feat: Port/Combinatorics.Hindman (#2280)

port of combinatorics.hindman



Co-authored-by: Johan Commelin <johan@commelin.net>

Mathlib.lean

@@ -387,6 +387,7 @@ import Mathlib.Combinatorics.Composition
 import Mathlib.Combinatorics.DoubleCounting
 import Mathlib.Combinatorics.HalesJewett
 import Mathlib.Combinatorics.Hall.Finite
+import Mathlib.Combinatorics.Hindman
 import Mathlib.Combinatorics.Partition
 import Mathlib.Combinatorics.Pigeonhole
 import Mathlib.Combinatorics.Quiver.Arborescence

Mathlib/Combinatorics/Hindman.lean

@@ -0,0 +1,310 @@
+/-
+Copyright (c) 2021 David Wärn. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Wärn
+
+! This file was ported from Lean 3 source module combinatorics.hindman
+! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Topology.StoneCech
+import Mathlib.Topology.Algebra.Semigroup
+import Mathlib.Data.Stream.Init
+
+/-!
+# Hindman's theorem on finite sums
+
+We prove Hindman's theorem on finite sums, using idempotent ultrafilters.
+
+Given an infinite sequence `a₀, a₁, a₂, …` of positive integers, the set `FS(a₀, …)` is the set
+of positive integers that can be expressed as a finite sum of `aᵢ`'s, without repetition. Hindman's
+theorem asserts that whenever the positive integers are finitely colored, there exists a sequence
+`a₀, a₁, a₂, …` such that `FS(a₀, …)` is monochromatic. There is also a stronger version, saying
+that whenever a set of the form `FS(a₀, …)` is finitely colored, there exists a sequence
+`b₀, b₁, b₂, …` such that `FS(b₀, …)` is monochromatic and contained in `FS(a₀, …)`. We prove both
+these versions for a general semigroup `M` instead of `ℕ+` since it is no harder, although this
+special case implies the general case.
+
+The idea of the proof is to extend the addition `(+) : M → M → M` to addition `(+) : βM → βM → βM`
+on the space `βM` of ultrafilters on `M`. One can prove that if `U` is an _idempotent_ ultrafilter,
+i.e. `U + U = U`, then any `U`-large subset of `M` contains some set `FS(a₀, …)` (see
+`exists_FS_of_large`). And with the help of a general topological argument one can show that any set
+of the form `FS(a₀, …)` is `U`-large according to some idempotent ultrafilter `U` (see
+`exists_idempotent_ultrafilter_le_FS`). This is enough to prove the theorem since in any finite
+partition of a `U`-large set, one of the parts is `U`-large.
+
+## Main results
+
+- `FS_partition_regular`: the strong form of Hindman's theorem
+- `exists_FS_of_finite_cover`: the weak form of Hindman's theorem
+
+## Tags
+
+Ramsey theory, ultrafilter
+
+-/
+
+
+open Filter
+
+/-- Multiplication of ultrafilters given by `∀ᶠ m in U*V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m*m')`. -/
+@[to_additive
+      "Addition of ultrafilters given by\n`∀ᶠ m in U+V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m+m')`."]
+def Ultrafilter.hasMul {M} [Mul M] : Mul (Ultrafilter M) where mul U V := (· * ·) <$> U <*> V
+#align ultrafilter.has_mul Ultrafilter.hasMul
+#align ultrafilter.has_add Ultrafilter.hasAdd
+
+attribute [local instance] Ultrafilter.hasMul Ultrafilter.hasAdd
+
+/- We could have taken this as the definition of `U * V`, but then we would have to prove that it
+defines an ultrafilter. -/
+@[to_additive]
+theorem Ultrafilter.eventually_mul {M} [Mul M] (U V : Ultrafilter M) (p : M → Prop) :
+    (∀ᶠ m in ↑(U * V), p m) ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m * m') :=
+  Iff.rfl
+#align ultrafilter.eventually_mul Ultrafilter.eventually_mul
+#align ultrafilter.eventually_add Ultrafilter.eventually_add
+
+-- porting note: slow to typecheck
+/-- Semigroup structure on `Ultrafilter M` induced by a semigroup structure on `M`. -/
+@[to_additive
+      "Additive semigroup structure on `Ultrafilter M` induced by an additive semigroup
+      structure on `M`."]
+def Ultrafilter.semigroup {M} [Semigroup M] : Semigroup (Ultrafilter M) :=
+  { Ultrafilter.hasMul with
+    mul_assoc := fun U V W =>
+      Ultrafilter.coe_inj.mp <|
+        Filter.ext' fun p => by simp only [Ultrafilter.eventually_mul, mul_assoc] }
+#align ultrafilter.semigroup Ultrafilter.semigroup
+#align ultrafilter.add_semigroup Ultrafilter.addSemigroup
+
+attribute [local instance] Ultrafilter.semigroup Ultrafilter.addSemigroup
+
+-- We don't prove `continuous_mul_right`, because in general it is false!
