# Theory Ramsey

theory Ramsey
imports Infinite_Set
```(*  Title:      HOL/Library/Ramsey.thy
Author:     Tom Ridge.  Converted to structured Isar by L C Paulson
*)

section ‹Ramsey's Theorem›

theory Ramsey
imports Infinite_Set
begin

subsection ‹Finite Ramsey theorem(s)›

text ‹
To distinguish the finite and infinite ones, lower and upper case
names are used.

This is the most basic version in terms of cliques and independent
sets, i.e. the version for graphs and 2 colours.
›

definition "clique V E ⟷ (∀v∈V. ∀w∈V. v ≠ w ⟶ {v, w} ∈ E)"
definition "indep V E ⟷ (∀v∈V. ∀w∈V. v ≠ w ⟶ {v, w} ∉ E)"

lemma ramsey2:
"∃r≥1. ∀(V::'a set) (E::'a set set). finite V ∧ card V ≥ r ⟶
(∃R ⊆ V. card R = m ∧ clique R E ∨ card R = n ∧ indep R E)"
(is "∃r≥1. ?R m n r")
proof (induct k ≡ "m + n" arbitrary: m n)
case 0
show ?case (is "∃r. ?Q r")
proof
from 0 show "?Q 1"
by (clarsimp simp: indep_def) (metis card.empty emptyE empty_subsetI)
qed
next
case (Suc k)
consider "m = 0 ∨ n = 0" | "m ≠ 0" "n ≠ 0" by auto
then show ?case (is "∃r. ?Q r")
proof cases
case 1
then have "?Q 1"
by (simp add: clique_def) (meson card_empty empty_iff empty_subsetI indep_def)
then show ?thesis ..
next
case 2
with Suc(2) have "k = (m - 1) + n" "k = m + (n - 1)" by auto
from this [THEN Suc(1)]
obtain r1 r2 where "r1 ≥ 1" "r2 ≥ 1" "?R (m - 1) n r1" "?R m (n - 1) r2" by auto
then have "r1 + r2 ≥ 1" by arith
moreover have "?R m n (r1 + r2)" (is "∀V E. _ ⟶ ?EX V E m n")
proof clarify
fix V :: "'a set"
fix E :: "'a set set"
assume "finite V" "r1 + r2 ≤ card V"
with ‹r1 ≥ 1› have "V ≠ {}" by auto
then obtain v where "v ∈ V" by blast
let ?M = "{w ∈ V. w ≠ v ∧ {v, w} ∈ E}"
let ?N = "{w ∈ V. w ≠ v ∧ {v, w} ∉ E}"
from ‹v ∈ V› have "V = insert v (?M ∪ ?N)" by auto
then have "card V = card (insert v (?M ∪ ?N))" by metis
also from ‹finite V› have "… = card ?M + card ?N + 1"
by (fastforce intro: card_Un_disjoint)
finally have "card V = card ?M + card ?N + 1" .
with ‹r1 + r2 ≤ card V› have "r1 + r2 ≤ card ?M + card ?N + 1" by simp
then consider "r1 ≤ card ?M" | "r2 ≤ card ?N" by arith
then show "?EX V E m n"
proof cases
case 1
from ‹finite V› have "finite ?M" by auto
with ‹?R (m - 1) n r1› and 1 have "?EX ?M E (m - 1) n" by blast
then obtain R where "R ⊆ ?M" "v ∉ R"
and CI: "card R = m - 1 ∧ clique R E ∨ card R = n ∧ indep R E" (is "?C ∨ ?I")
by blast
from ‹R ⊆ ?M› have "R ⊆ V" by auto
with ‹finite V› have "finite R" by (metis finite_subset)
from CI show ?thesis
proof
assume "?I"
with ‹R ⊆ V› show ?thesis by blast
next
assume "?C"
with ‹R ⊆ ?M› have *: "clique (insert v R) E"
by (auto simp: clique_def insert_commute)
from ‹?C› ‹finite R› ‹v ∉ R› ‹m ≠ 0› have "card (insert v R) = m" by simp
with ‹R ⊆ V› ‹v ∈ V› * show ?thesis by (metis insert_subset)
qed
next
case 2
from ‹finite V› have "finite ?N" by auto
with ‹?R m (n - 1) r2› 2 have "?EX ?N E m (n - 1)" by blast
then obtain R where "R ⊆ ?N" "v ∉ R"
and CI: "card R = m ∧ clique R E ∨ card R = n - 1 ∧ indep R E" (is "?C ∨ ?I")
by blast
from ‹R ⊆ ?N› have "R ⊆ V" by auto
with ‹finite V› have "finite R" by (metis finite_subset)
from CI show ?thesis
proof
assume "?C"
with ‹R ⊆ V› show ?thesis by blast
next
assume "?I"
with ‹R ⊆ ?N› have *: "indep (insert v R) E"
by (auto simp: indep_def insert_commute)
from ‹?I› ‹finite R› ‹v ∉ R› ‹n ≠ 0› have "card (insert v R) = n" by simp
with ‹R ⊆ V› ‹v ∈ V› * show ?thesis by (metis insert_subset)
qed
qed
qed
ultimately show ?thesis by blast
qed
qed

subsection ‹Preliminaries›

subsubsection ‹``Axiom'' of Dependent Choice›

