# Theory Semi_Mojmir

theory Semi_Mojmir
imports Preliminaries2 DTS
(*
Author:      Salomon Sickert
*)

section ‹Mojmir Automata (Without Final States)›

theory Semi_Mojmir
imports Main "Auxiliary/Preliminaries2" DTS
begin

subsection ‹Definitions›

locale semi_mojmir_def =
fixes
― ‹Alphapet›
Σ :: "'a set"
fixes
― ‹Transition Function›
δ :: "('b, 'a) DTS"
fixes
― ‹Initial State›
q⇩0 :: "'b"
fixes
― ‹$\omega$-Word›
w :: "'a word"
begin

definition sink :: "'b ⇒ bool"
where
"sink q ≡ (q⇩0 ≠ q) ∧ (∀ν ∈ Σ. δ q ν = q)"

declare sink_def [code]

fun token_run :: "nat ⇒ nat ⇒ 'b"
where
"token_run x n = run δ q⇩0 (suffix x w) (n - x)"

fun configuration :: "'b ⇒ nat ⇒ nat set"
where
"configuration q n = {x. x ≤ n ∧ token_run x n = q}"

fun oldest_token :: "'b ⇒ nat ⇒ nat option"
where
"oldest_token q n = (if configuration q n ≠ {} then Some (Min (configuration q n)) else None)"

fun senior :: "nat ⇒ nat ⇒ nat"
where
"senior x n = the (oldest_token (token_run x n) n)"

fun older_seniors :: "nat ⇒ nat ⇒ nat set"
where
"older_seniors x n = {s. ∃y. s = senior y n ∧ s < senior x n ∧ ¬ sink (token_run s n)}"

fun rank :: "nat ⇒ nat ⇒ nat option"
where
"rank x n =
(if x ≤ n ∧ ¬sink (token_run x n) then Some (card (older_seniors x n)) else None)"

fun senior_states :: "'b ⇒ nat ⇒ 'b set"
where
"senior_states q n =
{p. ∃x y. oldest_token p n = Some y ∧ oldest_token q n = Some x ∧ y < x ∧ ¬ sink p}"

fun state_rank :: "'b ⇒ nat ⇒ nat option"
where
"state_rank q n = (if configuration q n ≠ {} ∧ ¬sink q then Some (card (senior_states q n)) else None)"

definition max_rank :: "nat"
where
"max_rank = card (reach Σ δ q⇩0 - {q. sink q})"

subsubsection ‹Iterative Computation of State-Ranks›

fun initial :: "'b ⇒ nat option"
where
"initial q = (if q = q⇩0 then Some 0 else None)"

fun pre_ranks :: "('b ⇒ nat option) ⇒ 'a ⇒ 'b ⇒ nat set"
where
"pre_ranks r ν q = {i . ∃q'. r q' = Some i ∧ q = δ q' ν} ∪ (if q = q⇩0 then {max_rank} else {})"

fun step :: "('b ⇒ nat option) ⇒ 'a ⇒ ('b ⇒ nat option)"
where
"step r ν q = (
if
¬sink q ∧ pre_ranks r ν q ≠ {}
then
Some (card {q'. ¬sink q' ∧ pre_ranks r ν q' ≠ {} ∧ Min (pre_ranks r ν q') < Min (pre_ranks r ν q)})
else
None)"

subsubsection ‹Properties of Tokens›

definition token_squats :: "nat ⇒ bool"
where
"token_squats x = (∀n. ¬sink (token_run x n))"

end

locale semi_mojmir = semi_mojmir_def +
assumes
― ‹The alphabet is finite. Non-emptiness is derived from well-formed w›
finite_Σ: "finite Σ"
assumes
― ‹The set of reachable states is finite›
finite_reach: "finite (reach Σ δ q⇩0)"
assumes
― ‹w only contains letters from the alphabet›
bounded_w: "range w ⊆ Σ"
begin

lemma nonempty_Σ: "Σ ≠ {}"
using bounded_w by blast

lemma bounded_w': "w i ∈ Σ"
using bounded_w by blast

― ‹Naming Scheme:

This theory uses the following naming scheme to consistently name variables.

* Tokens: x, y, z
* Time: n, m
* Rank: i, j, k
* States: p, q›

lemma sink_rev_step:
"¬sink q ⟹ q = δ q' ν ⟹ ν ∈ Σ ⟹ ¬sink q'"
"¬sink q ⟹ q = δ q' (w i) ⟹ ¬sink q'"
using bounded_w' by (force simp only: sink_def)+

subsection ‹Token Run›

lemma token_stays_in_sink:
assumes "sink q"
assumes "token_run x n = q"
shows "token_run x (n + m) = q"
proof (cases "x ≤ n")
case True
show ?thesis
proof (induction m)
case 0
show ?case
using assms(2) by simp
next
case (Suc m)
have "x ≤ n + m"
using True by simp
moreover
have "⋀x. w x ∈ Σ"
using bounded_w by auto
ultimately
have "⋀t. token_run x (n + m)  = q ⟹ token_run x (n + m + 1) = q"
using ‹sink q›[unfolded sink_def] upt_add_eq_append[OF le0, of "n + m" 1]
using Suc_diff_le by simp
with Suc show ?case
by simp
qed
qed (insert assms, simp add: sink_def)

lemma token_is_not_in_sink:
"token_run x n ∉ A ⟹ token_run x (Suc n) ∈ A ⟹ ¬sink (token_run x n)"
by (metis Suc_eq_plus1 token_stays_in_sink)

lemma token_run_intial_state:
"token_run x x = q⇩0"
by simp

lemma token_run_P:
assumes "¬ P (token_run x n)"
assumes "P (token_run x (Suc (n + m)))"
shows "∃m' ≤ m. ¬ P (token_run x (n + m')) ∧ P (token_run x (Suc (n + m')))"
using assms by (induction m) (simp_all, metis add_Suc_right le_Suc_eq)

lemma token_run_merge_Suc:
assumes "x ≤ n"
assumes "y ≤ n"
assumes "token_run x n = token_run y n"
shows "token_run x (Suc n) = token_run y (Suc n)"
proof -
have "run δ q⇩0 (suffix x w) (Suc (n - x)) = run δ q⇩0 (suffix y w) (Suc (n - y))"
using assms by fastforce
thus ?thesis
using Suc_diff_le assms(1,2) by force
qed

lemma token_run_merge:
"⟦x ≤ n; y ≤ n; token_run x n = token_run y n⟧ ⟹ token_run x (n + m) = token_run y (n + m)"
using token_run_merge_Suc[of x _ y] by (induction m) auto

lemma token_run_mergepoint:
assumes "x < y"
assumes "token_run x (y + n) = token_run y (y + n)"
obtains m where "x ≤ (Suc m)" and "y ≤ (Suc m)"
and "y = Suc m ∨ token_run x m ≠ token_run y m"
and "token_run x (Suc m) = token_run y (Suc m)"
using assms by (induction n)

subsubsection ‹Step Lemmas›

lemma token_run_step:
assumes "x ≤ n"
assumes "token_run x n = q'"
assumes "q = δ q' (w n)"
shows "token_run x (Suc n) = q"
using assms unfolding token_run.simps Suc_diff_le[OF ‹x ≤ n›] by force

lemma token_run_step':
"x ≤ n ⟹ token_run x (Suc n) = δ (token_run x n) (w n)"
using token_run_step by simp

subsection ‹Configuration›

subsubsection ‹Properties›

lemma configuration_distinct:
"q ≠ q' ⟹ configuration q n ∩ configuration q' n = {}"
by auto

lemma configuration_finite:
"finite (configuration q n)"
by simp

lemma configuration_non_empty:
"x ≤ n ⟹ configuration (token_run x n) n ≠ {}"
by fastforce

lemma configuration_token:
"x ≤ n ⟹ x ∈ configuration (token_run x n) n"
by fastforce

lemmas configuration_Max_in = Max_in[OF configuration_finite]
lemmas configuration_Min_in = Min_in[OF configuration_finite]

subsubsection ‹Monotonicity›

lemma configuration_monotonic_Suc:
"x ≤ n ⟹ configuration (token_run x n) n ⊆ configuration (token_run x (Suc n)) (Suc n)"
proof
fix y
assume "y ∈ configuration (token_run x n) n"
hence "y ≤ n" and "token_run x n = token_run y n"
by simp_all
moreover
assume "x ≤ n"
ultimately
have "token_run x (Suc n) = token_run y (Suc n)"
using token_run_merge_Suc by blast
thus "y ∈ configuration (token_run x (Suc n)) (Suc n)"
using configuration_token ‹y ≤ n› by simp
qed

subsubsection ‹Pull-Up and Push-Down›

lemma pull_up_token_run_tokens:
"⟦x ≤ n; y ≤ n; token_run x n = token_run y n⟧ ⟹ ∃q. x ∈ configuration q n ∧ y ∈ configuration q n"
by force

lemma push_down_configuration_token_run:
"⟦x ∈ configuration q n; y ∈ configuration q n⟧ ⟹ x ≤ n ∧ y ≤ n ∧ token_run x n = token_run y n"
by simp

subsubsection ‹Step Lemmas›

lemma configuration_step:
"x ∈ configuration q' n ⟹ q = δ q' (w n) ⟹ x ∈ configuration q (Suc n)"
using Suc_diff_le by simp

lemma configuration_step_non_empty:
"configuration q' n ≠ {} ⟹ q = δ q' (w n) ⟹ configuration q (Suc n) ≠ {}"
by (blast dest: configuration_step)

lemma configuration_rev_step':
assumes "x ≠ Suc n"
assumes "x ∈ configuration q (Suc n)"
obtains q' where "q = δ q' (w n)" and "x ∈ configuration q' n"
using assms Suc_diff_le by force

lemma configuration_rev_step'':
assumes "x ∈ configuration q⇩0 (Suc n)"
shows "x = Suc n ∨ (∃q'. q⇩0 = δ q' (w n) ∧ x ∈ configuration q' n)"
using assms configuration_rev_step' by metis

lemma configuration_step_eq_q⇩0:
"configuration q⇩0 (Suc n) = {Suc n} ∪ ⋃{configuration q' n | q'. q⇩0 = δ q' (w n)}"
apply rule using configuration_rev_step'' apply fast using configuration_step[of _ _ n q⇩0] by fastforce

