# Theory HOL-Library.Diagonal_Subsequence

```(* Author: Fabian Immler, TUM *)

section ‹Sequence of Properties on Subsequences›

theory Diagonal_Subsequence
imports Complex_Main
begin

locale subseqs =
fixes P::"nat⇒(nat⇒nat)⇒bool"
assumes ex_subseq: "⋀n s. strict_mono (s::nat⇒nat) ⟹ ∃r'. strict_mono r' ∧ P n (s ∘ r')"
begin

definition reduce where "reduce s n = (SOME r'::nat⇒nat. strict_mono r' ∧ P n (s ∘ r'))"

lemma subseq_reduce[intro, simp]:
"strict_mono s ⟹ strict_mono (reduce s n)"
unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) auto

lemma reduce_holds:
"strict_mono s ⟹ P n (s ∘ reduce s n)"
unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) (auto simp: o_def)

primrec seqseq :: "nat ⇒ nat ⇒ nat" where
"seqseq 0 = id"
| "seqseq (Suc n) = seqseq n ∘ reduce (seqseq n) n"

lemma subseq_seqseq[intro, simp]: "strict_mono (seqseq n)"
proof (induct n)
case 0 thus ?case by (simp add: strict_mono_def)
next
case (Suc n) thus ?case by (subst seqseq.simps) (auto intro!: strict_mono_o)
qed

lemma seqseq_holds:
"P n (seqseq (Suc n))"
proof -
have "P n (seqseq n ∘ reduce (seqseq n) n)"
by (intro reduce_holds subseq_seqseq)
thus ?thesis by simp
qed

definition diagseq :: "nat ⇒ nat" where "diagseq i = seqseq i i"

lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
proof -
have "diagseq n < seqseq n (Suc n)"
using subseq_seqseq[of n] by (simp add: diagseq_def strict_mono_def)
also have "… ≤ seqseq n (reduce (seqseq n) n (Suc n))"
using strict_mono_less_eq seq_suble by blast
also have "… = diagseq (Suc n)" by (simp add: diagseq_def)
finally show ?thesis .
qed

lemma subseq_diagseq: "strict_mono diagseq"
using diagseq_mono by (simp add: strict_mono_Suc_iff diagseq_def)

primrec fold_reduce where
"fold_reduce n 0 = id"
| "fold_reduce n (Suc k) = fold_reduce n k ∘ reduce (seqseq (n + k)) (n + k)"

lemma subseq_fold_reduce[intro, simp]: "strict_mono (fold_reduce n k)"
proof (induct k)
case (Suc k) from strict_mono_o[OF this subseq_reduce] show ?case by (simp add: o_def)
qed (simp add: strict_mono_def)

lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n ∘ 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 ∘ fold_reduce m n"
by (induct n) simp_all

lemma diagseq_add: "diagseq (k + n) = (seqseq k ∘ (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 ∘ 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 ∘ (fold_reduce m (n - m))) n"
using diagseq_add[of m "n - m"] assms by simp

lemma subseq_diagonal_rest: "strict_mono (λx. fold_reduce k x (k + x))"
unfolding strict_mono_Suc_iff fold_reduce.simps o_def
proof
fix n
have "fold_reduce k n (k + n) < fold_reduce k n (k + Suc n)" (is "?lhs < _")
by (auto intro: strict_monoD)
also have "… ≤ fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"
by (auto intro: less_mono_imp_le_mono seq_suble strict_monoD)
finally show "?lhs < …" .
qed

lemma diagseq_seqseq: "diagseq ∘ ((+) k) = (seqseq k ∘ (λx. fold_reduce k x (k + x)))"
by (auto simp: o_def diagseq_add)

lemma diagseq_holds:
assumes subseq_stable: "⋀r s n. strict_mono r ⟹ P n s ⟹ P n (s ∘ r)"
shows "P k (diagseq ∘ ((+) (Suc k)))"
unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)

end

end
```