primrec choice :: "('a ⇒ bool) ⇒ ('a × 'a) set ⇒ nat ⇒ 'a"
where ― ‹An integer-indexed chain of choices›
choice_0: "choice P r 0 = (SOME x. P x)"
| choice_Suc: "choice P r (Suc n) = (SOME y. P y ∧ (choice P r n, y) ∈ r)"

lemma choice_n:
assumes P0: "P x0"
and Pstep: "⋀x. P x ⟹ ∃y. P y ∧ (x, y) ∈ r"
shows "P (choice P r n)"
proof (induct n)
case 0
show ?case by (force intro: someI P0)
next
case Suc
then show ?case by (auto intro: someI2_ex [OF Pstep])
qed

lemma dependent_choice:
assumes trans: "trans r"
and P0: "P x0"
and Pstep: "⋀x. P x ⟹ ∃y. P y ∧ (x, y) ∈ r"
obtains f :: "nat ⇒ 'a" where "⋀n. P (f n)" and "⋀n m. n < m ⟹ (f n, f m) ∈ r"
proof
fix n
show "P (choice P r n)"
by (blast intro: choice_n [OF P0 Pstep])
next
fix n m :: nat
assume "n < m"
from Pstep [OF choice_n [OF P0 Pstep]] have "(choice P r k, choice P r (Suc k)) ∈ r" for k
by (auto intro: someI2_ex)
then show "(choice P r n, choice P r m) ∈ r"
by (auto intro: less_Suc_induct [OF ‹n < m›] transD [OF trans])
qed

subsubsection ‹Partitions of a Set›

definition part :: "nat ⇒ nat ⇒ 'a set ⇒ ('a set ⇒ nat) ⇒ bool"
― ‹the function @{term f} partitions the @{term r}-subsets of the typically
infinite set @{term Y} into @{term s} distinct categories.›
where "part r s Y f ⟷ (∀X. X ⊆ Y ∧ finite X ∧ card X = r ⟶ f X < s)"

text ‹For induction, we decrease the value of @{term r} in partitions.›
lemma part_Suc_imp_part:
"⟦infinite Y; part (Suc r) s Y f; y ∈ Y⟧ ⟹ part r s (Y - {y}) (λu. f (insert y u))"
apply clarify
apply (drule_tac x="insert y X" in spec)
apply force
done

lemma part_subset: "part r s YY f ⟹ Y ⊆ YY ⟹ part r s Y f"
unfolding part_def by blast

subsection ‹Ramsey's Theorem: Infinitary Version›

lemma Ramsey_induction:
fixes s r :: nat
and YY :: "'a set"
and f :: "'a set ⇒ nat"
assumes "infinite YY" "part r s YY f"
shows "∃Y' t'. Y' ⊆ YY ∧ infinite Y' ∧ t' < s ∧ (∀X. X ⊆ Y' ∧ finite X ∧ card X = r ⟶ f X = t')"
using assms
proof (induct r arbitrary: YY f)
case 0
then show ?case
by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
next
case (Suc r)
show ?case
proof -
from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy ∈ YY"
by blast
let ?ramr = "{((y, Y, t), (y', Y', t')). y' ∈ Y ∧ Y' ⊆ Y}"
let ?propr = "λ(y, Y, t).
y ∈ YY ∧ y ∉ Y ∧ Y ⊆ YY ∧ infinite Y ∧ t < s
∧ (∀X. X⊆Y ∧ finite X ∧ card X = r ⟶ (f ∘ insert y) X = t)"
from Suc.prems have infYY': "infinite (YY - {yy})" by auto
from Suc.prems have partf': "part r s (YY - {yy}) (f ∘ insert yy)"
by (simp add: o_def part_Suc_imp_part yy)
have transr: "trans ?ramr" by (force simp add: trans_def)
from Suc.hyps [OF infYY' partf']
obtain Y0 and t0 where "Y0 ⊆ YY - {yy}" "infinite Y0" "t0 < s"
"X ⊆ Y0 ∧ finite X ∧ card X = r ⟶ (f ∘ insert yy) X = t0" for X
by blast
with yy have propr0: "?propr(yy, Y0, t0)" by blast
have proprstep: "∃y. ?propr y ∧ (x, y) ∈ ?ramr" if x: "?propr x" for x
proof (cases x)
case (fields yx Yx tx)
with x obtain yx' where yx': "yx' ∈ Yx"
by (blast dest: infinite_imp_nonempty)
from fields x have infYx': "infinite (Yx - {yx'})" by auto
with fields x yx' Suc.prems have partfx': "part r s (Yx - {yx'}) (f ∘ insert yx')"
by (simp add: o_def part_Suc_imp_part part_subset [where YY=YY and Y=Yx])
from Suc.hyps [OF infYx' partfx'] obtain Y' and t'
where Y': "Y' ⊆ Yx - {yx'}" "infinite Y'" "t' < s"
"X ⊆ Y' ∧ finite X ∧ card X = r ⟶ (f ∘ insert yx') X = t'" for X
by blast
from fields x Y' yx' have "?propr (yx', Y', t') ∧ (x, (yx', Y', t')) ∈ ?ramr"
by blast
then show ?thesis ..