lemma configuration_rev_step:
assumes "q ≠ q⇩0"
assumes "x ∈ configuration q (Suc n)"
obtains q' where "q = δ q' (w n)" and "x ∈ configuration q' n"
using configuration_rev_step'[OF _ assms(2)] assms by fastforce

lemma configuration_step_eq:
assumes "q ≠ q⇩0"
shows "configuration q (Suc n) = ⋃{configuration q' n | q'. q = δ q' (w n)}"
using configuration_rev_step[OF assms, of _ n] configuration_step by auto

lemma configuration_step_eq_unified:
shows "configuration q (Suc n) = ⋃{configuration q' n | q'. q = δ q' (w n)} ∪ (if q = q⇩0 then {Suc n} else {}) "
using configuration_step_eq configuration_step_eq_q⇩0 by force

subsection ‹Oldest Token›

subsubsection ‹Properties›

lemma oldest_token_always_def:
"∃i. i ≤ x ∧ oldest_token (token_run x n) n = Some i"
proof (cases "x ≤ n")
case False
let ?q = "token_run x n"
from False have "n ∈ configuration ?q n" and "configuration ?q n ≠ {}"
by auto
then obtain i where "i ≤ n" and "oldest_token ?q n = Some i"
by (metis Min.coboundedI oldest_token.simps configuration_finite)
moreover
hence "i ≤ x"
using False by linarith
ultimately
show ?thesis
by blast
qed fastforce

lemma oldest_token_bounded:
"oldest_token q n = Some x ⟹ x ≤ n"
by (metis oldest_token.simps configuration_Min_in option.distinct(1) option.inject push_down_configuration_token_run)

lemma oldest_token_distinct:
"q ≠ q' ⟹ oldest_token q n = Some i ⟹ oldest_token q' n = Some j ⟹ i ≠ j"
by (metis configuration_Min_in configuration_distinct disjoint_iff_not_equal option.distinct(1) oldest_token.simps option.sel)

lemma oldest_token_equal:
"oldest_token q n = Some i ⟹ oldest_token q' n = Some i ⟹ q = q'"
using oldest_token_distinct by blast

subsubsection ‹Monotonicity›

lemma oldest_token_monotonic_Suc:
assumes "x ≤ n"
assumes "oldest_token (token_run x n) n = Some i"
assumes "oldest_token (token_run x (Suc n)) (Suc n) = Some j"
shows "i ≥ j"
proof -
from assms have "i = Min (configuration (token_run x n) n)"
and "j = Min (configuration (token_run x (Suc n)) (Suc n))"
by (metis oldest_token.elims option.discI option.sel)+
thus ?thesis
using Min_antimono[OF configuration_monotonic_Suc[OF assms(1)] configuration_non_empty[OF assms(1)] configuration_finite] by blast
qed

subsubsection ‹Pull-Up and Push-Down›

lemma push_down_oldest_token_configuration:
"oldest_token q n = Some x ⟹ x ∈ configuration q n"
by (metis configuration_Min_in oldest_token.simps option.distinct(2) option.inject)

lemma push_down_oldest_token_token_run:
"oldest_token q n = Some x ⟹ token_run x n = q"
using push_down_oldest_token_configuration configuration.simps by blast

subsection ‹Senior Token›

subsubsection ‹Properties›

lemma senior_le_token:
"senior x n ≤ x"
using oldest_token_always_def[of x n] by fastforce

lemma senior_token_run:
"senior x n = senior y n ⟷ token_run x n = token_run y n"
by (metis oldest_token_always_def oldest_token_distinct option.sel senior.simps)

text ‹The senior of a token is always in the same state›

lemma senior_same_state:
"token_run (senior x n) n = token_run x n"
proof -
have X: "{t. t ≤ n ∧ token_run t n = token_run x n} ≠ {}"
by (cases "x ≤ n") auto
show ?thesis
using Min_in[OF _ X] by force
qed

lemma senior_senior:
"senior (senior x n) n = senior x n"
using senior_same_state senior_token_run by blast

subsubsection ‹Monotonicity›

lemma senior_monotonic_Suc:
"x ≤ n ⟹ senior x n ≥ senior x (Suc n)"
by (metis oldest_token_always_def oldest_token_monotonic_Suc option.sel senior.simps)

subsubsection ‹Pull-Up and Push-Down›

lemma pull_up_configuration_senior:
"⟦x ∈ configuration q n; y ∈ configuration q n⟧ ⟹ senior x n = senior y n"
by force

lemma push_down_senior_tokens:
"⟦x ≤ n; y ≤ n; senior x n = senior y n⟧ ⟹ ∃q. x ∈ configuration q n ∧ y ∈ configuration q n"
using senior_token_run pull_up_token_run_tokens by blast

subsection ‹Set of Older Seniors›

subsubsection ‹Properties›

lemma older_seniors_cases_subseteq [case_names le ge]:
assumes "older_seniors x n ⊆ older_seniors y n ⟹ P"
assumes "older_seniors x n ⊇ older_seniors y n ⟹ P"
shows "P" using assms by fastforce

lemma older_seniors_cases_subset [case_names less equal greater]:
assumes "older_seniors x n ⊂ older_seniors y n ⟹ P"
assumes "older_seniors x n = older_seniors y n ⟹ P"
assumes "older_seniors x n ⊃ older_seniors y n ⟹ P"
shows "P" using assms older_seniors_cases_subseteq by blast

lemma older_seniors_finite:
"finite (older_seniors x n)"
by fastforce

lemma older_seniors_older:
"y ∈ older_seniors x n ⟹ y < x"
using less_le_trans[OF _ senior_le_token, of y x n] by force

lemma older_seniors_senior_simp:
"older_seniors (senior x n) n = older_seniors x n"
unfolding older_seniors.simps senior_senior ..

lemma older_seniors_not_self_referential:
"senior x n ∉ older_seniors x n"
by simp

lemma older_seniors_not_self_referential_2:
"x ∉ older_seniors x n"
using older_seniors_older older_seniors_not_self_referential less_not_refl by blast

lemma older_seniors_subset:
"y ∈ older_seniors x n ⟹ older_seniors y n ⊂ older_seniors x n"
using older_seniors_not_self_referential_2 by (cases rule: older_seniors_cases_subset) blast+

lemma older_seniors_subset_2:
assumes "¬ sink (token_run x n)"
assumes "older_seniors x n ⊂ older_seniors y n"
shows "senior x n ∈ older_seniors y n"
proof -
have "senior x n < senior y n"
using assms(2) by fastforce
thus ?thesis
using assms(1)[unfolded senior_same_state[symmetric, of x n]]
unfolding older_seniors.simps by blast
qed

lemmas older_seniors_Max_in = Max_in[OF older_seniors_finite]
lemmas older_seniors_Min_in = Min_in[OF older_seniors_finite]
lemmas older_seniors_Max_coboundedI = Max.coboundedI[OF older_seniors_finite]
lemmas older_seniors_Min_coboundedI = Min.coboundedI[OF older_seniors_finite]
lemmas older_seniors_card_mono = card_mono[OF older_seniors_finite]
lemmas older_seniors_psubset_card_mono = psubset_card_mono[OF older_seniors_finite]

lemma older_seniors_recursive:
fixes x n
defines "os ≡ older_seniors x n"
assumes "os ≠ {}"
shows "os = {Max os} ∪ older_seniors (Max os) n"
(is "?lhs = ?rhs")
proof
show "?lhs ⊆ ?rhs"
proof
fix x
assume "x ∈ ?lhs"
show "x ∈ ?rhs"
proof (cases "x = Max os")
case False
hence "x < Max os"
by (metis older_seniors_Max_coboundedI os_def ‹x ∈ os› dual_order.order_iff_strict)
moreover
obtain y' where "Max os = senior y' n"
using older_seniors_Max_in assms(2)
unfolding os_def older_seniors.simps by blast
ultimately
have "x < senior (Max os) n"
using senior_senior by presburger
moreover
from ‹x ∈ ?lhs› obtain y where "x = senior y n" and "¬ sink (token_run x n)"
unfolding os_def older_seniors.simps by blast
ultimately
show ?thesis
unfolding older_seniors.simps by blast
qed blast
qed
next
show "?lhs ⊇ ?rhs"
using older_seniors_subset older_seniors_Max_in assms(2)
unfolding os_def by blast
qed

lemma older_seniors_recursive_card:
fixes x n
defines "os ≡ older_seniors x n"
assumes "os ≠ {}"
shows "card os = Suc (card (older_seniors (Max os) n))"
by (metis older_seniors_recursive assms Un_empty_left Un_insert_left card_insert_disjoint older_seniors_finite older_seniors_not_self_referential_2)

lemma older_seniors_card:
"card (older_seniors x n) = card (older_seniors y n) ⟷ older_seniors x n = older_seniors y n"
by (metis less_not_refl older_seniors_cases_subset older_seniors_psubset_card_mono)

lemma older_seniors_card_le:
"card (older_seniors x n) < card (older_seniors y n) ⟷ older_seniors x n ⊂ older_seniors y n"
by (metis card_mono card_psubset not_le older_seniors_cases_subseteq older_seniors_finite psubset_card_mono)

lemma older_seniors_card_less:
"card (older_seniors x n) ≤ card (older_seniors y n) ⟷ older_seniors x n ⊆ older_seniors y n"
by (metis not_le older_seniors_card_mono older_seniors_cases_subseteq older_seniors_psubset_card_mono subset_not_subset_eq)

subsubsection ‹Monotonicity›

lemma older_seniors_monotonic_Suc:
assumes "x ≤ n"
shows "older_seniors x n ⊇ older_seniors x (Suc n)"
proof
fix y
assume "y ∈ older_seniors x (Suc n)"
then obtain ox where "y = senior ox (Suc n)"
and "y < senior x (Suc n)"
and "¬ sink (token_run y (Suc n))"
unfolding older_seniors.simps by blast

