(* *)
(**************************************************************************)
+include "delayed_updating/unwind/unwind2_rmap_lift.ma".
include "delayed_updating/unwind/unwind2_rmap_eq.ma".
+include "delayed_updating/substitution/lift_rmap_structure.ma".
include "delayed_updating/syntax/path_closed.ma".
include "delayed_updating/syntax/path_depth.ma".
include "ground/relocation/nap.ma".
+include "ground/relocation/tr_pushs_tls.ma".
include "ground/lib/stream_tls_plus.ma".
include "ground/lib/stream_eq_eq.ma".
(* Destructions with cpp ****************************************************)
-lemma nap_plus_unwind2_rmap_closed (o) (f) (q) (m) (n):
- q ϵ 𝐂❨o,n❩ →
- f@§❨m❩+♭q = ▶[f]q@§❨m+n❩.
-#o #f #q #m #n #Hq elim Hq -q -n //
+lemma nap_plus_unwind2_rmap_closed (o) (e) (f) (q) (m) (n):
+ q ϵ 𝐂❨o,n,e❩ →
+ f@§❨m+e❩+♭q = ▶[f]q@§❨m+n❩.
+#o #e #f #q #m #n #Hq elim Hq -q -n //
#q #n [ #k #_ ] #_ #IH
[ <depth_d_dx <unwind2_rmap_d_dx
<tr_compose_nap <tr_uni_nap //
]
qed-.
+lemma nap_unwind2_rmap_closed (o) (f) (q) (n):
+ q ϵ 𝐂❨o,n,𝟎❩ →
+ f@§❨𝟎❩+♭q = ▶[f]q@§❨n❩.
+#o #f #q #n #Hn
+/2 width=2 by nap_plus_unwind2_rmap_closed/
+qed-.
+
lemma nap_plus_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (m) (n):
- q ϵ 𝐂❨o,n❩ →
+ q ϵ 𝐂❨o,n,𝟎❩ →
(⫯▶[f]p)@§❨m❩+♭q = ▶[f](p●𝗟◗q)@§❨m+n❩.
#o #f #p #q #m #n #Hn
/2 width=2 by nap_plus_unwind2_rmap_closed/
qed-.
lemma nap_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
- q ϵ 𝐂❨o,n❩ →
+ q ϵ 𝐂❨o,n,𝟎❩ →
♭q = ▶[f](p●𝗟◗q)@§❨n❩.
#o #f #p #q #n #Hn
>(nplus_zero_sn n)
qed-.
lemma tls_succ_plus_unwind2_rmap_push_closed (o) (f) (q) (n):
- q ϵ 𝐂❨o,n❩ →
+ q ϵ 𝐂❨o,n,𝟎❩ →
∀m. ⇂*[m]f ≗ ⇂*[↑(m+n)]▶[⫯f]q.
#o #f #q #n #Hn elim Hn -q -n //
#q #n #k #_ #_ #IH #m
qed-.
lemma tls_succ_unwind2_rmap_push_closed (o) (f) (q) (n):
- q ϵ 𝐂❨o,n❩ →
- f ≗ ⇂*[↑n]▶[⫯f](q).
+ q ϵ 𝐂❨o,n,𝟎❩ →
+ f ≗ ⇂*[↑n]▶[⫯f]q.
#o #f #q #n #Hn
/2 width=2 by tls_succ_plus_unwind2_rmap_push_closed/
qed-.
lemma tls_succ_plus_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
- q ϵ 𝐂❨o,n❩ →
+ q ϵ 𝐂❨o,n,𝟎❩ →
∀m. ⇂*[m]▶[f]p ≗ ⇂*[↑(m+n)]▶[f](p●𝗟◗q).
#o #f #p #q #n #Hn #m
/2 width=2 by tls_succ_plus_unwind2_rmap_push_closed/
qed-.
+
+lemma tls_succ_unwind2_rmap_closed (f) (q) (n):
+ q ϵ 𝐂❨Ⓕ,n,𝟎❩ →
+ ⇂f ≗ ⇂*[↑n]▶[f]q.
+#f #q #n #Hn
+@(stream_eq_canc_dx … (stream_tls_eq_repl …))
+[| @(unwind2_rmap_eq_repl … (tr_compose_id_dx …)) | skip ]
+@(stream_eq_trans … (stream_tls_eq_repl …))
+[| @(lift_unwind2_rmap_after … ) | skip ]
+<tr_compose_tls <tr_id_unfold
+@(stream_eq_trans … (tr_compose_eq_repl …))
+[| @(tls_succ_unwind2_rmap_push_closed … Hn) | skip | // | skip ]
+@(stream_eq_trans ????? (tr_compose_id_dx …))
+<tr_pap_succ_nap <(nap_unwind2_rmap_closed … Hn) <nplus_zero_sn
+<lift_rmap_structure <stream_tls_succ <tr_tls_pushs //
+qed-.