(* Main properties **********************************************************)
-(* Basic_2A1: includes: lift_inj *)
-theorem lifts_inj: ∀t,T1,U. ⬆*[t] T1 ≡ U → ∀T2. ⬆*[t] T2 ≡ U → T1 = T2.
-#t #T1 #U #H elim H -t -T1 -U
-[ /2 width=2 by lifts_inv_sort2/
-| #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref2 … HX) -HX
- /4 width=4 by at_inj, eq_f/
-| /2 width=2 by lifts_inv_gref2/
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind2 … HX) -HX
- #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat2 … HX) -HX
- #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-]
-qed-.
-
(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div: ∀T,T2,t2. ⬆*[t2] T2 ≡ T → ∀T1,t. ⬆*[t] T1 ≡ T →
- ∀t1. t2 ⊚ t1 ≡ t → ⬆*[t1] T1 ≡ T2.
-#T #T2 #t2 #H elim H -T -T2 -t2
-[ #k #t2 #T1 #t #H >(lifts_inv_sort2 … H) -T1 //
-| #i2 #i #t2 #Hi2 #T1 #t #H #t1 #Ht21 elim (lifts_inv_lref2 … H) -H
+theorem lifts_div: ∀T,T2,f2. ⬆*[f2] T2 ≡ T → ∀T1,f. ⬆*[f] T1 ≡ T →
+ ∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
+#T #T2 #f2 #H elim H -T -T2 -f2
+[ #s #f2 #T1 #f #H >(lifts_inv_sort2 … H) -T1 //
+| #i2 #i #f2 #Hi2 #T1 #f #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
#i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
-| #p #t2 #T1 #t #H >(lifts_inv_gref2 … H) -T1 //
-| #a #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H
+| #l #f2 #T1 #f #H >(lifts_inv_gref2 … H) -T1 //
+| #p #I #W2 #W #U2 #U #f2 #_ #_ #IHW #IHU #T1 #f #H
elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
- /4 width=3 by lifts_bind, after_true/
-| #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H
+ /4 width=3 by lifts_bind, after_O2/
+| #I #W2 #W #U2 #U #f2 #_ #_ #IHW #IHU #T1 #f #H
elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
/3 width=3 by lifts_flat/
]
qed-.
-(* Basic_2A1: includes: lift_mono *)
-theorem lifts_mono: ∀t,T,U1. ⬆*[t] T ≡ U1 → ∀U2. ⬆*[t] T ≡ U2 → U1 = U2.
-#t #T #U1 #H elim H -t -T -U1
-[ /2 width=2 by lifts_inv_sort1/
-| #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref1 … HX) -HX
- /4 width=4 by at_mono, eq_f/
-| /2 width=2 by lifts_inv_gref1/
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind1 … HX) -HX
- #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat1 … HX) -HX
- #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-]
-qed-.
-
(* 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: ∀T1,T,t1. ⬆*[t1] T1 ≡ T → ∀T2,t2. ⬆*[t2] T ≡ T2 →
- ∀t. t2 ⊚ t1 ≡ t → ⬆*[t] T1 ≡ T2.
-#T1 #T #t1 #H elim H -T1 -T -t1
-[ #k #t1 #T2 #t2 #H >(lifts_inv_sort1 … H) -T2 //
-| #i1 #i #t1 #Hi1 #T2 #t2 #H #t #Ht21 elim (lifts_inv_lref1 … H) -H
+theorem lifts_trans: ∀T1,T,f1. ⬆*[f1] T1 ≡ T → ∀T2,f2. ⬆*[f2] T ≡ T2 →
+ ∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1 ≡ T2.
+#T1 #T #f1 #H elim H -T1 -T -f1
+[ #s #f1 #T2 #f2 #H >(lifts_inv_sort1 … H) -T2 //
+| #i1 #i #f1 #Hi1 #T2 #f2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
#i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
-| #p #t1 #T2 #t2 #H >(lifts_inv_gref1 … H) -T2 //
-| #a #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H
+| #l #f1 #T2 #f2 #H >(lifts_inv_gref1 … H) -T2 //
+| #p #I #W1 #W #U1 #U #f1 #_ #_ #IHW #IHU #T2 #f2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /4 width=3 by lifts_bind, after_true/
-| #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H
+ /4 width=3 by lifts_bind, after_O2/
+| #I #W1 #W #U1 #U #f1 #_ #_ #IHW #IHU #T2 #f2 #H
elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/3 width=3 by lifts_flat/
]
qed-.
(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-theorem lifts_conf: ∀T,T1,t1. ⬆*[t1] T ≡ T1 → ∀T2,t. ⬆*[t] T ≡ T2 →
- ∀t2. t2 ⊚ t1 ≡ t → ⬆*[t2] T1 ≡ T2.
-#T #T1 #t1 #H elim H -T -T1 -t1
-[ #k #t1 #T2 #t #H >(lifts_inv_sort1 … H) -T2 //
-| #i #i1 #t1 #Hi1 #T2 #t #H #t2 #Ht21 elim (lifts_inv_lref1 … H) -H
+theorem lifts_conf: ∀T,T1,f1. ⬆*[f1] T ≡ T1 → ∀T2,f. ⬆*[f] T ≡ T2 →
+ ∀f2. f2 ⊚ f1 ≡ f → ⬆*[f2] T1 ≡ T2.
+#T #T1 #f1 #H elim H -T -T1 -f1
+[ #s #f1 #T2 #f #H >(lifts_inv_sort1 … H) -T2 //
+| #i #i1 #f1 #Hi1 #T2 #f #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
#i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
-| #p #t1 #T2 #t #H >(lifts_inv_gref1 … H) -T2 //
-| #a #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H
+| #l #f1 #T2 #f #H >(lifts_inv_gref1 … H) -T2 //
+| #p #I #W #W1 #U #U1 #f1 #_ #_ #IHW #IHU #T2 #f #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /4 width=3 by lifts_bind, after_true/
-| #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H
+ /4 width=3 by lifts_bind, after_O2/
+| #I #W #W1 #U #U1 #f1 #_ #_ #IHW #IHU #T2 #f #H
elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/3 width=3 by lifts_flat/
]
qed-.
+
+(* Advanced proprerties *****************************************************)
+
+(* Basic_2A1: includes: lift_inj *)
+lemma lifts_inj: ∀T1,U,f. ⬆*[f] T1 ≡ U → ∀T2. ⬆*[f] T2 ≡ U → T1 = T2.
+#T1 #U #f #H1 #T2 #H2 lapply (isid_after_dx 𝐈𝐝 f ?)
+/3 width=6 by lifts_div, lifts_fwd_isid/
+qed-.
+
+(* Basic_2A1: includes: lift_mono *)
+lemma lifts_mono: ∀T,U1,f. ⬆*[f] T ≡ U1 → ∀U2. ⬆*[f] T ≡ U2 → U1 = U2.
+#T #U1 #f #H1 #U2 #H2 lapply (isid_after_sn 𝐈𝐝 f ?)
+/3 width=6 by lifts_conf, lifts_fwd_isid/
+qed-.
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