qed
from dependent_choice [OF transr propr0 proprstep]
obtain g where pg: "?propr (g n)" and rg: "n < m ⟹ (g n, g m) ∈ ?ramr" for n m :: nat
by blast
let ?gy = "fst ∘ g"
let ?gt = "snd ∘ snd ∘ g"
have rangeg: "∃k. range ?gt ⊆ {..<k}"
proof (intro exI subsetI)
fix x
assume "x ∈ range ?gt"
then obtain n where "x = ?gt n" ..
with pg [of n] show "x ∈ {..<s}" by (cases "g n") auto
qed
have "finite (range ?gt)"
then obtain s' and n' where s': "s' = ?gt n'" and infeqs': "infinite {n. ?gt n = s'}"
by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: infinite_UNIV_nat)
with pg [of n'] have less': "s'<s" by (cases "g n'") auto
have inj_gy: "inj ?gy"
proof (rule linorder_injI)
fix m m' :: nat
assume "m < m'"
from rg [OF this] pg [of m] show "?gy m ≠ ?gy m'"
by (cases "g m", cases "g m'") auto
qed
show ?thesis
proof (intro exI conjI)
from pg show "?gy ` {n. ?gt n = s'} ⊆ YY"
by (auto simp add: Let_def split_beta)
from infeqs' show "infinite (?gy ` {n. ?gt n = s'})"
by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
show "s' < s" by (rule less')
show "∀X. X ⊆ ?gy ` {n. ?gt n = s'} ∧ finite X ∧ card X = Suc r ⟶ f X = s'"
proof -
have "f X = s'"
if X: "X ⊆ ?gy ` {n. ?gt n = s'}"
and cardX: "finite X" "card X = Suc r"
for X
proof -
from X obtain AA where AA: "AA ⊆ {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
with cardX have "AA ≠ {}" by auto
then have AAleast: "(LEAST x. x ∈ AA) ∈ AA" by (auto intro: LeastI_ex)
show ?thesis
proof (cases "g (LEAST x. x ∈ AA)")
case (fields ya Ya ta)
with AAleast Xeq have ya: "ya ∈ X" by (force intro!: rev_image_eqI)
then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
also have "… = ta"
proof -
have *: "X - {ya} ⊆ Ya"
proof
fix x assume x: "x ∈ X - {ya}"
then obtain a' where xeq: "x = ?gy a'" and a': "a' ∈ AA"
with fields x have "a' ≠ (LEAST x. x ∈ AA)" by auto
with Least_le [of "λx. x ∈ AA", OF a'] have "(LEAST x. x ∈ AA) < a'"
by arith
from xeq fields rg [OF this] show "x ∈ Ya" by auto
qed
have "card (X - {ya}) = r"
with pg [of "LEAST x. x ∈ AA"] fields cardX * show ?thesis
by (auto simp del: insert_Diff_single)
qed
also from AA AAleast fields have "… = s'" by auto
finally show ?thesis .
qed
qed
then show ?thesis by blast
qed
qed
qed
qed

theorem Ramsey:
fixes s r :: nat
and Z :: "'a set"
and f :: "'a set ⇒ nat"
shows
"⟦infinite Z;
∀X. X ⊆ Z ∧ finite X ∧ card X = r ⟶ f X < s⟧
⟹ ∃Y t. Y ⊆ Z ∧ infinite Y ∧ t < s
∧ (∀X. X ⊆ Y ∧ finite X ∧ card X = r ⟶ f X = t)"
by (blast intro: Ramsey_induction [unfolded part_def])

corollary Ramsey2:
fixes s :: nat
and Z :: "'a set"
and f :: "'a set ⇒ nat"
assumes infZ: "infinite Z"
and part: "∀x∈Z. ∀y∈Z. x ≠ y ⟶ f {x, y} < s"
shows "∃Y t. Y ⊆ Z ∧ infinite Y ∧ t < s ∧ (∀x∈Y. ∀y∈Y. x≠y ⟶ f {x, y} = t)"
proof -
from part have part2: "∀X. X ⊆ Z ∧ finite X ∧ card X = 2 ⟶ f X < s"
by (fastforce simp add: eval_nat_numeral card_Suc_eq)
obtain Y t where *:
"Y ⊆ Z" "infinite Y" "t < s" "(∀X. X ⊆ Y ∧ finite X ∧ card X = 2 ⟶ f X = t)"
by (insert Ramsey [OF infZ part2]) auto
then have "∀x∈Y. ∀y∈Y. x ≠ y ⟶ f {x, y} = t" by auto
with * show ?thesis by iprover
qed

subsection ‹Disjunctive Well-Foundedness›

text ‹
An application of Ramsey's theorem to program termination. See
@{cite "Podelski-Rybalchenko"}.
