(* Constructions with pcc ***************************************************)
-lemma lift_path_closed (o) (f) (q) (n):
- q ϵ 𝐂❨o,n❩ → 🠡[f]q ϵ 𝐂❨o,🠢[f]q@❨n❩❩.
-#o #f #q #n #H0 elim H0 -q -n //
+lemma lift_path_closed (o) (e) (f) (q) (n):
+ q ϵ 𝐂❨o,n,e❩ → 🠡[f]q ϵ 𝐂❨o,🠢[f]q@❨n❩,f@❨e❩❩.
+#o #e #f #q #n #H0 elim H0 -q -n //
#q #n [ #k #Ho ] #_ #IH
/2 width=1 by pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
/4 width=1 by pcc_d_dx, tr_xap_pos/
qed.
lemma lift_path_rmap_closed (o) (f) (p) (q) (n):
- q ϵ 𝐂❨o,n❩ → 🠡[🠢[f]p]q ϵ 𝐂❨o,🠢[f](p●q)@❨n❩❩.
+ q ϵ 𝐂❨o,n,𝟎❩ → 🠡[🠢[f]p]q ϵ 𝐂❨o,🠢[f](p●q)@❨n❩,𝟎❩.
/2 width=1 by lift_path_closed/
qed.
lemma lift_path_rmap_closed_L (o) (f) (p) (q) (n):
- q ϵ 𝐂❨o,n❩ → 🠡[🠢[f](p◖𝗟)]q ϵ 𝐂❨o,🠢[f](p●𝗟◗q)@§❨n❩❩.
+ q ϵ 𝐂❨o,n,𝟎❩ → 🠡[🠢[f](p◖𝗟)]q ϵ 𝐂❨o,🠢[f](p●𝗟◗q)@§❨n❩,𝟎❩.
#o #f #p #q #n #Hq
lapply (lift_path_closed … (🠢[f](p◖𝗟)) … Hq) #Hq0
lapply (pcc_L_sn … Hq) -Hq #Hq1