partial commit in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / relocation / lifts_lifts.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma
index 0581f6d1c3a28dc4b3b07b25473737a83e54a537..406436b9fb115c465e44de28936c73a85cc6be2c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_lifts.ma
@@ -20,9 +20,9 @@ include "static_2/relocation/lifts.ma".
 
 (* Basic_1: includes: lift_gen_lift *)
 (* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div4: â\88\80f2,Tf,T. â¬\86*[f2] Tf â\89\98 T â\86\92 â\88\80g2,Tg. â¬\86*[g2] Tg ≘ T →
-                    ∀f1,g1. H_at_div f2 g2 f1 g1 →
-                    â\88\83â\88\83T0. â¬\86*[f1] T0 â\89\98 Tf & â¬\86*[g1] T0 ≘ Tg.
+theorem lifts_div4: â\88\80f2,Tf,T. â\87§*[f2] Tf â\89\98 T â\86\92 â\88\80g2,Tg. â\87§*[g2] Tg ≘ T →
+                    ∀f1,g1. H_pr_pat_div f2 g2 f1 g1 →
+                    â\88\83â\88\83T0. â\87§*[f1] T0 â\89\98 Tf & â\87§*[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
@@ -36,7 +36,7 @@ theorem lifts_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≘ T → ∀g2,Tg. ⬆*[g2] Tg ≘
 | #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/ ]
+  elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by pr_pat_div_push_bi/ ]
   /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
@@ -46,17 +46,17 @@ theorem lifts_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≘ T → ∀g2,Tg. ⬆*[g2] Tg ≘
 ]
 qed-.
 
-lemma lifts_div4_one: â\88\80f,Tf,T. â¬\86*[⫯f] Tf ≘ T →
-                      â\88\80T1. â¬\86*[1] T1 ≘ T →
-                      â\88\83â\88\83T0. â¬\86*[1] T0 â\89\98 Tf & â¬\86*[f] T0 ≘ T1.
-/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
+lemma lifts_div4_one: â\88\80f,Tf,T. â\87§*[⫯f] Tf ≘ T →
+                      â\88\80T1. â\87§[1] T1 ≘ T →
+                      â\88\83â\88\83T0. â\87§[1] T0 â\89\98 Tf & â\87§*[f] T0 ≘ T1.
+/4 width=6 by lifts_div4, pr_pat_div_id_dx, pr_pat_div_push_next/ qed-.
 
-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 ≘ T →
-                    â\88\80f1. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f1] T1 ≘ T2.
+theorem lifts_div3: â\88\80f2,T,T2. â\87§*[f2] T2 â\89\98 T â\86\92 â\88\80f,T1. â\87§*[f] T1 ≘ T →
+                    â\88\80f1. f2 â\8a\9a f1 â\89\98 f â\86\92 â\87§*[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/
+  #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat_dx/
 | #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
@@ -70,12 +70,12 @@ 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\98 T â\86\92 â\88\80f2,T2. â¬\86*[f2] T ≘ T2 →
-                     â\88\80f. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f] T1 ≘ T2.
+theorem lifts_trans: â\88\80f1,T1,T. â\87§*[f1] T1 â\89\98 T â\86\92 â\88\80f2,T2. â\87§*[f2] T ≘ T2 →
+                     â\88\80f. f2 â\8a\9a f1 â\89\98 f â\86\92 â\87§*[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/
+  #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat/
 | #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
@@ -87,17 +87,17 @@ theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≘ T → ∀f2,T2. ⬆*[f2] T ≘
 qed-.
 
 lemma lifts_trans4_one (f) (T1) (T2):
-                       â\88\80T. â¬\86*[1]T1 â\89\98 T â\86\92 â¬\86*[⫯f]T ≘ T2 →
-                       â\88\83â\88\83T0. â¬\86*[f]T1 â\89\98 T0 & â¬\86*[1]T0 ≘ T2.
-/4 width=6 by lifts_trans, lifts_split_trans, after_uni_one_dx/ qed-.
+                       â\88\80T. â\87§[1]T1 â\89\98 T â\86\92 â\87§*[⫯f]T ≘ T2 →
+                       â\88\83â\88\83T0. â\87§*[f]T1 â\89\98 T0 & â\87§[1]T0 ≘ T2.
+/4 width=6 by lifts_trans, lifts_split_trans, pr_after_push_unit/ qed-.
 
 (* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-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 ≘ T2 →
-                    â\88\80f2. f2 â\8a\9a f1 â\89\98 f â\86\92 â¬\86*[f2] T1 ≘ T2.
+theorem lifts_conf: â\88\80f1,T,T1. â\87§*[f1] T â\89\98 T1 â\86\92 â\88\80f,T2. â\87§*[f] T ≘ T2 →
+                    â\88\80f2. f2 â\8a\9a f1 â\89\98 f â\86\92 â\87§*[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/
+  #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, pr_after_des_pat_sn/
 | #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
@@ -112,13 +112,13 @@ qed-.
 
 (* 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)
+#f #T1 #U #H1 #T2 #H2 lapply (pr_after_isi_dx 𝐢  … 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)
+#f #T #U1 #H1 #U2 #H2 lapply (pr_after_isi_sn 𝐢  … f)
 /3 width=6 by lifts_conf, lifts_fwd_isid/
 qed-.
 
@@ -133,3 +133,10 @@ lemma deliftable2_sn_bi: ∀R. deliftable2_sn R → deliftable2_bi R.
 elim (HR … HU12 … HTU1) -HR -U1 #X #HUX #HTX
 <(lifts_inj … HUX … HTU2) -U2 -T2 -f //
 qed-.
+
+lemma lifts_trans_uni (T):
+      ∀l1,T1. ⇧[l1] T1 ≘ T →
+      ∀l2,T2. ⇧[l2] T ≘ T2 → ⇧[l1+l2] T1 ≘ T2.
+#T #l1 #T1 #HT1 #l2 #T2 #HT2
+@(lifts_trans … HT1 … HT2) //
+qed-.