Diophantine functions and Matiyasevic's theorem

Hilbert's tenth problem asked whether there exists an algorithm which for a given integer polynomial determines whether this polynomial has integer solutions. It was answered in the negative in 1970, the final step being completed by Matiyasevic who showed that the power function is Diophantine.

Here a function is called Diophantine if its graph is Diophantine as a set. A subset S ⊆ ℕ ^ α in turn is called Diophantine if there exists an integer polynomial on α ⊕ β such that v ∈ S iff there exists t : ℕ^β with p (v, t) = 0.

Main definitions

• IsPoly: a predicate stating that a function is a multivariate integer polynomial.
• Poly: the type of multivariate integer polynomial functions.
• Dioph: a predicate stating that a set is Diophantine, i.e. a set S ⊆ ℕ^α is Diophantine if there exists a polynomial on α ⊕ β such that v ∈ S iff there exists t : ℕ^β with p (v, t) = 0.
• DiophFn: a predicate on a function stating that it is Diophantine in the sense that its graph is Diophantine as a set.

Main statements

• pell_dioph states that solutions to Pell's equation form a Diophantine set.
• pow_dioph states that the power function is Diophantine, a version of Matiyasevic's theorem.

References

• [M. Carneiro, A Lean formalization of Matiyasevic's theorem][carneiro2018matiyasevic]
• [M. Davis, Hilbert's tenth problem is unsolvable][MR317916]

Tags

Matiyasevic's theorem, Hilbert's tenth problem

TODO

• Finish the solution of Hilbert's tenth problem.
• Connect Poly to MvPolynomial

Multivariate integer polynomials

Note that this duplicates MvPolynomial.

inductive IsPoly {α : Type u_1} : ((α → ℕ) → ℤ) → Prop

A predicate asserting that a function is a multivariate integer polynomial. (We are being a bit lazy here by allowing many representations for multiplication, rather than only allowing monomials and addition, but the definition is equivalent and this is easier to use.)

• proj {α : Type u_1} (i : α) : IsPoly fun (x : α → ℕ) => ↑(x i)
• const {α : Type u_1} (n : ℤ) : IsPoly fun (x : α → ℕ) => n
• sub {α : Type u_1} {f g : (α → ℕ) → ℤ} : IsPoly f → IsPoly g → IsPoly fun (x : α → ℕ) => f x - g x
• mul {α : Type u_1} {f g : (α → ℕ) → ℤ} : IsPoly f → IsPoly g → IsPoly fun (x : α → ℕ) => f x * g x

theorem IsPoly.neg {α : Type u_1} {f : (α → ℕ) → ℤ} : IsPoly f → IsPoly (-f)

theorem IsPoly.add {α : Type u_1} {f g : (α → ℕ) → ℤ} (hf : IsPoly f) (hg : IsPoly g) : IsPoly (f + g)

def Poly (α : Type u) : Type u

The type of multivariate integer polynomials

instance Poly.instFunLike {α : Type u_1} : FunLike (Poly α) (α → ℕ) ℤ

theorem Poly.isPoly {α : Type u_1} (f : Poly α) : IsPoly ⇑f

The underlying function of a Poly is a polynomial

theorem Poly.ext {α : Type u_1} {f g : Poly α} : (∀ (x : α → ℕ), f x = g x) → f = g

Extensionality for Poly α

theorem Poly.ext_iff {α : Type u_1} {f g : Poly α} : f = g ↔ ∀ (x : α → ℕ), f x = g x

def Poly.proj {α : Type u_1} (i : α) : Poly α

The ith projection function, x_i.

@[simp]

theorem Poly.proj_apply {α : Type u_1} (i : α) (x : α → ℕ) : (proj i) x = ↑(x i)

def Poly.const {α : Type u_1} (n : ℤ) : Poly α

The constant function with value n : ℤ.

@[simp]

theorem Poly.const_apply {α : Type u_1} (n : ℤ) (x : α → ℕ) : (const n) x = n

instance Poly.instZero {α : Type u_1} : Zero (Poly α)

instance Poly.instOne {α : Type u_1} : One (Poly α)

instance Poly.instNeg {α : Type u_1} : Neg (Poly α)

instance Poly.instAdd {α : Type u_1} : Add (Poly α)

instance Poly.instSub {α : Type u_1} : Sub (Poly α)

instance Poly.instMul {α : Type u_1} : Mul (Poly α)

@[simp]

theorem Poly.coe_zero {α : Type u_1} : ⇑0 = ⇑(const 0)

@[simp]

theorem Poly.coe_one {α : Type u_1} : ⇑1 = ⇑(const 1)

