- ground_2: support for lifts_div4

[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..8477714f3d2c9ff7e30c3b8e2b319523a6bbdb88 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 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.
+#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 lift_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: ∀f1,T1,T. ⬆*[f1] T1 ≡ T → ∀T2,f2. ⬆*[f2] T ≡ T2 →
+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 #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: ∀f1,T,T1. ⬆*[f1] T ≡ T1 → ∀T2,f. ⬆*[f] T ≡ T2 →
+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 #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/
 ]
@@ -77,7 +108,7 @@ qed-.
 (* 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/
+/3 width=6 by lifts_div3, lifts_fwd_isid/
 qed-.
 
 (* Basic_2A1: includes: lift_mono *)