (lifts f).
-#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
+#f #T1 #U #H1 #T2 #H2 lapply (pr_after_isi_dx 𝐢 … f)
/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).
-#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
+#f #T #U1 #H1 #U2 #H2 lapply (pr_after_isi_sn 𝐢 … f)
/3 width=6 by lifts_conf, lifts_fwd_isid/
qed-.
qed-.
lemma lifts_trans_uni (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.
#T #l1 #T1 #HT1 #l2 #T2 #HT2
@(lifts_trans … HT1 … HT2) //
qed-.