hence "y = senior y n"
using senior_senior senior_le_token senior_monotonic_Suc assms
moreover
have "y < senior x n"
using assms less_le_trans[OF ‹y < senior x (Suc n)› senior_monotonic_Suc]
by blast
moreover
have "¬ sink (token_run y n)"
using ‹¬ sink (token_run y (Suc n))› token_stays_in_sink
unfolding Suc_eq_plus1 by metis

ultimately
show "y ∈ older_seniors x n"
unfolding older_seniors.simps by blast
qed

lemma older_seniors_monotonic:
"x ≤ n ⟹ older_seniors x n ⊇ older_seniors x (n + m)"

lemma older_seniors_stable:
"x ≤ n ⟹ older_seniors x n = older_seniors x (n + m + m') ⟹ older_seniors x n = older_seniors x (n + m)"
by (induction m') (simp, unfold set_eq_subset, metis dual_order.trans le_add1 older_seniors_monotonic)

lemma card_older_seniors_monotonic:
"x ≤ n ⟹ card (older_seniors x n) ≥ card (older_seniors x (n + m))"
using older_seniors_monotonic older_seniors_card_mono by meson

subsubsection ‹Pull-Up and Push-Down›

lemma pull_up_senior_older_seniors:
"senior x n = senior y n ⟹ older_seniors x n = older_seniors y n"
unfolding older_seniors.simps senior.simps senior_token_run by presburger

lemma pull_up_senior_older_seniors_less:
"senior x n < senior y n ⟹ older_seniors x n ⊆ older_seniors y n"
by force

lemma pull_up_senior_older_seniors_less_2:
assumes "¬ sink (token_run x n)"
assumes "senior x n < senior y n"
shows "older_seniors x n ⊂ older_seniors y n"
proof -
from assms have "senior x n ∈ older_seniors y n"
unfolding senior_same_state[of x n, symmetric] older_seniors.simps by blast
thus ?thesis
using older_seniors_not_self_referential pull_up_senior_older_seniors_less[OF assms(2)] by blast
qed

lemma pull_up_senior_older_seniors_le:
"senior x n ≤ senior y n ⟹ older_seniors x n ⊆ older_seniors y n"
using pull_up_senior_older_seniors pull_up_senior_older_seniors_less
unfolding dual_order.order_iff_strict by blast

lemma push_down_older_seniors_senior:
assumes "¬ sink (token_run x n)"
assumes "¬ sink (token_run y n)"
assumes "older_seniors x n = older_seniors y n"
shows "senior x n = senior y n"
using assms by (cases "senior x n" " senior y n" rule: linorder_cases) (fast dest: pull_up_senior_older_seniors_less_2)+

subsubsection ‹Tower Lemma›

lemma older_seniors_tower'':
assumes "x ≤ n"
assumes "y ≤ n"
assumes "¬sink (token_run x n)"
assumes "¬sink (token_run y n)"
assumes "older_seniors x n = older_seniors x (Suc n)"
assumes "older_seniors y n ⊆ older_seniors x n"
shows "older_seniors y n = older_seniors y (Suc n)"
proof
{
fix s
assume "s ∈ older_seniors y n" and "older_seniors y n ⊂ older_seniors x n"
hence "s ∈ older_seniors x n"
using assms by blast
hence "¬sink (token_run s (Suc n))" and "∃z. s = senior z (Suc n)"
unfolding assms by simp+
moreover
have "senior y n ≤ senior y (Suc n)"
proof (rule ccontr)
assume "¬senior y n ≤ senior y (Suc n)"
moreover
have "senior y n ≤ n"
by (metis assms(2) senior_le_token le_trans)
ultimately
have "∀z. senior y n ≠ senior z (Suc n)"
using token_run_merge_Suc[unfolded senior_token_run[symmetric], OF ‹y ≤ n›]
by (metis senior_senior le_refl)
hence "senior y n ∉ older_seniors x (Suc n)"
using assms by simp
moreover
have "senior y n ∈ older_seniors x n"
using assms ‹older_seniors y n ⊂ older_seniors x n› older_seniors_subset_2 by meson
ultimately
show False
unfolding assms ..
qed
hence "s < senior y (Suc n)"
using ‹s ∈ older_seniors y n› by fastforce
ultimately
have "s ∈ older_seniors y (Suc n)"
unfolding older_seniors.simps by blast
}
moreover
{
fix s
assume "s ∈ older_seniors y n" and "older_seniors y n = older_seniors x n"
moreover
hence "senior y n = senior x n"
using assms(3-4) push_down_older_seniors_senior by blast
hence "senior y (Suc n) = senior x (Suc n)"
using token_run_merge_Suc[OF assms(2,1)] unfolding senior_token_run by blast
ultimately
have "s ∈ older_seniors y (Suc n)"
by (metis assms(5) older_seniors_senior_simp)
}
ultimately
show "older_seniors y n ⊆ older_seniors y (Suc n)"
using assms by blast
qed (metis older_seniors_monotonic_Suc assms(2))

lemma older_seniors_tower''2:
assumes "x ≤ n"
assumes "y ≤ n"
assumes "¬sink (token_run x (n + m))"
assumes "¬sink (token_run y (n + m))"
assumes "older_seniors x n = older_seniors x (n + m)"
assumes "older_seniors y n ⊆ older_seniors x n"
shows "older_seniors y n = older_seniors y (n + m)"
using assms
proof (induction m arbitrary: n)
case (Suc m)
have "¬sink (token_run x (n + m))" and "¬sink (token_run y (n + m))"
using ‹¬sink (token_run x (n + Suc m))› ‹¬sink (token_run y (n + Suc m))›
using token_stays_in_sink[of _ _ "n + m" 1]
moreover
have "older_seniors x n = older_seniors x (n + m)"
using Suc.prems(5) older_seniors_stable[OF ‹x ≤ n›]
moreover
hence "older_seniors x (n + m) = older_seniors x (Suc (n + m))"
ultimately
have "older_seniors y n = older_seniors y (n + m)"
using Suc by meson
also
have "… = older_seniors y (Suc (n + m))"
using older_seniors_tower''[OF _ _ ‹¬sink (token_run x (n + m))› ‹¬sink (token_run y (n + m))› ‹older_seniors x (n + m) = older_seniors x (Suc (n + m))›] Suc
finally
show ?case
qed simp

lemma older_seniors_tower':
assumes "y ∈ older_seniors x n"
assumes "older_seniors x n = older_seniors x (Suc n)"
shows "older_seniors y n = older_seniors y (Suc n)"
(is "?lhs = ?rhs")
using assms
proof (induction "card (older_seniors x n)" arbitrary: x y)
case 0
hence "older_seniors x n = {}"
using older_seniors_finite card_eq_0_iff by metis
thus ?case
using "0.prems" by blast
next
case (Suc c)
let ?os = "older_seniors x n"
have "?os ≠ {}"
using Suc.prems(1) by blast

hence "y = Max ?os ∨ y ∈ older_seniors (Max ?os) n"
using Suc.prems(1) older_seniors_recursive by blast
moreover
have "older_seniors (Max ?os) n = older_seniors (Max ?os) (Suc n)"
using Suc.prems(2) older_seniors_recursive ‹?os ≠ {}› older_seniors_not_self_referential_2
by (metis Un_empty_left Un_insert_left insert_ident)
moreover
{
fix s
assume "s ∈ older_seniors (Max ?os) n"
moreover
from Suc.hyps(2) have "card (older_seniors (Max ?os) n) = c"
unfolding older_seniors_recursive_card[OF ‹?os ≠ {}›] by blast
ultimately
have "older_seniors s n = older_seniors s (Suc n)"
by (metis Suc.hyps(1) ‹older_seniors (Max ?os) n = older_seniors (Max ?os) (Suc n)›)
}
ultimately
show ?case
by blast
qed

lemma older_seniors_tower:
"⟦x ≤ n; y ∈ older_seniors x n; older_seniors x n = older_seniors x (n + m)⟧ ⟹ older_seniors y n = older_seniors y (n + m)"
proof (induction m)
case (Suc m)
hence "older_seniors x n = older_seniors x (n + m)"
using older_seniors_monotonic older_seniors_monotonic_Suc subset_antisym
hence "older_seniors y n = older_seniors y (n + m)"
using Suc.IH[OF Suc.prems(1,2)] by blast
also
have "… = older_seniors y (n + Suc m)"
using older_seniors_tower'[of y x "n + m"] Suc.prems unfolding add_Suc_right
by (metis ‹older_seniors x n = older_seniors x (n + m)›)
finally
show ?case .
qed simp

subsection ‹Rank›

subsubsection ‹Properties›

lemma rank_None_before:
"x > n ⟹ rank x n = None"
by simp

lemma rank_None_Suc:
assumes "x ≤ n"
assumes "rank x n = None"
shows "rank x (Suc n) = None"
proof -
have "sink (token_run x n)"
using assms by (metis option.distinct(1) rank.simps)
hence "sink (token_run x (Suc n))"
using token_stays_in_sink by (metis (erased, hide_lams) Suc_leD le_Suc_ex not_less_eq_eq)
thus ?thesis
by simp
qed

lemma rank_Some_time:
"rank x n = Some j ⟹ x ≤ n"
by (metis option.distinct(1) rank.simps)

lemma rank_Some_sink:
"rank x n = Some j ⟹ ¬sink (token_run x n)"
by fastforce

lemma rank_Some_card:
"rank x n = Some j ⟹ card (older_seniors x n) = j"
by (metis option.distinct(1) option.inject rank.simps)

lemma rank_initial:
"∃i. rank x x = Some i"
unfolding rank.simps sink_def by force

lemma rank_continuous:
assumes "rank x n = Some i"
assumes "rank x (n + m) = Some j"
assumes "m' ≤ m"
shows "∃k. rank x (n + m') = Some k"
using assms
proof (induction m arbitrary: j m')
case (Suc m)
thus ?case
proof (cases "m' = Suc m")
case False
with Suc.prems have "m' ≤ m"
by linarith
moreover
obtain j' where "rank x (n + m) = Some j'"
using Suc.prems(1,2) rank_Some_time rank_None_Suc
ultimately
show ?thesis
using Suc.IH[OF Suc.prems(1)] by blast
qed simp
qed simp

lemma rank_token_squats:
"token_squats x ⟹ x ≤ n ⟹ ∃i. rank x n = Some i"
unfolding token_squats_def by simp

