(* Author: Fabian Immler, TUM *)

header {* 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. subseq s ==> ∃r'. subseq r' ∧ P n (s o r')"

begin

definition reduce where "reduce s n = (SOME r'. subseq r' ∧ P n (s o r'))"

lemma subseq_reduce[intro, simp]:

"subseq s ==> subseq (reduce s n)"

unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) auto

lemma reduce_holds:

"subseq s ==> P n (s o reduce s n)"

unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) (auto simp: o_def)

primrec seqseq where

"seqseq 0 = id"

| "seqseq (Suc n) = seqseq n o reduce (seqseq n) n"

lemma subseq_seqseq[intro, simp]: "subseq (seqseq n)"

proof (induct n)

case (Suc n) thus ?case by (subst seqseq.simps) (auto simp: subseq_reduce intro!: subseq_o)

qed (simp add: subseq_def)

lemma seqseq_holds:

"P n (seqseq (Suc n))"

proof -

have "P n (seqseq n o reduce (seqseq n) n)"

by (intro reduce_holds subseq_seqseq)

thus ?thesis by simp

qed

definition diagseq where "diagseq i = seqseq i i"

lemma subseq_mono: "subseq f ==> a ≤ b ==> f a ≤ f b"

by (metis le_eq_less_or_eq subseq_mono)

lemma subseq_strict_mono: "subseq f ==> a < b ==> f a < f b"

by (simp add: subseq_def)

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 subseq_def)

also have "… ≤ seqseq n (reduce (seqseq n) n (Suc n))"

by (auto intro: subseq_mono seq_suble)

also have "… = diagseq (Suc n)" by (simp add: diagseq_def)

finally show ?thesis .

qed

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 reduce (seqseq (n + k)) (n + k)"

lemma subseq_fold_reduce[intro, simp]: "subseq (fold_reduce n k)"

proof (induct k)

case (Suc k) from subseq_o[OF this subseq_reduce] 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 (λx. fold_reduce k x (k + x))"

unfolding subseq_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: subseq_strict_mono)

also have "… ≤ fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"

by (rule subseq_mono) (auto intro!: seq_suble subseq_mono)

finally show "?lhs < …" .

qed

lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (λx. fold_reduce k x (k + x)))"

by (auto simp: o_def diagseq_add)

lemma diagseq_holds:

assumes subseq_stable: "!!r s n. subseq r ==> P n s ==> P n (s o r)"

shows "P k (diagseq o (op + (Suc k)))"

unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)

end

end