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 79c2000872024cb32a4282b396a39fe88055a84d..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
@@ -20,17 +20,48 @@ include "basic_2/relocation/lifts.ma".
 
 (* Basic_1: includes: lift_gen_lift *)
 (* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div: ∀f2,T,T2. ⬆*[f2] T2 ≡ T → ∀T1,f. ⬆*[f] T1 ≡ T →
-                   ∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
+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 #T1 #f #H >(lifts_inv_sort2 … H) -T1 //
-| #f2 #i2 #i #Hi2 #T1 #f #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
+[ #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 #T1 #f #H >(lifts_inv_gref2 … H) -T1 //
-| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #T1 #f #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_O2/
-| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #T1 #f #H
+| #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/
 ]
@@ -39,34 +70,34 @@ 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\80T2,f2. 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 #T2 #f2 #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i1 #i #Hi1 #T2 #f2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
+[ #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 #T2 #f2 #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #T2 #f2 #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_O2/
-| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #T2 #f2 #H
+| #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: â\88\80f1,T,T1. â¬\86*[f1] T â\89¡ T1 â\86\92 â\88\80T2,f. 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 #T2 #f #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i #i1 #Hi1 #T2 #f #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
+[ #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 #T2 #f #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #T2 #f #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_O2/
-| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #T2 #f #H
+| #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/
 ]
@@ -75,13 +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)
-/3 width=6 by lifts_div, lifts_fwd_isid/
+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-.