lemma rank_older_seniors_bounded:
assumes "y ∈ older_seniors x n"
assumes "rank x n = Some j"
shows "∃j' < j. rank y n = Some j'"
proof -
from assms(1) have "¬sink (token_run y n)"
by simp
moreover
from assms have "y ≤ n"
by (metis dual_order.trans linear not_less older_seniors_older option.distinct(1) rank.simps)
moreover
have "older_seniors y n ⊂ older_seniors x n"
using older_seniors_subset assms(1) by presburger
hence "card (older_seniors y n) < card (older_seniors x n)"
by (rule older_seniors_psubset_card_mono)
ultimately
show ?thesis
using rank_Some_card[OF assms(2)] rank.simps by meson
qed

subsubsection ‹Bounds›

lemma max_rank_lowerbound:
"0 < max_rank"
proof -
obtain a where "a ∈ Σ"
using nonempty_Σ by blast
hence "range (λ_. a) ⊆ Σ" and "q⇩0 = run δ q⇩0 (λ_. a) 0"
by auto
hence "q⇩0 ∈ reach Σ δ q⇩0"
unfolding reach_def by blast
thus ?thesis
using reach_card_0[OF nonempty_Σ] finite_reach max_rank_def sink_def by force
qed

lemma older_seniors_card_bounded:
assumes "¬sink (token_run x n)" and "x ≤ n"
shows "card (older_seniors x n) < card (reach Σ δ q⇩0 - {q. sink q})"
(is "card ?S4 < card ?S0")
proof -
let ?S1 = "{token_run x n | x n. True} - {q. sink q}"
let ?S2 = "(λq. the (oldest_token q n))  ?S1"
let ?S3 = "{s. ∃x. s = senior x n ∧ ¬(sink (token_run s n))}"

have "?S1 ⊆ ?S0"
unfolding reach_def token_run.simps using bounded_w by fastforce
hence "finite ?S1" and C1: "card ?S1 ≤ card ?S0"
using finite_reach card_mono finite_subset
apply (simp add: finite_subset) by (metis ‹{token_run x n |x n. True} - Collect sink ⊆ reach Σ δ q⇩0 - Collect sink› card_mono finite_Diff local.finite_reach)
hence "finite ?S2" and C2: "card ?S2 ≤ card ?S1"
using finite_imageI card_image_le by blast+
moreover
have "?S3 ⊆ ?S2"
proof
fix s
assume "s ∈ ?S3"
hence "s = senior s n" and "¬sink (token_run s n)"
using senior_senior by fastforce+
thus "s ∈ ?S2"
by auto
qed
ultimately
have "finite ?S3" and C3: "card ?S3 ≤ card ?S2"
using card_mono finite_subset by blast+
moreover
have "senior x n ∈ ?S3" and "senior x n ∉ ?S4" and "?S4 ⊆ ?S3"
using assms older_seniors_not_self_referential senior_same_state by auto
hence "?S4 ⊂ ?S3"
by blast
ultimately
have "finite ?S4" and C4: "card ?S4 < card ?S3"
using psubset_card_mono finite_subset by blast+
show ?thesis
using C1 C2 C3 C4 by linarith
qed

lemma rank_upper_bound:
"rank x n = Some i ⟹ i < max_rank"
using older_seniors_card_bounded unfolding max_rank_def
by (fast dest: rank_Some_card rank_Some_time rank_Some_sink )

lemma rank_range:
"∃i. range (rank x) ⊆ {None} ∪ Some  {0..<i}"
proof
{
fix i_option
assume "i_option ∈ range (rank x)"
hence "i_option ∈ {None} ∪ Some  {0..<max_rank}"
proof (cases i_option)
case (Some i)
hence "i ∈ {0..<max_rank}"
using ‹i_option ∈ range (rank x)› rank_upper_bound by force
thus ?thesis
using Some by blast
qed blast
}
thus "range (rank x) ⊆ ({None} ∪ Some  {0..<max_rank})" ..
qed

subsubsection ‹Monotonicity›

lemma rank_monotonic:
"⟦rank x n = Some i; rank x (n + m) = Some j⟧ ⟹ i ≥ j"
using card_older_seniors_monotonic rank_Some_card rank_Some_time by metis

subsubsection ‹Pull-Up and Push-Down›

lemma pull_up_senior_rank:
"⟦x ≤ n; y ≤ n; senior x n = senior y n⟧ ⟹ rank x n = rank y n"
by (metis senior_token_run rank.simps pull_up_senior_older_seniors)

lemma pull_up_configuration_rank:
"⟦x ∈ configuration q n; y ∈ configuration q n⟧ ⟹ rank x n = rank y n"
by force

lemma push_down_rank_older_seniors:
"⟦rank x n = rank y n; rank x n = Some i⟧ ⟹ older_seniors x n = older_seniors y n"
by (metis older_seniors_card option.distinct(2) option.sel rank.simps)

lemma push_down_rank_senior:
"⟦rank x n = rank y n; rank x n = Some i⟧ ⟹ senior x n = senior y n"
by (metis push_down_rank_older_seniors push_down_older_seniors_senior option.distinct(1) rank.elims)

lemma push_down_rank_tokens:
"⟦rank x n = rank y n; rank x n = Some i⟧ ⟹ (∃q. x ∈ configuration q n ∧ y ∈ configuration q n)"
by (metis push_down_senior_tokens rank_Some_time push_down_rank_senior)

subsubsection ‹Pulled-Up Lemmas›

lemma rank_senior_senior:
"x ≤ n ⟹ rank (senior x n) n = rank x n"

subsubsection ‹Stable Rank›

definition stable_rank :: "nat ⇒ nat ⇒ bool"
where
"stable_rank x i = (∀⇩∞n. rank x n = Some i)"

lemma stable_rank_unique:
assumes "stable_rank x i"
assumes "stable_rank x j"
shows "i = j"
proof -
from assms obtain n m where "⋀n'. n' ≥ n ⟹ rank x n' = Some i"
and "⋀m'. m' ≥ m ⟹ rank x m' = Some j"
unfolding stable_rank_def MOST_nat_le by blast
hence "rank x (n + m) = Some i" and "rank x (n + m) = Some j"
thus ?thesis
by simp
qed

lemma stable_rank_equiv_token_squats:
"token_squats x = (∃i. stable_rank x i)"
(is "?lhs = ?rhs")
proof
assume ?lhs
define ranks where "ranks = {j | j n. rank x n = Some j}"
hence "ranks ⊆ {0..<max_rank}" and "the (rank x x) ∈ ranks"
using rank_upper_bound rank_initial[of x] unfolding ranks_def by fastforce+ (* Takes roughly 10s *)
hence "finite ranks" and "ranks ≠ {}"
using finite_reach finite_atLeastAtMost infinite_super by fast+

define i where "i = Min ranks"
obtain n where "rank x n = Some i"
using Min_in[OF ‹finite ranks› ‹ranks ≠ {}›]
unfolding i_def ranks_def by blast

have "⋀j. j ∈ ranks ⟹ j ≥ i"
using Min_in[OF ‹finite ranks› ‹ranks ≠ {}›] unfolding i_def
by (metis Min.coboundedI ‹finite ranks›)
hence "⋀m j. rank x (n + m) = Some j ⟹ j ≥ i"
unfolding ranks_def by blast
moreover
have "⋀m j. rank x (n + m) = Some j ⟹ j ≤ i"
using rank_monotonic[OF ‹rank x n = Some i›] by blast
moreover
have "⋀m. ∃j. rank x (n + m) = Some j"
using rank_token_squats[OF ‹?lhs›] rank_Some_time[OF ‹rank x n = Some i›] by simp
ultimately
have "⋀m. rank x (n + m) = Some i"
by (metis le_antisym)
thus ?rhs
unfolding stable_rank_def MOST_nat_le by (metis le_iff_add)
next
assume ?rhs
thus ?lhs
unfolding token_squats_def stable_rank_def MOST_nat_le
qed

lemma stable_rank_same_tokens:
assumes "stable_rank x i"
assumes "stable_rank y j"
assumes "x ∈ configuration q n"
assumes "y ∈ configuration q n"
shows "i = j"
proof -
from assms(1) obtain n_i where "n_i ≥ n" and "∀t ≥ n_i. rank x t = Some i"
unfolding stable_rank_def MOST_nat_le by (metis linear order_trans)
moreover
from assms(2) obtain n_j where "n_j ≥ n" and "∀t ≥ n_j. rank y t = Some j"
unfolding stable_rank_def MOST_nat_le by (metis linear order_trans)
moreover
define m where "m = max n_i n_j"
ultimately
have "rank x m = Some i" and "rank y m = Some j"
by (metis max.bounded_iff order_refl)+
moreover
have "m ≥ n"
by (metis ‹n ≤ n_j› le_trans max.cobounded2 m_def)
have "∃q'. x ∈ configuration q' m ∧ y ∈ configuration q' m"
using push_down_configuration_token_run[OF assms(3,4)]
using token_run_merge[of x n y]
using pull_up_token_run_tokens[of x m y]
using ‹m ≥ n›[unfolded le_iff_add] by force
ultimately
show ?thesis
using pull_up_configuration_rank by (metis option.inject)
qed

