include "basic_2/relocation/lifts.ma".
-(* GENERIC RELOCATION *******************************************************)
+(* GENERIC RELOCATION FOR TERMS *********************************************)
(* 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/
+(* 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 →
+ ∃∃T0. ⬆*[f1] T0 ≘ Tf & ⬆*[g1] T0 ≘ 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
+ /2 width=3 by ex2_intro/
+| #f2 #jf #j #Hf2 #g2 #Tg #H #f1 #g1 #H0
+ elim (lifts_inv_lref2 … H) -H #jg #Hg2 #H destruct
+ elim (H0 … Hf2 Hg2) -H0 -j /3 width=3 by lifts_lref, ex2_intro/
+| #f2 #l #g2 #Tg #H #f1 #g1 #_
+ lapply (lifts_inv_gref2 … H) -H #H destruct
+ /2 width=3 by ex2_intro/
+| #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/ ]
+ /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
+ elim (IHV … HVg … H0) -IHV -HVg
+ elim (IHT … HTg … H0) -IHT -HTg -H0
+ /3 width=5 by lifts_flat, ex2_intro/
]
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
+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-.
+
+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/
-| #p #t2 #T1 #t #H >(lifts_inv_gref2 … H) -T1 //
-| #a #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #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
- /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/
+| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #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: ∀f1,T1,T. ⬆*[f1] T1 ≘ T → ∀f2,T2. ⬆*[f2] T ≘ T2 →
+ ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] T1 ≘ 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
#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
+| #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
- /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/
+| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #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: ∀f1,T,T1. ⬆*[f1] T ≘ T1 → ∀f,T2. ⬆*[f] T ≘ T2 →
+ ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] T1 ≘ 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/
-| #p #t1 #T2 #t #H >(lifts_inv_gref1 … H) -T2 //
-| #a #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #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
- /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/
+| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #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: ∀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. 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-.
+
+lemma liftable2_sn_bi: ∀R. liftable2_sn R → liftable2_bi R.
+#R #HR #T1 #T2 #HT12 #f #U1 #HTU1 #U2 #HTU2
+elim (HR … HT12 … HTU1) -HR -T1 #X #HTX #HUX
+<(lifts_mono … HTX … HTU2) -T2 -U2 -f //
+qed-.
+
+lemma deliftable2_sn_bi: ∀R. deliftable2_sn R → deliftable2_bi R.
+#R #HR #U1 #U2 #HU12 #f #T1 #HTU1 #T2 #HTU2
+elim (HR … HU12 … HTU1) -HR -U1 #X #HUX #HTX
+<(lifts_inj … HUX … HTU2) -U2 -T2 -f //
+qed-.
projects / helm.git / blobdiff