include "delayed_updating/unwind/unwind2_rmap_eq.ma".
include "delayed_updating/syntax/path_closed.ma".
include "delayed_updating/syntax/path_depth.ma".
-include "ground/relocation/xap.ma".
+include "ground/relocation/nap.ma".
include "ground/lib/stream_tls_plus.ma".
include "ground/lib/stream_eq_eq.ma".
-include "ground/arith/nat_le_plus.ma".
-include "ground/arith/nat_le_pred.ma".
(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
(* Destructions with cpp ****************************************************)
-lemma xap_le_unwind2_rmap_append_closed_dx (o) (f) (p) (q) (n):
- q ϵ 𝐂❨o,n❩ → ∀m. m ≤ n →
- ▶[f]q＠❨m❩ = ▶[f](p●q)＠❨m❩.
-#o #f #p #q #n #Hq elim Hq -q -n
-[|*: #q #n [ #k #_ ] #_ #IH ] #m #Hm
-[ <(nle_inv_zero_dx … Hm) -m //
-| <unwind2_rmap_d_dx <unwind2_rmap_d_dx
- <tr_compose_xap <tr_compose_xap
- @IH -IH (**) (* auto too slow *)
- @nle_trans [| @tr_uni_xap ]
- /2 width=1 by nle_plus_bi_dx/
-| <unwind2_rmap_m_dx <unwind2_rmap_m_dx
- /2 width=2 by/
-| <unwind2_rmap_L_dx <unwind2_rmap_L_dx
- elim (nle_inv_succ_dx … Hm) -Hm // * #Hm #H0
- >H0 -H0 <tr_xap_push <tr_xap_push
- /3 width=2 by eq_f/
-| <unwind2_rmap_A_dx <unwind2_rmap_A_dx
- /2 width=2 by/
-| <unwind2_rmap_S_dx <unwind2_rmap_S_dx
- /2 width=2 by/
-]
-qed-.
-
-lemma nap_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
+lemma nap_plus_unwind2_rmap_closed (o) (f) (q) (m) (n):
q ϵ 𝐂❨o,n❩ →
- ▶[f](𝗟◗q)＠§❨n❩ = ▶[f](p●𝗟◗q)＠§❨n❩.
-#o #f #p #q #n #Hq
-lapply (pcc_L_sn … Hq) -Hq #Hq
-lapply (xap_le_unwind2_rmap_append_closed_dx o f p … Hq (↑n) ?) -Hq //
-<tr_xap_succ_nap <tr_xap_succ_nap #Hq
-/2 width=1 by eq_inv_nsucc_bi/
+ f＠§❨m❩+♭q = ▶[f]q＠§❨m+n❩.
+#o #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 //
+| <depth_L_dx <unwind2_rmap_L_dx
+ <tr_nap_push <nplus_succ_dx //
+]
qed-.
-lemma nap_unwind2_rmap_push_closed_depth (o) (f) (q) (n):
+lemma nap_plus_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (m) (n):
q ϵ 𝐂❨o,n❩ →
- ♭q = ▶[⫯f]q＠§❨n❩.
-#o #f #q #n #Hq elim Hq -q -n
-[|*: #q #n [ #k #_ ] #_ #IH ] //
-<unwind2_rmap_d_dx <tr_compose_nap //
+ (⫯▶[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_depth (o) (f) (p) (q) (n):
+lemma nap_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
q ϵ 𝐂❨o,n❩ →
♭q = ▶[f](p●𝗟◗q)＠§❨n❩.
-#o #f #p #q #n #Hq
-<nap_unwind2_rmap_append_closed_Lq_dx //
-/2 width=2 by nap_unwind2_rmap_push_closed_depth/
-qed-.
-
-lemma xap_unwind2_rmap_append_closed_true_dx_depth (f) (p) (q) (n):
- q ϵ 𝐂❨Ⓣ,n❩ → ♭q = ▶[f](p●q)＠❨n❩.