subsubsection ‹Tower Lemma›

lemma rank_tower:
assumes "i ≤ j"
assumes "rank x n = Some j"
assumes "rank x (n + m) = Some j"
assumes "rank y n = Some i"
shows "rank y (n + m) = Some i"
proof (cases i j rule: linorder_cases)
case less
{
hence "card (older_seniors (senior y n) n) < card (older_seniors x n)"
using assms rank_Some_card senior_same_state by force
hence "senior y n ∈ older_seniors x n"
by (metis older_seniors_card_le rank_Some_sink assms(4) older_seniors_senior_simp older_seniors_subset_2)
moreover
have "older_seniors x n = older_seniors x (n + m)"
by (metis assms(2,3) rank_Some_card rank_Some_time card_subset_eq[OF older_seniors_finite] older_seniors_monotonic)
ultimately
have "older_seniors (senior y n) n = older_seniors (senior y n) (n + m)" and "senior y n ∈ older_seniors x (n + m)"
using older_seniors_tower rank_Some_time assms(2) by blast+
}
moreover
have "rank (senior y n) n = Some i"
by (metis assms(4) rank_Some_time rank_senior_senior)
ultimately
have "rank (senior y n) (n + m) = Some i"
by (metis rank_older_seniors_bounded[OF _ assms(3)] rank_Some_card)
moreover
have "senior y n ≤ n"
by (metis ‹rank (senior y n) n = Some i› rank_Some_time)
hence "senior y n ∈ configuration (token_run y (n + m)) (n + m)"
by (metis (full_types) token_run_merge[OF _ rank_Some_time[OF assms(4)] senior_same_state] configuration_token trans_le_add1)
ultimately
show ?thesis
next
case equal
hence "x ≤ n" and "y ≤ n" and "token_run x n = token_run y n"
using assms(2-4) push_down_rank_tokens by force+
moreover
hence "token_run x (n + m) = token_run y (n + m)"
using token_run_merge by blast
ultimately
show ?thesis
qed (insert ‹i ≤ j›, linarith)

lemma stable_rank_alt_def:
"rank x n = Some j ∧ stable_rank x j ⟷ (∀m ≥ n. rank x m = Some j)"
(is "?rhs ⟷ ?lhs")
proof
assume ?rhs
then obtain m' where "∀m ≥ m'. rank x m = Some j"
unfolding stable_rank_def MOST_nat_le by blast
moreover
hence "rank x n = Some j" and "rank x m' = Some j"
using ‹?rhs› by blast+
{
fix m
assume "n ≤ n + m" and "n + m < m'"
then obtain j' where "rank x (n + m) = Some j'"
by (metis ‹?rhs› stable_rank_equiv_token_squats rank_Some_time rank_token_squats trans_le_add1)
moreover
hence "j' ≤ j"
using ‹rank x n = Some j› rank_monotonic by blast
moreover
have "j ≤ j'"
using ‹rank x (n + m) = Some j'› ‹rank x m' = Some j›  ‹n + m < m'› rank_monotonic
ultimately
have "rank x (n + m) = Some j"
by simp
}
ultimately
show ?lhs
qed (unfold stable_rank_def MOST_nat_le, blast)

lemma stable_rank_tower:
assumes "j ≤ i"
assumes "rank x n = Some j"
assumes "rank y n = Some i"
assumes "stable_rank y i"
shows "stable_rank x j"
using assms rank_tower[OF ‹j ≤ i›] stable_rank_alt_def[of y n i]
unfolding stable_rank_def[of x j, unfolded MOST_nat_le] by (metis le_Suc_ex)

subsection ‹Senior States›

lemma senior_states_initial:
"senior_states q 0 = {}"
by simp

lemma senior_states_cases_subseteq [case_names le ge]:
assumes "senior_states p n ⊆ senior_states q n ⟹ P"
assumes "senior_states p n ⊇ senior_states q n ⟹ P"
shows "P" using assms by force

lemma senior_states_cases_subset [case_names less equal greater]:
assumes "senior_states p n ⊂ senior_states q n ⟹ P"
assumes "senior_states p n = senior_states q n ⟹ P"
assumes "senior_states p n ⊃ senior_states q n ⟹ P"
shows "P" using assms senior_states_cases_subseteq by blast

lemma senior_states_finite:
"finite (senior_states q n)"
by fastforce

lemmas senior_states_card_mono = card_mono[OF senior_states_finite]
lemmas senior_states_psubset_card_mono = psubset_card_mono[OF senior_states_finite]

lemma senior_states_card:
"card (senior_states p n) = card (senior_states q n) ⟷ senior_states p n = senior_states q n"
by (metis less_not_refl senior_states_cases_subset senior_states_psubset_card_mono)

lemma senior_states_card_le:
"card (senior_states p n) < card (senior_states q n) ⟷ senior_states p n ⊂ senior_states q n"
by (metis card_mono not_less senior_states_cases_subseteq senior_states_finite senior_states_psubset_card_mono subset_not_subset_eq)

lemma senior_states_card_less:
"card (senior_states p n) ≤ card (senior_states q n) ⟷ senior_states p n ⊆ senior_states q n"
by (metis card_mono card_seteq senior_states_cases_subseteq senior_states_finite)

lemma senior_states_older_seniors:
"(λy. token_run y n)  older_seniors x n = senior_states (token_run x n) n"
(is "?lhs = ?rhs")
proof -
have "?lhs = {q'. ∃ost ot. q' = token_run ost n ∧ ost = senior ot n ∧ ost < senior x n ∧ ¬ sink q'}"
by auto
also
have "… = {q'. ∃t ot. oldest_token q' n = Some t ∧ t = senior ot n ∧ t < senior x n ∧ ¬ sink q'}"
unfolding senior.simps by (metis (erased, hide_lams) oldest_token_always_def push_down_oldest_token_token_run option.sel)
also
have "… = {q'. ∃t. oldest_token q' n = Some t ∧ t < senior x n ∧ ¬ sink q'}"
by auto
also
have "… = ?rhs"
unfolding senior_states.simps senior.simps by (metis (erased, hide_lams) oldest_token_always_def option.sel)
finally
show "?lhs = ?rhs"
.
qed

lemma card_older_senior_senior_states:
assumes "x ∈ configuration q n"
shows "card (older_seniors x n) = card (senior_states q n)"
(is "?lhs = ?rhs")
proof -
have "inj_on (λt. token_run t n) (older_seniors x n)"
unfolding inj_on_def using senior_same_state
by (fastforce simp del: token_run.simps)
moreover
have "token_run x n = q"
using assms by simp
ultimately
show "?lhs = ?rhs"
using card_image[of "(λt. token_run t n)" "older_seniors x n"]
unfolding senior_states_older_seniors by presburger
qed

subsection ‹Rank of States›

subsubsection ‹Alternative Definitions›

lemma state_rank_eq_rank:
"state_rank q n = (case oldest_token q n of None ⇒ None | Some t ⇒ rank t n) "
(is "?lhs = ?rhs")
proof (cases "oldest_token q n")
case (None)
thus ?thesis
by (metis not_Some_eq oldest_token.elims option.simps(4) state_rank.elims)
next
case (Some x)
hence "?lhs = (if ¬sink q then Some (card (older_seniors x n)) else None)"
by (metis emptyE push_down_oldest_token_configuration[OF Some] card_older_senior_senior_states state_rank.simps)
also
have "… = rank x n"
using oldest_token_bounded[OF Some] push_down_oldest_token_token_run[OF Some] by auto
also
have "… = ?rhs"
using Some by force
finally
show ?thesis .
qed

lemma state_rank_eq_rank_SOME:
"state_rank q n = (if configuration q n ≠ {} then rank (SOME x. x ∈ configuration q n) n else None)"
proof (cases "oldest_token q n")
case (Some x)
thus ?thesis
unfolding state_rank_eq_rank Some option.simps(5)
by (metis Some ex_in_conv pull_up_configuration_rank push_down_oldest_token_configuration someI_ex)
qed (unfold state_rank_eq_rank; metis not_Some_eq oldest_token.elims option.simps(4))

lemma rank_eq_state_rank:
"x ≤ n ⟹ rank x n = state_rank (token_run x n) n"
unfolding state_rank_eq_rank_SOME[of "token_run x n"]
by (metis all_not_in_conv configuration_token pull_up_configuration_rank someI_ex)

subsubsection ‹Pull-Up and Push-Down›

lemma pull_up_configuration_state_rank:
"configuration q n = {} ⟹ state_rank q n = None"
by force

lemma push_down_state_rank_tokens:
"state_rank q n = Some i ⟹ configuration q n ≠ {}"
by (metis not_Some_eq state_rank.elims)

lemma push_down_state_rank_configuration_None:
"state_rank q n = None ⟹ ¬sink q ⟹ configuration q n = {}"
unfolding state_rank.simps by (metis option.distinct(1))

lemma push_down_state_rank_oldest_token:
"state_rank q n = Some i ⟹ ∃x. oldest_token q n = Some x"
by (metis oldest_token.elims state_rank.elims)

lemma push_down_state_rank_token_run:
"state_rank q n = Some i ⟹ ∃x. token_run x n = q ∧ x ≤ n"
by (blast dest: push_down_state_rank_oldest_token push_down_oldest_token_token_run oldest_token_bounded)

subsubsection ‹Properties›

lemma state_rank_distinct:
assumes distinct: "p ≠ q"
assumes ranked_1: "state_rank p n = Some i"
assumes ranked_2: "state_rank q n = Some j"
shows "i ≠ j"
proof
assume "i = j"
obtain x y where "x ∈ configuration p n" and "y ∈ configuration q n"
using assms push_down_state_rank_tokens by blast
hence "rank x n = Some i" and "rank y n = Some j"
using assms pull_up_configuration_rank unfolding state_rank_eq_rank_SOME
by (metis all_not_in_conv someI_ex)+
hence "x ∈ configuration q n"
using ‹y ∈ configuration q n› push_down_rank_tokens
unfolding ‹i = j› by auto
hence "p = q"
using ‹x ∈ configuration p n› by fastforce
thus "False"
using distinct by blast
qed

lemma state_rank_initial_state:
obtains i where "state_rank q⇩0 n = Some i"
unfolding state_rank.simps sink_def by fastforce

lemma state_rank_sink:
"sink q ⟹ state_rank q n = None"
by simp

lemma state_rank_upper_bound:
"state_rank q n = Some i ⟹ i < max_rank"
by (metis option.simps(5) rank_upper_bound push_down_state_rank_oldest_token state_rank_eq_rank)

lemma state_rank_range:
"state_rank q n ∈ {None} ∪ Some  {0..<max_rank}"
by (cases "state_rank q n") (simp add: state_rank_upper_bound[of q n])+

lemma state_rank_None:
"¬sink q ⟹ state_rank q n = None ⟷ oldest_token q n = None"
by simp

lemma state_rank_Some:
"¬sink q ⟹ (∃i. state_rank q n = Some i) ⟷ (∃j. oldest_token q n = Some j)"
by simp