+@[to_additive]
+theorem Ultrafilter.continuous_mul_left {M} [Semigroup M] (V : Ultrafilter M) :
+    Continuous (· * V) :=
+  TopologicalSpace.IsTopologicalBasis.continuous ultrafilterBasis_is_basis _ <|
+    Set.forall_range_iff.mpr fun s => ultrafilter_isOpen_basic { m : M | ∀ᶠ m' in V, m * m' ∈ s }
+#align ultrafilter.continuous_mul_left Ultrafilter.continuous_mul_left
+#align ultrafilter.continuous_add_left Ultrafilter.continuous_add_left
+
+namespace Hindman
+
+-- porting note: mathport wants these names to be `fS`, `fP`, etc, but this does violence to
+-- mathematical naming conventions, as does `fs`, `fp`, so we just followed `mathlib` 3 here
+
+/-- `FS a` is the set of finite sums in `a`, i.e. `m ∈ FS a` if `m` is the sum of a nonempty
+subsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/
+inductive FS {M} [AddSemigroup M] : Stream' M → Set M
+  | head (a : Stream' M) : FS a a.head
+  | tail (a : Stream' M) (m : M) (h : FS a.tail m) : FS a m
+  | cons (a : Stream' M) (m : M) (h : FS a.tail m) : FS a (a.head + m)
+set_option linter.uppercaseLean3 false in
+#align hindman.FS Hindman.FS
+
+/-- `FP a` is the set of finite products in `a`, i.e. `m ∈ FP a` if `m` is the product of a nonempty
+subsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/
+@[to_additive FS]
+inductive FP {M} [Semigroup M] : Stream' M → Set M
+  | head (a : Stream' M) : FP a a.head
+  | tail (a : Stream' M) (m : M) (h : FP a.tail m) : FP a m
+  | cons (a : Stream' M) (m : M) (h : FP a.tail m) : FP a (a.head * m)
+set_option linter.uppercaseLean3 false in
+#align hindman.FP Hindman.FP
+
+/-- If `m` and `m'` are finite products in `M`, then so is `m * m'`, provided that `m'` is obtained
+from a subsequence of `M` starting sufficiently late. -/
+@[to_additive
+      "If `m` and `m'` are finite sums in `M`, then so is `m + m'`, provided that `m'`
+      is obtained from a subsequence of `M` starting sufficiently late."]
+theorem FP.mul {M} [Semigroup M] {a : Stream' M} {m : M} (hm : m ∈ FP a) :
+    ∃ n, ∀ m' ∈ FP (a.drop n), m * m' ∈ FP a := by
+  induction' hm with a a m hm ih a m hm ih
+  · exact ⟨1, fun m hm => FP.cons a m hm⟩
+  · cases' ih with n hn
+    use n + 1
+    intro m' hm'
+    exact FP.tail _ _ (hn _ hm')
+  · cases' ih with n hn
+    use n + 1
+    intro m' hm'
+    rw [mul_assoc]
+    exact FP.cons _ _ (hn _ hm')
+set_option linter.uppercaseLean3 false in
+#align hindman.FP.mul Hindman.FP.mul
+set_option linter.uppercaseLean3 false in
+#align hindman.FS.add Hindman.FS.add
+
+@[to_additive exists_idempotent_ultrafilter_le_FS]
+theorem exists_idempotent_ultrafilter_le_FP {M} [Semigroup M] (a : Stream' M) :
+    ∃ U : Ultrafilter M, U * U = U ∧ ∀ᶠ m in U, m ∈ FP a := by
+  let S : Set (Ultrafilter M) := ⋂ n, { U | ∀ᶠ m in U, m ∈ FP (a.drop n) }
+  have h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_
+  rcases h with ⟨U, hU, U_idem⟩
+  · refine' ⟨U, U_idem, _⟩
+    convert Set.mem_interᵢ.mp hU 0
+  · exact Ultrafilter.continuous_mul_left
+  · apply IsCompact.nonempty_interᵢ_of_sequence_nonempty_compact_closed
+    · intro n U hU
+      apply Eventually.mono hU
+      rw [add_comm, ← Stream'.drop_drop, ← Stream'.tail_eq_drop]
+      exact FP.tail _
+    · intro n
+      exact ⟨pure _, mem_pure.mpr <| FP.head _⟩
+    · exact (ultrafilter_isClosed_basic _).isCompact
+    · intro n
+      apply ultrafilter_isClosed_basic
+  · exact IsClosed.isCompact (isClosed_interᵢ fun i => ultrafilter_isClosed_basic _)
+  · intro U hU V hV
+    rw [Set.mem_interᵢ] at *
+    intro n
+    rw [Set.mem_setOf_eq, Ultrafilter.eventually_mul]
+    apply Eventually.mono (hU n)
+    intro m hm
+    obtain ⟨n', hn⟩ := FP.mul hm
+    apply Eventually.mono (hV (n' + n))
+    intro m' hm'
+    apply hn
+    simpa only [Stream'.drop_drop] using hm'
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_idempotent_ultrafilter_le_FP Hindman.exists_idempotent_ultrafilter_le_FP
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_idempotent_ultrafilter_le_FS Hindman.exists_idempotent_ultrafilter_le_FS
+
+@[to_additive exists_FS_of_large]
+theorem exists_FP_of_large {M} [Semigroup M] (U : Ultrafilter M) (U_idem : U * U = U) (s₀ : Set M)
+    (sU : s₀ ∈ U) : ∃ a, FP a ⊆ s₀ := by
+  /- Informally: given a `U`-large set `s₀`, the set `s₀ ∩ { m | ∀ᶠ m' in U, m * m' ∈ s₀ }` is also
+  `U`-large (since `U` is idempotent). Thus in particular there is an `a₀` in this intersection. Now
+  let `s₁` be the intersection `s₀ ∩ { m | a₀ * m ∈ s₀ }`. By choice of `a₀`, this is again
+  `U`-large, so we can repeat the argument starting from `s₁`, obtaining `a₁`, `s₂`, etc.