@[simp]

theorem Poly.coe_neg {α : Type u_1} (f : Poly α) : ⇑(-f) = -⇑f

@[simp]

theorem Poly.coe_add {α : Type u_1} (f g : Poly α) : ⇑(f + g) = ⇑f + ⇑g

@[simp]

theorem Poly.coe_sub {α : Type u_1} (f g : Poly α) : ⇑(f - g) = ⇑f - ⇑g

@[simp]

theorem Poly.coe_mul {α : Type u_1} (f g : Poly α) : ⇑(f * g) = ⇑f * ⇑g

@[simp]

theorem Poly.zero_apply {α : Type u_1} (x : α → ℕ) : 0 x = 0

@[simp]

theorem Poly.one_apply {α : Type u_1} (x : α → ℕ) : 1 x = 1

@[simp]

theorem Poly.neg_apply {α : Type u_1} (f : Poly α) (x : α → ℕ) : (-f) x = -f x

@[simp]

theorem Poly.add_apply {α : Type u_1} (f g : Poly α) (x : α → ℕ) : (f + g) x = f x + g x

@[simp]

theorem Poly.sub_apply {α : Type u_1} (f g : Poly α) (x : α → ℕ) : (f - g) x = f x - g x

@[simp]

theorem Poly.mul_apply {α : Type u_1} (f g : Poly α) (x : α → ℕ) : (f * g) x = f x * g x

instance Poly.instInhabited (α : Type u_3) : Inhabited (Poly α)

instance Poly.instAddCommGroup {α : Type u_1} : AddCommGroup (Poly α)

instance Poly.instAddGroupWithOne {α : Type u_1} : AddGroupWithOne (Poly α)

instance Poly.instCommRing {α : Type u_1} : CommRing (Poly α)

theorem Poly.induction {α : Type u_1} {C : Poly α → Prop} (H1 : ∀ (i : α), C (proj i)) (H2 : ∀ (n : ℤ), C (const n)) (H3 : ∀ (f g : Poly α), C f → C g → C (f - g)) (H4 : ∀ (f g : Poly α), C f → C g → C (f * g)) (f : Poly α) : C f

def Poly.sumsq {α : Type u_1} : List (Poly α) → Poly α

The sum of squares of a list of polynomials. This is relevant for Diophantine equations, because it means that a list of equations can be encoded as a single equation: x = 0 ∧ y = 0 ∧ z = 0 is equivalent to x^2 + y^2 + z^2 = 0.

theorem Poly.sumsq_nonneg {α : Type u_1} (x : α → ℕ) (l : List (Poly α)) : 0 ≤ (sumsq l) x

theorem Poly.sumsq_eq_zero {α : Type u_1} (x : α → ℕ) (l : List (Poly α)) : (sumsq l) x = 0 ↔ List.Forall (fun (a : Poly α) => a x = 0) l

def Poly.map {α : Type u_3} {β : Type u_4} (f : α → β) (g : Poly α) : Poly β

Map the index set of variables, replacing x_i with x_(f i).

@[simp]

theorem Poly.map_apply {α : Type u_3} {β : Type u_4} (f : α → β) (g : Poly α) (v : β → ℕ) : (map f g) v = g (v ∘ f)

Diophantine sets

def Dioph {α : Type u} (S : Set (α → ℕ)) : Prop

A set S ⊆ ℕ^α is Diophantine if there exists a polynomial on α ⊕ β such that v ∈ S iff there exists t : ℕ^β with p (v, t) = 0.

theorem Dioph.ext {α : Type u} {S S' : Set (α → ℕ)} (d : Dioph S) (H : ∀ (v : α → ℕ), v ∈ S ↔ v ∈ S') : Dioph S'

theorem Dioph.of_no_dummies {α : Type u} (S : Set (α → ℕ)) (p : Poly α) (h : ∀ (v : α → ℕ), S v ↔ p v = 0) : Dioph S

theorem Dioph.inject_dummies_lem {α β γ : Type u} (f : β → γ) (g : γ → Option β) (inv : ∀ (x : β), g (f x) = some x) (p : Poly (α ⊕ β)) (v : α → ℕ) : (∃ (t : β → ℕ), p (Sum.elim v t) = 0) ↔ ∃ (t : γ → ℕ), (Poly.map (Sum.elim Sum.inl (Sum.inr ∘ f)) p) (Sum.elim v t) = 0

theorem Dioph.inject_dummies {α β γ : Type u} {S : Set (α → ℕ)} (f : β → γ) (g : γ → Option β) (inv : ∀ (x : β), g (f x) = some x) (p : Poly (α ⊕ β)) (h : ∀ (v : α → ℕ), S v ↔ ∃ (t : β → ℕ), p (Sum.elim v t) = 0) : ∃ (q : Poly (α ⊕ γ)), ∀ (v : α → ℕ), S v ↔ ∃ (t : γ → ℕ), q (Sum.elim v t) = 0

theorem Dioph.reindex_dioph {α : Type u} (β : Type u) {S : Set (α → ℕ)} (f : α → β) : Dioph S → Dioph {v : β → ℕ | v ∘ f ∈ S}

theorem Dioph.DiophList.forall {α : Type u} (l : List (Set (α → ℕ))) (d : List.Forall Dioph l) : Dioph {v : α → ℕ | List.Forall (fun (S : Set (α → ℕ)) => v ∈ S) l}

theorem Dioph.inter {α : Type u} {S S' : Set (α → ℕ)} (d : Dioph S) (d' : Dioph S') : Dioph (S ∩ S')