lemma state_rank_oldest_token:
assumes "state_rank p n = Some i"
assumes "state_rank q n = Some j"
assumes "oldest_token p n = Some x"
assumes "oldest_token q n = Some y"
shows "i < j ⟷ x < y"
proof -
have "configuration p n ≠ {}" and "configuration q n ≠ {}"
using assms(3,4) by (metis oldest_token.simps option.distinct(1))+
moreover
have "¬sink p" and "¬sink q"
using assms(1,2) state_rank_sink by auto
ultimately
have i_def: "i = card (senior_states p n)" and j_def: "j = card (senior_states q n)"
using assms(1,2) option.sel by simp_all
hence "i < j ⟷ senior_states p n ⊂ senior_states q n"
using senior_states_card_le by presburger
also
with assms(3,4) have "… ⟷ x < y"
proof (cases rule: senior_states_cases_subset[of p n q])
case equal
thus ?thesis
using assms state_rank_distinct i_def j_def
by (metis less_irrefl option.sel)
qed auto
ultimately
show ?thesis
by meson
qed

lemma state_rank_oldest_token_le:
assumes "state_rank p n = Some i"
assumes "state_rank q n = Some j"
assumes "oldest_token p n = Some x"
assumes "oldest_token q n = Some y"
shows "i ≤ j ⟷ x ≤ y"
using state_rank_oldest_token[OF assms] assms state_rank_distinct oldest_token_equal
by (cases "x = y") ((metis option.sel order_refl), (metis le_eq_less_or_eq option.inject))

lemma state_rank_in_function_set:
shows "(λq. state_rank q t) ∈ {f. (∀x. x ∉ reach Σ δ q⇩0 ⟶ f x = None) ∧
(∀x. x ∈ reach Σ δ q⇩0 ⟶ f x ∈ {None} ∪ Some  {0..<max_rank})}"
proof -
{
fix x
assume "x ∉ reach Σ δ q⇩0"
hence "⋀token. x ≠ token_run token t"
unfolding reach_def token_run.simps using bounded_w by fastforce
hence "state_rank x t = None"
using pull_up_configuration_state_rank by auto
}
with state_rank_range show ?thesis
by blast
qed

subsection ‹Step Function›

fun pre_oldest_tokens :: "'b ⇒ nat ⇒ nat set"
where
"pre_oldest_tokens q n = {x. ∃q'. oldest_token q' n = Some x ∧ q = δ q' (w n)} ∪ (if q = q⇩0 then {Suc n} else {})"

lemma pre_oldest_configuration_range:
"pre_oldest_tokens q n ⊆ {0..Suc n}"
proof -
have "{x. ∃q'. oldest_token q' n = Some x ∧ q = δ q' (w n)} ⊆ {0..n}"
(is "?lhs ⊆ ?rhs")
proof
fix x
assume "x ∈ ?lhs"
then obtain q' where "oldest_token q' n = Some x"
by blast
thus "x ∈ ?rhs"
unfolding atLeastAtMost_iff using oldest_token_bounded[of q' n x] by blast
qed
thus ?thesis
by (cases "q = q⇩0") fastforce+
qed

lemma pre_oldest_configuration_finite:
"finite (pre_oldest_tokens q n)"
using pre_oldest_configuration_range finite_atLeastAtMost by (rule finite_subset)

lemmas pre_oldest_configuration_Min_in = Min_in[OF pre_oldest_configuration_finite]

lemma pre_oldest_configuration_obtain:
assumes "x ∈ pre_oldest_tokens q n - {Suc n}"
obtains q' where "oldest_token q' n = Some x" and "q = δ q' (w n)"
using assms by (cases "q = q⇩0", auto)

lemma pre_oldest_configuration_element:
assumes "oldest_token q' n = Some ot"
assumes "q = δ q' (w n)"
shows "ot ∈ pre_oldest_tokens q n"
proof
show "ot ∈ {ot. ∃q'. oldest_token q' n = Some ot ∧ q = δ q' (w n)}"
(is "_ ∈ ?A")
using assms by blast
show "?A ⊆ pre_oldest_tokens q n"
by simp
qed

lemma pre_oldest_configuration_initial_state:
"Suc n ∈ pre_oldest_tokens q n ⟹ q = q⇩0"
using oldest_token_bounded[of _ n "Suc n"]
by (cases "q = q⇩0") auto

lemma pre_oldest_configuration_initial_state_2:
"q = q⇩0 ⟹ Suc n ∈ pre_oldest_tokens q n"
by fastforce

lemma pre_oldest_configuration_tokens:
"pre_oldest_tokens q n ≠ {} ⟷ configuration q (Suc n) ≠ {}"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then obtain ot where ot_def: "ot ∈ pre_oldest_tokens q n"
by blast
thus ?rhs
proof (cases "ot = Suc n")
case True
thus ?thesis
using pre_oldest_configuration_initial_state configuration_non_empty[of "Suc n" "Suc n"] ‹ot ∈ pre_oldest_tokens q n› unfolding token_run_intial_state  by blast
next
case False
then obtain q' where "oldest_token q' n = Some ot" and "q = δ q' (w n)"
using ot_def pre_oldest_configuration_obtain by blast
moreover
hence "configuration q' n ≠ {}"
by (metis oldest_token.simps option.distinct(2))
ultimately
show ?rhs
by (elim configuration_step_non_empty)
qed
next
assume ?rhs
then obtain token where "token ∈ configuration q (Suc n)" and "token ≤ Suc n" and "token_run token (Suc n) = q"
by auto
moreover
{
assume "token ≤ n"
then obtain q' where "token_run token n = q'" and "q = δ q' (w n)"
using ‹token_run token (Suc n) = q› unfolding token_run.simps Suc_diff_le[OF ‹token ≤ n›] by fastforce
then obtain ot where "oldest_token q' n = Some ot"
using oldest_token_always_def by blast
with ‹q = δ q' (w n)› have ?lhs
using pre_oldest_configuration_element by blast
}
ultimately
show ?lhs
using pre_oldest_configuration_initial_state_2 by fastforce
qed

lemma oldest_token_rec:
"oldest_token q (Suc n) = (if pre_oldest_tokens q n ≠ {} then Some (Min (pre_oldest_tokens q n)) else None)"
proof (cases "oldest_token q (Suc n)")
case (Some ot)
moreover
hence "ot ∈ configuration q (Suc n)"
by (rule push_down_oldest_token_configuration)
hence "configuration q (Suc n) ≠ {}"
by blast
hence "pre_oldest_tokens q n ≠ {}"
unfolding pre_oldest_configuration_tokens .
let ?ot = "Min (pre_oldest_tokens q n)"
{
{
{
assume "ot < Suc n"
hence "ot ≠ Suc n"
by blast
then obtain q' where "ot ∈ configuration q' n" and "q = δ q' (w n)"
using configuration_rev_step' ‹ot ∈ configuration q (Suc n)› by metis
{
fix token
assume "token ∈ configuration q' n"
hence "token ∈ configuration q (Suc n)"
using ‹q = δ q' (w n)› by (rule configuration_step)
hence "ot ≤ token"
using Some by (metis Min.coboundedI ‹configuration q (Suc n) ≠ {}› configuration_finite oldest_token.simps option.inject)
}
hence "Min (configuration q' n) = ot"
by (metis Min_eqI ‹ot ∈ configuration q' n› configuration_finite)
hence "oldest_token q' n = Some ot"
using ‹ot ∈ configuration q' n› unfolding oldest_token.simps by auto
hence "ot ∈ pre_oldest_tokens q n"
using ‹q = δ q' (w n)› by (rule pre_oldest_configuration_element)
}
moreover
{
assume "ot = Suc n"
moreover
hence "q = q⇩0"
using Some by (metis push_down_oldest_token_token_run token_run_intial_state)
ultimately
have "ot ∈ pre_oldest_tokens q n"
by simp
}
ultimately
have "ot ∈ pre_oldest_tokens q n"
using Some[THEN oldest_token_bounded] by linarith
}
moreover
{
fix ot' q'
assume "oldest_token q' n = Some ot'" and "q = δ q' (w n)"
moreover
hence "ot' ∈ configuration q (Suc n)"
using push_down_oldest_token_configuration configuration_step by blast
hence "ot ≤ ot'"
using Some by (metis Min.coboundedI ‹configuration q (Suc n) ≠ {}› configuration_finite oldest_token.simps option.inject)
}
hence "⋀y. y ∈ pre_oldest_tokens q n - {Suc n} ⟹ ot ≤ y"
using pre_oldest_configuration_obtain by metis
hence "⋀y. y ∈ pre_oldest_tokens q n ⟹ ot ≤ y"
using Some[THEN oldest_token_bounded] by force
ultimately
have "?ot = ot"
using Min_eqI[OF pre_oldest_configuration_finite, of q n ot] by fast
}
ultimately
show ?thesis
unfolding pre_oldest_configuration_tokens oldest_token.simps
by (metis ‹configuration q (Suc n) ≠ {}›)
qed (unfold pre_oldest_configuration_tokens oldest_token.simps, metis option.distinct(2))

lemma pre_ranks_range:
"pre_ranks (λq. state_rank q n) ν q  ⊆ {0..max_rank}"
proof -
have "{i | q' i. state_rank q' n = Some i ∧ q = δ q' ν} ⊆ {0..max_rank}"
using state_rank_upper_bound by fastforce
thus ?thesis
by auto
qed

lemma pre_ranks_finite:
"finite (pre_ranks (λq. state_rank q n) ν q)"
using pre_ranks_range finite_atLeastAtMost by (rule finite_subset)

lemmas pre_ranks_Min_in = Min_in[OF pre_ranks_finite]

lemma pre_ranks_state_obtain:
assumes "r⇩q ∈ pre_ranks r ν q - {max_rank}"
obtains q' where "r q' = Some r⇩q" and "q = δ q' ν"
using assms by (cases "q = q⇩0", auto)

lemma pre_ranks_element:
assumes "state_rank q' n = Some r"
assumes "q = δ q' (w n)"
shows "r ∈ pre_ranks (λq. state_rank q n) (w n) q"
proof
show "r ∈ {i. ∃q'. (λq. state_rank q n) q' = Some i ∧ q = δ q' (w n)}"
(is "_ ∈ ?A")
using assms by blast
show "?A ⊆ pre_ranks (λq. state_rank q n) (w n) q"
by simp
qed