›

definition disj_wf :: "('a × 'a) set ⇒ bool"
where "disj_wf r ⟷ (∃T. ∃n::nat. (∀i<n. wf (T i)) ∧ r = (⋃i<n. T i))"

definition transition_idx :: "(nat ⇒ 'a) ⇒ (nat ⇒ ('a × 'a) set) ⇒ nat set ⇒ nat"
where "transition_idx s T A = (LEAST k. ∃i j. A = {i, j} ∧ i < j ∧ (s j, s i) ∈ T k)"

lemma transition_idx_less:
assumes "i < j" "(s j, s i) ∈ T k" "k < n"
shows "transition_idx s T {i, j} < n"
proof -
from assms(1,2) have "transition_idx s T {i, j} ≤ k"
by (simp add: transition_idx_def, blast intro: Least_le)
with assms(3) show ?thesis by simp
qed

lemma transition_idx_in:
assumes "i < j" "(s j, s i) ∈ T k"
shows "(s j, s i) ∈ T (transition_idx s T {i, j})"
using assms
by (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR cong: conj_cong) (erule LeastI)

text ‹To be equal to the union of some well-founded relations is equivalent
to being the subset of such a union.›
lemma disj_wf: "disj_wf r ⟷ (∃T. ∃n::nat. (∀i<n. wf(T i)) ∧ r ⊆ (⋃i<n. T i))"
apply (rule_tac x="λi. T i Int r" in exI)
apply (rule_tac x=n in exI)
done

theorem trans_disj_wf_implies_wf:
assumes "trans r"
and "disj_wf r"
shows "wf r"
proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
assume "∃s. ∀i. (s (Suc i), s i) ∈ r"
then obtain s where sSuc: "∀i. (s (Suc i), s i) ∈ r" ..
from ‹disj_wf r› obtain T and n :: nat where wfT: "∀k<n. wf(T k)" and r: "r = (⋃k<n. T k)"
have s_in_T: "∃k. (s j, s i) ∈ T k ∧ k<n" if "i < j" for i j
proof -
from ‹i < j› have "(s j, s i) ∈ r"
proof (induct rule: less_Suc_induct)
case 1
then show ?case by (simp add: sSuc)
next
case 2
with ‹trans r› show ?case
unfolding trans_def by blast
qed
then show ?thesis by (auto simp add: r)
qed
have trless: "i ≠ j ⟹ transition_idx s T {i, j} < n" for i j
apply (blast dest: s_in_T transition_idx_less)
apply (subst insert_commute)
apply (blast dest: s_in_T transition_idx_less)
done
have "∃K k. K ⊆ UNIV ∧ infinite K ∧ k < n ∧
(∀i∈K. ∀j∈K. i ≠ j ⟶ transition_idx s T {i, j} = k)"
by (rule Ramsey2) (auto intro: trless infinite_UNIV_nat)
then obtain K and k where infK: "infinite K" and "k < n"
and allk: "∀i∈K. ∀j∈K. i ≠ j ⟶ transition_idx s T {i, j} = k"
by auto
have "(s (enumerate K (Suc m)), s (enumerate K m)) ∈ T k" for m :: nat
proof -
let ?j = "enumerate K (Suc m)"
let ?i = "enumerate K m"
have ij: "?i < ?j" by (simp add: enumerate_step infK)
have "?j ∈ K" "?i ∈ K" by (simp_all add: enumerate_in_set infK)
with ij have k: "k = transition_idx s T {?i, ?j}" by (simp add: allk)
from s_in_T [OF ij] obtain k' where "(s ?j, s ?i) ∈ T k'" "k'<n" by blast
then show "(s ?j, s ?i) ∈ T k" by (simp add: k transition_idx_in ij)
qed
then have "¬ wf (T k)"
unfolding wf_iff_no_infinite_down_chain by fast
with wfT ‹k < n› show False by blast
qed

end
```