(* Author: Fabian Immler, TUM *)
header {* Sequence of Properties on Subsequences *}
theory Diagonal_Subsequence
imports SEQ
begin
locale subseqs =
fixes P::"nat\(nat\nat)\bool"
assumes ex_subseq: "\n s. subseq s \ \r'. subseq r' \ P n (s o r')"
begin
primrec seqseq where
"seqseq 0 = id"
| "seqseq (Suc n) = seqseq n o (SOME r'. subseq r' \ P n (seqseq n o r'))"
lemma seqseq_ex:
shows "subseq (seqseq n) \
(\r'. seqseq (Suc n) = seqseq n o r' \ subseq r' \ P n (seqseq n o r'))"
proof (induct n)
case 0
let ?P = "\r'. subseq r' \ P 0 r'"
let ?r = "Eps ?P"
have "?P ?r" using ex_subseq[of id 0] by (intro someI_ex[of ?P]) (auto simp: subseq_def)
thus ?case by (auto simp: subseq_def)
next
case (Suc n)
then obtain r' where
Suc': "seqseq (Suc n) = seqseq n \ r'" "subseq (seqseq n)" "subseq r'"
"P n (seqseq n o r')"
by blast
let ?P = "\r'a. subseq (r'a ) \ P (Suc n) (seqseq n o r' o r'a)"
let ?r = "Eps ?P"
have "?P ?r" using ex_subseq[of "seqseq n o r'" "Suc n"] Suc'
by (intro someI_ex[of ?P]) (auto intro: subseq_o simp: o_assoc)
moreover have "seqseq (Suc (Suc n)) = seqseq n \ r' \ ?r"
by (subst seqseq.simps) (simp only: Suc' o_assoc)
moreover note subseq_o[OF `subseq (seqseq n)` `subseq r'`]
ultimately show ?case unfolding Suc' by (auto simp: o_def)
qed
lemma subseq_seqseq:
shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto
definition reducer where "reducer n = (SOME r'. subseq r' \ P n (seqseq n o r'))"
lemma subseq_reducer: "subseq (reducer n)" and reducer_reduces: "P n (seqseq n o reducer n)"
unfolding atomize_conj unfolding reducer_def using subseq_seqseq
by (rule someI_ex[OF ex_subseq])
lemma seqseq_reducer[simp]:
"seqseq (Suc n) = seqseq n o reducer n"
by (simp add: reducer_def)
declare seqseq.simps(2)[simp del]
definition diagseq where "diagseq i = seqseq i i"
lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
unfolding diagseq_def seqseq_reducer o_def
by (metis subseq_mono[OF subseq_seqseq] less_le_trans lessI seq_suble subseq_reducer)
lemma subseq_diagseq: "subseq diagseq"
using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
primrec fold_reduce where
"fold_reduce n 0 = id"
| "fold_reduce n (Suc k) = fold_reduce n k o reducer (n + k)"
lemma subseq_fold_reduce: "subseq (fold_reduce n k)"
proof (induct k)
case (Suc k) from subseq_o[OF this subseq_reducer] show ?case by (simp add: o_def)
qed (simp add: subseq_def)
lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
by (induct k) simp_all
lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
by (induct n) (simp_all)
lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
using seqseq_fold_reduce by (simp add: diagseq_def)
lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n"
by (induct n) simp_all
lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)"
proof -
have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
by (simp add: diagseq_fold_reduce)
also have "\ = (seqseq k o fold_reduce k n) (k + n)"
unfolding fold_reduce_add seqseq_fold_reduce ..
finally show ?thesis .
qed
lemma diagseq_sub:
assumes "m \ n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n"
using diagseq_add[of m "n - m"] assms by simp
lemma subseq_diagonal_rest: "subseq (\x. fold_reduce k x (k + x))"
unfolding subseq_Suc_iff fold_reduce.simps o_def
by (metis subseq_mono[OF subseq_fold_reduce] less_le_trans lessI add_Suc_right seq_suble
subseq_reducer)
lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\x. fold_reduce k x (k + x)))"
by (auto simp: o_def diagseq_add)
end
end