lemma pre_ranks_initial_state:
"max_rank ∈ pre_ranks (λq. state_rank q n) ν q ⟹ q = q⇩0"
using state_rank_upper_bound by (cases "q = q⇩0") auto

lemma pre_ranks_initial_state_2:
"q = q⇩0 ⟹ max_rank ∈ pre_ranks r ν q"
by fastforce

lemma pre_ranks_tokens:
assumes "¬sink q"
shows "pre_ranks (λq. state_rank q n) (w n) q ≠ {} ⟷ configuration q (Suc n) ≠ {}"
(is "?lhs = ?rhs")
proof
assume ?lhs
thus ?rhs
proof (cases "q ≠ q⇩0")
case True
hence "{i. ∃q'. state_rank q' n = Some i ∧ q = δ q' (w n)} ≠ {}"
using ‹?lhs› by simp
then obtain q' where "state_rank q' n ≠ None" and "q = δ q' (w n)"
by blast
moreover
hence "configuration q' n ≠ {}"
unfolding state_rank.simps by meson
ultimately
show ?rhs
by (elim configuration_step_non_empty)
qed auto
next
assume ?rhs
then obtain token where "token ∈ configuration q (Suc n)" and "token ≤ Suc n" and "token_run token (Suc n) = q"
by auto
moreover
{
assume "token ≤ n"
then obtain q' where "token_run token n = q'" and "q = δ q' (w n)"
using ‹token_run token (Suc n) = q› unfolding token_run.simps Suc_diff_le[OF ‹token ≤ n›] by fastforce
hence "¬sink q'"
using ‹¬sink q› sink_rev_step bounded_w by blast
then obtain r where "state_rank q' n = Some r"
using ‹¬sink q› configuration_non_empty[OF ‹token ≤ n›] unfolding ‹token_run token n = q'› by simp
with ‹q = δ q' (w n)› have ?lhs
using pre_ranks_element by blast
}
ultimately
show ?lhs
by fastforce
qed

lemma pre_ranks_pre_oldest_token_Min_state_special:
assumes "¬sink q"
assumes "configuration q (Suc n) ≠ {}"
shows "Min (pre_ranks (λq. state_rank q n) (w n) q) = max_rank ⟷ Min (pre_oldest_tokens q n) = Suc n"
(is "?lhs ⟷ ?rhs")
proof
from assms have "pre_oldest_tokens q n ≠ {}"
and "pre_ranks  (λq. state_rank q n) (w n) q ≠ {}"
using pre_ranks_tokens pre_oldest_configuration_tokens by simp_all

{
assume ?lhs
have "q = q⇩0"
apply (rule ccontr)
using state_rank_upper_bound pre_ranks_Min_in[OF ‹pre_ranks (λq. state_rank q n) (w n) q ≠ {}›] ‹?lhs›
by auto
moreover
{
fix q'
assume "q = δ q' (w n)"
hence "¬sink q'"
using ‹¬sink q› bounded_w unfolding sink_def
using calculation by blast
{
fix i
assume "state_rank q' n = Some i"
hence "False"
using ‹q = δ q' (w n)›
using Min.coboundedI[OF pre_ranks_finite, of _ n "(w n)" q]
unfolding ‹?lhs› using state_rank_upper_bound[of q' n] by fastforce
}
hence "state_rank q' n = None"
by fastforce
hence "oldest_token q' n = None"
using ‹¬sink q'› by (metis state_rank_None)
}
hence "{ot. ∃q'. oldest_token q' n = Some ot ∧ q = δ q' (w n)} = {}"
by fastforce
ultimately
show "?rhs"
by auto
}

{
assume ?rhs
{
fix q'
assume "q = δ q' (w n)"
have "state_rank q' n = None"
proof (cases "oldest_token q' n")
case (Some t)
hence "t ≤ n"
using oldest_token_bounded[of q' n] by blast
moreover
have "Suc n ≤ t"
using ‹q = δ q' (w n)›
using Min.coboundedI[OF pre_oldest_configuration_finite, of _ q n]
unfolding ‹?rhs› using ‹oldest_token q' n = Some t› by auto
ultimately
have "False"
by linarith
thus ?thesis
..
qed (unfold state_rank_eq_rank, auto)
}
hence X: "{i. ∃q'.  (λq. state_rank q n) q' = Some i ∧ q = δ q' (w n)} = {}"
by fastforce

have "q = q⇩0"
apply (rule ccontr)
using ‹pre_ranks (λq. state_rank q n) (w n) q ≠ {}›
unfolding pre_ranks.simps X by simp
hence "pre_ranks (λq. state_rank q n) (w n) q = {max_rank}"
unfolding pre_ranks.simps X by force
thus ?lhs
by fastforce
}
qed

lemma pre_ranks_pre_oldest_token_Min_state:
assumes "¬sink q"
assumes "q = δ q' (w n)"
assumes "configuration q (Suc n) ≠ {}"
defines "min_r ≡ Min (pre_ranks (λq. state_rank q n) (w n) q)"
defines "min_ot ≡ Min (pre_oldest_tokens q n)"
shows "state_rank q' n = Some min_r ⟷ oldest_token q' n = Some min_ot"
(is "?lhs ⟷ ?rhs")
proof
from assms have "pre_oldest_tokens q n ≠ {}" and "¬sink q'"
and "pre_ranks (λq. state_rank q n) (w n) q ≠ {}"
using pre_ranks_tokens pre_oldest_configuration_tokens bounded_w unfolding sink_def
by (simp_all, metis rangeI subset_iff)

{
assume ?lhs
thus ?rhs
proof (cases min_r max_rank rule: linorder_cases)
case less
then obtain ot where "oldest_token q' n = Some ot"
by (metis push_down_state_rank_oldest_token ‹?lhs›)
moreover
{
{
fix q'' ot''
assume "q = δ q'' (w n)"
assume "oldest_token q'' n = Some ot''"
moreover
have "¬sink q''"
using ‹q = δ q'' (w n)› assms unfolding sink_def
by (metis rangeI subset_eq bounded_w)
then obtain r'' where "state_rank q'' n = Some r''"
using ‹oldest_token q'' n = Some ot''› by (metis state_rank_Some)
moreover
hence "r'' ∈ pre_ranks (λq. state_rank q n) (w n) q" (* Move to special lemma *)
using ‹q = δ q'' (w n)› unfolding pre_ranks.simps by blast
then have "min_r ≤ r''"
unfolding min_r_def by (metis Min.coboundedI pre_ranks_finite)
ultimately
have "ot ≤ ot''"
using state_rank_oldest_token_le[OF ‹?lhs› _ ‹oldest_token q' n = Some ot›] by blast
}
hence "⋀x. x ∈ {ot. ∃q'. oldest_token q' n = Some ot ∧ q = δ q' (w n)} ⟹ ot ≤ x"
by blast
moreover
have "ot ≤ Suc n"
using oldest_token_bounded[OF ‹oldest_token q' n = Some ot›] by simp
ultimately
have "⋀x. x ∈ pre_oldest_tokens q n ⟹ ot ≤ x"
unfolding pre_oldest_tokens.simps apply (cases "q⇩0 = q") apply auto done
hence "ot ≤ min_ot"
unfolding min_ot_def
unfolding Min_ge_iff[OF pre_oldest_configuration_finite ‹pre_oldest_tokens q n ≠ {}›, of ot]
by simp
}
moreover
have "ot ≥ min_ot"
using Min.coboundedI[OF pre_oldest_configuration_finite] pre_oldest_configuration_element
unfolding min_ot_def by (metis assms(2) calculation(1))
ultimately
show ?thesis
by simp
qed (insert not_less, blast intro: state_rank_upper_bound less_imp_le_nat)+
}

{
assume ?rhs
thus ?lhs
proof (cases min_ot "Suc n" rule: linorder_cases)
case less
then obtain r where "state_rank q' n = Some r"
using ‹?rhs› ‹¬sink q'› by (metis state_rank_Some)
moreover
{
{
fix r''
assume "r'' ∈ pre_ranks (λq. state_rank q n) (w n) q - {max_rank}"
then obtain q'' where "state_rank q'' n = Some r''"
and "q = δ q'' (w n)"
using pre_ranks_state_obtain by blast
moreover
then obtain ot'' where "oldest_token q'' n = Some ot''"
using push_down_state_rank_oldest_token by fastforce
moreover
hence "min_ot ≤ ot''"
using ‹q = δ q'' (w n)› pre_oldest_configuration_element Min.coboundedI pre_oldest_configuration_finite
unfolding min_ot_def by metis
ultimately
have "r ≤ r''"
using state_rank_oldest_token_le[OF ‹state_rank q' n = Some r› _ ‹?rhs›] by blast
}
moreover
have "r ≤ max_rank"
using state_rank_upper_bound[OF ‹state_rank q' n = Some r›] by linarith
ultimately
have "⋀x. x ∈  pre_ranks (λq. state_rank q n) (w n) q ⟹ r ≤ x"
unfolding pre_ranks.simps apply (cases "q⇩0 = q") apply auto done
hence "r ≤ min_r"
unfolding min_r_def Min_ge_iff[OF pre_ranks_finite ‹pre_ranks (λq. state_rank q n) (w n) q ≠ {}›]
by simp
}
moreover
have "r ≥ min_r"
using Min.coboundedI[OF pre_ranks_finite] pre_ranks_element
unfolding min_r_def by (metis assms(2) calculation(1))
ultimately
show ?thesis
by simp
qed (insert not_less, blast intro: oldest_token_bounded Suc_lessD)+
}
qed

lemma Min_pre_ranks_pre_oldest_tokens:
fixes n
defines "r ≡ (λq. state_rank q n)"
assumes "configuration p (Suc n) ≠ {}"
and "configuration q (Suc n) ≠ {}"
assumes "¬sink q"
and "¬sink p"
shows "Min (pre_ranks r (w n) p) < Min (pre_ranks r (w n) q) ⟷ Min (pre_oldest_tokens p n) < Min (pre_oldest_tokens q n)"
(is "?lhs ⟷ ?rhs")
proof
have pre_ranks_Min: "⋀x ν. (x < Min (pre_ranks r (w n) q)) = (∀a ∈ pre_ranks r (w n) q. x < a)"
using assms pre_ranks_finite Min.bounded_iff pre_ranks_tokens by simp
have pre_oldest_configuration_Min: "⋀x. (x < Min (pre_oldest_tokens q n)) = (∀a∈pre_oldest_tokens q n. x < a)"
using assms pre_oldest_configuration_finite Min.bounded_iff pre_oldest_configuration_tokens by simp
have "⋀x. w x ∈ Σ"
using bounded_w by auto

