renaming in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lifts_lifts.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma
index 8477714f3d2c9ff7e30c3b8e2b319523a6bbdb88..9d7fc61bfee120789fca2059f3d13885132e1c00 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma
@@ -20,9 +20,9 @@ include "basic_2/relocation/lifts.ma".
 
 (* Basic_1: includes: lift_gen_lift *)
 (* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lift_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.
+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
@@ -46,13 +46,13 @@ theorem lift_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≡ T → ∀g2,Tg. ⬆*[g2] Tg ≡ T
 ]
 qed-.
 
-lemma lifts_div4_one: â\88\80f,Tf,T. â¬\86*[â\86\91f] Tf â\89¡ T →
-                      â\88\80T1. â¬\86*[1] T1 â\89¡ T →
-                      â\88\83â\88\83T0. â¬\86*[1] T0 â\89¡ Tf & â¬\86*[f] T0 â\89¡ T1.
-/4 width=6 by lift_div4, at_div_id_dx, at_div_pn/ qed-.
+lemma lifts_div4_one: â\88\80f,Tf,T. â¬\86*[⫯f] Tf â\89\98 T →
+                      â\88\80T1. â¬\86*[1] T1 â\89\98 T →
+                      â\88\83â\88\83T0. â¬\86*[1] T0 â\89\98 Tf & â¬\86*[f] T0 â\89\98 T1.
+/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
 
-theorem lifts_div3: â\88\80f2,T,T2. â¬\86*[f2] T2 â\89¡ T â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89¡ T →
-                    â\88\80f1. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f1] T1 â\89¡ T2.
+theorem lifts_div3: â\88\80f2,T,T2. â¬\86*[f2] T2 â\89\98 T â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89\98 T →
+                    â\88\80f1. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f1] T1 â\89\98 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
@@ -70,8 +70,8 @@ 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: â\88\80f1,T1,T. â¬\86*[f1] T1 â\89¡ T â\86\92 â\88\80f2,T2. â¬\86*[f2] T â\89¡ T2 →
-                     â\88\80f. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f] T1 â\89¡ T2.
+theorem lifts_trans: â\88\80f1,T1,T. â¬\86*[f1] T1 â\89\98 T â\86\92 â\88\80f2,T2. â¬\86*[f2] T â\89\98 T2 →
+                     â\88\80f. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f] T1 â\89\98 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
@@ -87,8 +87,8 @@ theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≡ T → ∀f2,T2. ⬆*[f2] T ≡
 qed-.
 
 (* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-theorem lifts_conf: â\88\80f1,T,T1. â¬\86*[f1] T â\89¡ T1 â\86\92 â\88\80f,T2. â¬\86*[f] T â\89¡ T2 →
-                    â\88\80f2. f2 â\8a\9a f1 â\89¡ f â\86\92 â¬\86*[f2] T1 â\89¡ T2.
+theorem lifts_conf: â\88\80f1,T,T1. â¬\86*[f1] T â\89\98 T1 â\86\92 â\88\80f,T2. â¬\86*[f] T â\89\98 T2 →
+                    â\88\80f2. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f2] T1 â\89\98 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
@@ -106,13 +106,25 @@ qed-.
 (* Advanced proprerties *****************************************************)
 
 (* Basic_2A1: includes: lift_inj *)
-lemma lifts_inj: ∀f,T1,U. ⬆*[f] T1 ≡ U → ∀T2. ⬆*[f] T2 ≡ U → T1 = T2.
-#f #T1 #U #H1 #T2 #H2 lapply (isid_after_dx 𝐈𝐝  … f)
+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,U1. ⬆*[f] T ≡ U1 → ∀U2. ⬆*[f] T ≡ U2 → U1 = U2.
-#f #T #U1 #H1 #U2 #H2 lapply (isid_after_sn 𝐈𝐝  … f)
+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-.