+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lifts.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Main properties **********************************************************)
-
-(* 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-.
-
-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/
-| #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_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_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: ∀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/
-| #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_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: ∀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/
-| #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_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-.