{
let ?min_i = "Min (pre_ranks r (w n) p)"
let ?min_j = "Min (pre_ranks r (w n) q)"

assume ?lhs

have "?min_i ∈ pre_ranks r (w n) p" and "?min_j ∈ pre_ranks r (w n) q"
using Min_in[OF pre_ranks_finite] assms pre_ranks_tokens by presburger+
hence "?min_i ≤ max_rank" and "?min_j ≤ max_rank"
using pre_ranks_range atLeastAtMost_iff unfolding r_def by blast+
with ‹?lhs› have "?min_i ≠ max_rank"
by linarith
then obtain p' i' where "i' = ?min_i" and "r p' = Some i'" and "p = δ p' (w n)"
using ‹?min_i ∈ pre_ranks r (w n) p› apply (cases "p = q⇩0") apply auto by fastforce
then obtain ot' where "oldest_token p' n = Some ot'"
unfolding assms by (metis push_down_state_rank_oldest_token)
have "state_rank p' n = Some ?min_i"
using ‹i' = ?min_i› ‹r p' = Some i'› unfolding assms by simp
hence "ot' = Min (pre_oldest_tokens p n)"
using pre_ranks_pre_oldest_token_Min_state[OF ‹¬sink p› ‹p = δ p' (w n)› ‹configuration p (Suc n) ≠ {}›] ‹oldest_token p' n = Some ot'›
unfolding r_def by (metis option.inject)
moreover
have "ot' < Suc n"
proof (cases ot' "Suc n" rule: linorder_cases)
case equal
hence "?min_i = max_rank"
using pre_ranks_pre_oldest_token_Min_state_special[of p n, OF ‹¬sink p› ‹configuration p (Suc n) ≠ {}›] assms
unfolding ‹ot' = Min (pre_oldest_tokens p n)› by simp
thus ?thesis
using ‹?min_i ≠ max_rank› by simp
next
case greater
moreover
have "ot' ∈ {0..Suc n}"
using ‹oldest_token p' n = Some ot'›[THEN oldest_token_bounded] by fastforce
ultimately
show ?thesis
by simp
qed simp
moreover
{
fix ot⇩q
assume "ot⇩q ∈ pre_oldest_tokens q n - {Suc n}"
then obtain q' where "oldest_token q' n = Some ot⇩q" and "q = δ q' (w n)"
using pre_oldest_configuration_obtain by blast
moreover
hence "¬sink q'"
using ‹¬sink q› ‹⋀x. w x ∈ Σ› unfolding sink_def by auto
then obtain r⇩q where "state_rank q' n = Some r⇩q"
unfolding assms state_rank.simps using ‹oldest_token q' n = Some ot⇩q›
by (metis oldest_token.simps option.distinct(2))
moreover
hence "r⇩q ∈ pre_ranks r (w n) q"
using ‹q = δ q' (w n)›
unfolding pre_ranks.simps assms by blast
hence "?min_j ≤ r⇩q"
using Min.coboundedI[OF pre_ranks_finite] unfolding assms by blast
hence "?min_i < r⇩q"
using ‹?lhs› by linarith
hence "ot' < ot⇩q"
using state_rank_oldest_token[OF ‹state_rank p' n = Some ?min_i› ‹state_rank q' n = Some r⇩q› ‹oldest_token p' n = Some ot'› ‹oldest_token q' n = Some ot⇩q›]
unfolding assms by simp
}
ultimately
show ?rhs
using pre_oldest_configuration_Min by blast
}

{
define ot_p where "ot_p = Min (pre_oldest_tokens p n)"
define ot_q where "ot_q = Min (pre_oldest_tokens q n)"
assume ?rhs
hence "ot_p < ot_q"
unfolding ot_p_def ot_q_def .

have "oldest_token p (Suc n) = Some ot_p" and "oldest_token q (Suc n) = Some ot_q"
unfolding ot_p_def ot_q_def oldest_token_rec pre_oldest_configuration_tokens by (metis assms)+

(* Min oldest ⟷ Min rank *)
define min_r⇩p where "min_r⇩p = Min (pre_ranks r (w n) p)"
hence "min_r⇩p ∈ pre_ranks r (w n) p"
using pre_ranks_Min_in assms pre_ranks_tokens by simp
hence *: "min_r⇩p < max_rank"
proof (cases min_r⇩p max_rank rule: linorder_cases)
case equal
hence "ot_p = Suc n"
using pre_ranks_pre_oldest_token_Min_state_special[of p n, OF _ ‹configuration p (Suc n) ≠ {}›] assms
unfolding ot_p_def min_r⇩p_def by simp
moreover
have "Min (pre_oldest_tokens q n) ∈ pre_oldest_tokens q n"
using Min_in[OF pre_oldest_configuration_finite ] assms pre_oldest_configuration_tokens by presburger
hence "ot_q ∈ {0..Suc n}"
using pre_oldest_configuration_range[of q n]
unfolding ot_q_def by blast
hence "ot_q ≤ Suc n"
by simp
ultimately
show ?thesis
using ‹ot_p < ot_q› by simp
next
case greater
moreover
have "min_r⇩p ∈ {0..max_rank}"
using pre_ranks_range ‹min_r⇩p ∈ pre_ranks r (w n) p›
unfolding r_def ..
ultimately
show ?thesis
by simp
qed simp
moreover
from * have "min_r⇩p ∈ pre_ranks r (w n) p - {max_rank}"
using ‹min_r⇩p ∈ pre_ranks r (w n) p› by simp
then obtain p' where "r p' = Some min_r⇩p" and "p = δ p' (w n)"
using pre_ranks_state_obtain by blast
hence "oldest_token p' n = Some ot_p"
using pre_ranks_pre_oldest_token_Min_state[OF ‹¬sink p› ‹p = δ p' (w n)› ‹configuration p (Suc n) ≠ {}›]
unfolding r_def[symmetric] min_r⇩p_def[symmetric] ot_p_def[symmetric] by (metis r_def)
{
fix r⇩q
assume "r⇩q ∈ pre_ranks r (w n) q - {max_rank}"
then obtain q' where q': "r q' = Some r⇩q" "q = δ q' (w n)"
using pre_ranks_state_obtain by blast
moreover
from q' obtain ot_q' where ot_q': "oldest_token q' n = Some ot_q'"
unfolding assms by (metis push_down_state_rank_oldest_token)
moreover
from ot_q' have "ot_q' ∈ pre_oldest_tokens q n"
using ‹q = δ q' (w n)›
unfolding pre_oldest_tokens.simps by blast
hence "ot_q ≤ ot_q'"
unfolding ot_q_def
by (rule Min.coboundedI[OF pre_oldest_configuration_finite])
hence "ot_p < ot_q'"
using ‹ot_p < ot_q› by linarith
ultimately
have "min_r⇩p < r⇩q"
using state_rank_oldest_token ‹r p' = Some min_r⇩p› ‹oldest_token p' n = Some ot_p›
unfolding assms by blast
}
ultimately
show ?lhs
using pre_ranks_Min unfolding min_r⇩p_def by blast
}
qed

subsubsection ‹Definition of initial and step›

lemma state_rank_initial:
"state_rank q 0 = initial q"
using state_rank_initial_state by force

lemma state_rank_step:
"state_rank q (Suc n) = step (λq. state_rank q n) (w n) q"
(is "?lhs = ?rhs")
proof (cases "sink q")
case False
{
assume "configuration q (Suc n) = {}"
hence ?thesis
using False pull_up_configuration_state_rank pre_ranks_tokens
unfolding step.simps by presburger
}
moreover
{
assume "configuration q (Suc n) ≠ {}"
hence "?lhs = Some (card (senior_states q (Suc n)))"
using False unfolding state_rank.simps by presburger
also
have "… = ?rhs"
proof -
let ?r = "λq. state_rank q n"
have "{q'. ¬sink q' ∧ pre_ranks ?r (w n) q' ≠ {} ∧ Min (pre_ranks ?r (w n) q') < Min (pre_ranks ?r (w n) q)} = senior_states q (Suc n)"
(is "?S = ?S'")
proof (rule set_eqI)
fix q'
have "q' ∈ ?S ⟷ ¬sink q' ∧ configuration q' (Suc n) ≠ {} ∧ Min (pre_ranks  ?r (w n) q') < Min (pre_ranks ?r (w n) q)"
using pre_ranks_tokens by blast
also
have "… ⟷ ¬sink q' ∧ configuration q' (Suc n) ≠ {} ∧ Min (pre_oldest_tokens q' n) < Min (pre_oldest_tokens q n)"
by (metis ‹configuration q (Suc n) ≠ {}› ‹¬sink q› Min_pre_ranks_pre_oldest_tokens)
also
have "… ⟷ ¬sink q' ∧ (∃x y. oldest_token q' (Suc n) = Some y ∧ oldest_token q (Suc n) = Some x ∧ y < x)"
unfolding oldest_token_rec by (metis pre_oldest_configuration_tokens ‹configuration q (Suc n) ≠ {}› option.distinct(2) option.sel)
finally
show "q' ∈ ?S ⟷ q' ∈ ?S'"
unfolding senior_states.simps by blast
qed
thus ?thesis
using ‹¬sink q› ‹configuration q (Suc n) ≠ {}›
unfolding step.simps pre_ranks_tokens[OF ‹¬sink q›] by presburger
qed
finally
have ?thesis .
}
ultimately
show ?thesis
by blast
qed auto

lemma state_rank_step_foldl:
"(λq. state_rank q n) = foldl step initial (map w [0..<n])"
by (induction n) (unfold state_rank_initial state_rank_step, simp_all)

end

end
`