+  This gives the desired infinite sequence. -/
+  have exists_elem : ∀ {s : Set M} (_hs : s ∈ U), (s ∩ { m | ∀ᶠ m' in U, m * m' ∈ s }).Nonempty :=
+    fun {s} hs =>
+    Ultrafilter.nonempty_of_mem
+      (inter_mem hs <| by
+        rw [← U_idem] at hs
+        exact hs)
+  let elem : { s // s ∈ U } → M := fun p => (exists_elem p.property).some
+  let succ : {s // s ∈ U} → {s // s ∈ U} := fun (p : {s // s ∈ U}) =>
+        ⟨p.val ∩ {m : M | elem p * m ∈ p.val},
+         inter_mem p.property
+           (show (exists_elem p.property).some ∈ {m : M | ∀ᶠ (m' : M) in ↑U, m * m' ∈ p.val} from
+              p.val.inter_subset_right {m : M | ∀ᶠ (m' : M) in ↑U, m * m' ∈ p.val}
+                (exists_elem p.property).some_mem)⟩
+  use Stream'.corec elem succ (Subtype.mk s₀ sU)
+  suffices ∀ (a : Stream' M), ∀ m ∈ FP a, ∀ p, a = Stream'.corec elem succ p → m ∈ p.val
+    by
+    intro m hm
+    exact this _ m hm ⟨s₀, sU⟩ rfl
+  clear sU s₀
+  intro a m h
+  induction' h with b b n h ih b n h ih
+  · rintro p rfl
+    rw [Stream'.corec_eq, Stream'.head_cons]
+    exact Set.inter_subset_left _ _ (Set.Nonempty.some_mem _)
+  · rintro p rfl
+    refine' Set.inter_subset_left _ _ (ih (succ p) _)
+    rw [Stream'.corec_eq, Stream'.tail_cons]
+  · rintro p rfl
+    have := Set.inter_subset_right _ _ (ih (succ p) ?_)
+    · simpa only using this
+    rw [Stream'.corec_eq, Stream'.tail_cons]
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_FP_of_large Hindman.exists_FP_of_large
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_FS_of_large Hindman.exists_FS_of_large
+
+/-- The strong form of **Hindman's theorem**: in any finite cover of an FP-set, one the parts
+contains an FP-set. -/
+@[to_additive FS_partition_regular
+      "The strong form of **Hindman's theorem**: in any finite cover of
+      an FS-set, one the parts contains an FS-set."]
+theorem FP_partition_regular {M} [Semigroup M] (a : Stream' M) (s : Set (Set M)) (sfin : s.Finite)
+    (scov : FP a ⊆ ⋃₀ s) : ∃ c ∈ s, ∃ b : Stream' M, FP b ⊆ c :=
+  let ⟨U, idem, aU⟩ := exists_idempotent_ultrafilter_le_FP a
+  let ⟨c, cs, hc⟩ := (Ultrafilter.finite_unionₛ_mem_iff sfin).mp (mem_of_superset aU scov)
+  ⟨c, cs, exists_FP_of_large U idem c hc⟩
+set_option linter.uppercaseLean3 false in
+#align hindman.FP_partition_regular Hindman.FP_partition_regular
+set_option linter.uppercaseLean3 false in
+#align hindman.FS_partition_regular Hindman.FS_partition_regular
+
+/-- The weak form of **Hindman's theorem**: in any finite cover of a nonempty semigroup, one of the
+parts contains an FP-set. -/
+@[to_additive exists_FS_of_finite_cover
+      "The weak form of **Hindman's theorem**: in any finite cover
+      of a nonempty additive semigroup, one of the parts contains an FS-set."]
