X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Flifts_lifts.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Flifts_lifts.ma;h=41c641716a280339f1c67ae399d534dead1e23aa;hb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;hp=0000000000000000000000000000000000000000;hpb=222044da28742b24584549ba86b1805a87def070;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma new file mode 100644 index 000000000..41c641716 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma @@ -0,0 +1,130 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "static_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-.