-#f #p #q #n #Hq elim Hq -q -n //
-#q #n #k #Ho #_ #IH
-<unwind2_rmap_d_dx <tr_compose_xap
->Ho // <tr_uni_xap_succ <Ho //
+#o #f #p #q #n #Hn
+>(nplus_zero_sn n)
+<(nap_plus_unwind2_rmap_append_closed_Lq_dx … Hn) -Hn
+<nplus_zero_sn //
qed-.
lemma tls_succ_plus_unwind2_rmap_push_closed (o) (f) (q) (n):
q ϵ 𝐂❨o,n❩ →
∀m. ⇂*[m]f ≗ ⇂*[↑(m+n)]▶[⫯f]q.
-#o #f #q #n #Hq elim Hq -q -n //
+#o #f #q #n #Hn elim Hn -q -n //
#q #n #k #_ #_ #IH #m
@(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
>nrplus_inj_dx >nrplus_inj_sn >nsucc_unfold //
qed-.
-lemma tls_plus_unwind2_rmap_closed_true (f) (q) (n):
- q ϵ 𝐂❨Ⓣ,n❩ →
- ∀m. ⇂*[m]f ≗ ⇂*[m+n]▶[f]q.
-#f #q #n #Hq elim Hq -q -n //
-#q #n #k #Ho #_ #IH #m
->Ho // <nplus_succ_dx
-@(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
->nrplus_inj_dx >nrplus_inj_sn >nsucc_unfold
->nplus_succ_dx <Ho //
-qed-.
-
-lemma tls_succ_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
+lemma tls_succ_unwind2_rmap_push_closed (o) (f) (q) (n):
q ϵ 𝐂❨o,n❩ →
- ▶[f]p ≗ ⇂*[↑n]▶[f](p●𝗟◗q).
+ f ≗ ⇂*[↑n]▶[⫯f](q).
+#o #f #q #n #Hn
/2 width=2 by tls_succ_plus_unwind2_rmap_push_closed/
qed-.
-lemma tls_unwind2_rmap_append_closed_true_dx (f) (p) (q) (n):
- q ϵ 𝐂❨Ⓣ,n❩ →
- ▶[f]p ≗ ⇂*[n]▶[f](p●q).
-/2 width=1 by tls_plus_unwind2_rmap_closed_true/
-qed-.
-
-lemma nap_plus_unwind2_rmap_append_closed_Lq_dx_depth (o) (f) (p) (q) (m) (n):
+lemma tls_succ_plus_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
q ϵ 𝐂❨o,n❩ →
- ▶[f]p＠❨m❩+♭q = ▶[f](p●𝗟◗q)＠§❨m+n❩.
-#o #f #p #q #m #n #Hq
-<tr_nap_plus @eq_f2
-[ <(tr_xap_eq_repl … (tls_succ_unwind2_rmap_append_closed_Lq_dx …)) //
-| /2 width=2 by nap_unwind2_rmap_append_closed_Lq_dx_depth/
-]
-qed-.
-
-lemma nap_plus_unwind2_rmap_append_closed_bLq_dx_depth (o) (f) (p) (b) (q) (m) (n):
- b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
- ♭b+♭q = ▶[f](p●b●𝗟◗q)＠§❨m+n❩.
-#o #f #p #b #q #m #n #Hb #Hq
->(xap_unwind2_rmap_append_closed_true_dx_depth f p … Hb) -Hb
->(nap_plus_unwind2_rmap_append_closed_Lq_dx_depth … Hq) -Hq //
-qed-.
-
-lemma tls_succ_plus_unwind2_rmap_append_closed_bLq_dx (o) (f) (p) (b) (q) (m) (n):
- b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
- ▶[f]p ≗ ⇂*[↑(m+n)]▶[f](p●b●𝗟◗q).
-#o #f #p #b #q #m #n #Hb #Hq
->nplus_succ_dx <stream_tls_plus >list_append_assoc
-@(stream_eq_trans … (tls_unwind2_rmap_append_closed_true_dx … Hb)) -Hb
-/3 width=2 by stream_tls_eq_repl, tls_succ_unwind2_rmap_append_closed_Lq_dx/
+ ∀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-.
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