+theorem exists_FP_of_finite_cover {M} [Semigroup M] [Nonempty M] (s : Set (Set M)) (sfin : s.Finite)
+    (scov : ⊤ ⊆ ⋃₀ s) : ∃ c ∈ s, ∃ a : Stream' M, FP a ⊆ c :=
+  let ⟨U, hU⟩ :=
+    exists_idempotent_of_compact_t2_of_continuous_mul_left (@Ultrafilter.continuous_mul_left M _)
+  let ⟨c, c_s, hc⟩ := (Ultrafilter.finite_unionₛ_mem_iff sfin).mp (mem_of_superset univ_mem scov)
+  ⟨c, c_s, exists_FP_of_large U hU c hc⟩
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_FP_of_finite_cover Hindman.exists_FP_of_finite_cover
+set_option linter.uppercaseLean3 false in
+#align hindman.exists_FS_of_finite_cover Hindman.exists_FS_of_finite_cover
+
+@[to_additive FS_iter_tail_sub_FS]
+theorem FP_drop_subset_FP {M} [Semigroup M] (a : Stream' M) (n : ℕ) : FP (a.drop n) ⊆ FP a := by
+  induction' n with n ih; · rfl
+  rw [Nat.succ_eq_one_add, ← Stream'.drop_drop]
+  exact _root_.trans (FP.tail _) ih
+set_option linter.uppercaseLean3 false in
+#align hindman.FP_drop_subset_FP Hindman.FP_drop_subset_FP
+set_option linter.uppercaseLean3 false in
+#align hindman.FS_iter_tail_sub_FS Hindman.FS_iter_tail_sub_FS
+
+@[to_additive]
+theorem FP.singleton {M} [Semigroup M] (a : Stream' M) (i : ℕ) : a.nth i ∈ FP a := by
+  induction' i with i ih generalizing a
+  · apply FP.head
+  · apply FP.tail
+    apply ih
+set_option linter.uppercaseLean3 false in
+#align hindman.FP.singleton Hindman.FP.singleton
+set_option linter.uppercaseLean3 false in
+#align hindman.FS.singleton Hindman.FS.singleton
+
+@[to_additive]
+theorem FP.mul_two {M} [Semigroup M] (a : Stream' M) (i j : ℕ) (ij : i < j) :
+    a.nth i * a.nth j ∈ FP a := by
+  refine' FP_drop_subset_FP _ i _
+  rw [← Stream'.head_drop]
+  apply FP.cons
+  rcases le_iff_exists_add.mp (Nat.succ_le_of_lt ij) with ⟨d, hd⟩
+  have := FP.singleton (a.drop i).tail d
+  rw [Stream'.tail_eq_drop, Stream'.nth_drop, Stream'.nth_drop] at this
+  convert this
+  rw [hd, add_comm, Nat.succ_add, Nat.add_succ]
+set_option linter.uppercaseLean3 false in
+#align hindman.FP.mul_two Hindman.FP.mul_two
+set_option linter.uppercaseLean3 false in
+#align hindman.FS.add_two Hindman.FS.add_two
+
+@[to_additive]
+theorem FP.finset_prod {M} [CommMonoid M] (a : Stream' M) (s : Finset ℕ) (hs : s.Nonempty) :
+    (s.prod fun i => a.nth i) ∈ FP a := by
+  refine' FP_drop_subset_FP _ (s.min' hs) _
+  induction' s using Finset.strongInduction with s ih
+  rw [← Finset.mul_prod_erase _ _ (s.min'_mem hs), ← Stream'.head_drop]
+  cases' (s.erase (s.min' hs)).eq_empty_or_nonempty with h h
+  · rw [h, Finset.prod_empty, mul_one]
+    exact FP.head _
+  · apply FP.cons
+    rw [Stream'.tail_eq_drop, Stream'.drop_drop, add_comm]
+    refine' Set.mem_of_subset_of_mem _ (ih _ (Finset.erase_ssubset <| s.min'_mem hs) h)
+    have : s.min' hs + 1 ≤ (s.erase (s.min' hs)).min' h :=
+      Nat.succ_le_of_lt (Finset.min'_lt_of_mem_erase_min' _ _ <| Finset.min'_mem _ _)
+    cases' le_iff_exists_add.mp this with d hd
+    rw [hd, add_comm, ← Stream'.drop_drop]
+    apply FP_drop_subset_FP
+set_option linter.uppercaseLean3 false in
+#align hindman.FP.finset_prod Hindman.FP.finset_prod
+set_option linter.uppercaseLean3 false in
+#align hindman.FS.finset_sum Hindman.FS.finset_sum
+
+end Hindman