Diophantine sets are closed under intersection.

theorem Dioph.union {α : Type u} {S S' : Set (α → ℕ)} : Dioph S → Dioph S' → Dioph (S ∪ S')

Diophantine sets are closed under union.

def Dioph.DiophPFun {α : Type u} (f : (α → ℕ) →. ℕ) : Prop

A partial function is Diophantine if its graph is Diophantine.

def Dioph.DiophFn {α : Type u} (f : (α → ℕ) → ℕ) : Prop

A function is Diophantine if its graph is Diophantine.

theorem Dioph.reindex_diophFn {α β : Type u} {f : (α → ℕ) → ℕ} (g : α → β) (d : DiophFn f) : DiophFn fun (v : β → ℕ) => f (v ∘ g)

theorem Dioph.ex_dioph {α β : Type u} {S : Set (α ⊕ β → ℕ)} : Dioph S → Dioph {v : α → ℕ | ∃ (x : β → ℕ), Sum.elim v x ∈ S}

theorem Dioph.ex1_dioph {α : Type u} {S : Set (Option α → ℕ)} : Dioph S → Dioph {v : α → ℕ | ∃ (x : ℕ), Option.elim' x v ∈ S}

theorem Dioph.dom_dioph {α : Type u} {f : (α → ℕ) →. ℕ} (d : DiophPFun f) : Dioph f.Dom

theorem Dioph.diophFn_iff_pFun {α : Type u} (f : (α → ℕ) → ℕ) : DiophFn f = DiophPFun ↑f

theorem Dioph.abs_poly_dioph {α : Type u} (p : Poly α) : DiophFn fun (v : α → ℕ) => (p v).natAbs

theorem Dioph.proj_dioph {α : Type u} (i : α) : DiophFn fun (v : α → ℕ) => v i

theorem Dioph.diophPFun_comp1 {α : Type u} {S : Set (Option α → ℕ)} (d : Dioph S) {f : (α → ℕ) →. ℕ} (df : DiophPFun f) : Dioph {v : α → ℕ | ∃ (h : f.Dom v), Option.elim' (f.fn v h) v ∈ S}

theorem Dioph.diophFn_comp1 {α : Type u} {S : Set (Option α → ℕ)} (d : Dioph S) {f : (α → ℕ) → ℕ} (df : DiophFn f) : Dioph {v : α → ℕ | Option.elim' (f v) v ∈ S}

theorem Dioph.diophFn_vec_comp1 {n : ℕ} {S : Set (Vector3 ℕ n.succ)} (d : Dioph S) {f : Vector3 ℕ n → ℕ} (df : DiophFn f) : Dioph {v : Vector3 ℕ n | Vector3.cons (f v) v ∈ S}

theorem Dioph.vec_ex1_dioph (n : ℕ) {S : Set (Vector3 ℕ n.succ)} (d : Dioph S) : Dioph {v : Fin2 n → ℕ | ∃ (x : ℕ), Vector3.cons x v ∈ S}

Deleting the first component preserves the Diophantine property.

theorem Dioph.diophFn_vec {n : ℕ} (f : Vector3 ℕ n → ℕ) : DiophFn f ↔ Dioph {v : Fin2 (n + 1) → ℕ | f (v ∘ Fin2.fs) = v Fin2.fz}

theorem Dioph.diophPFun_vec {n : ℕ} (f : Vector3 ℕ n →. ℕ) : DiophPFun f ↔ Dioph {v : Fin2 (n + 1) → ℕ | f.graph (v ∘ Fin2.fs, v Fin2.fz)}

theorem Dioph.diophFn_compn {α : Type} {n : ℕ} {S : Set (α ⊕ Fin2 n → ℕ)} : Dioph S → ∀ {f : Vector3 ((α → ℕ) → ℕ) n}, VectorAllP DiophFn f → Dioph {v : α → ℕ | (Sum.elim v fun (i : Fin2 n) => f i v) ∈ S}

theorem Dioph.dioph_comp {α : Type} {n : ℕ} {S : Set (Vector3 ℕ n)} (d : Dioph S) (f : Vector3 ((α → ℕ) → ℕ) n) (df : VectorAllP DiophFn f) : Dioph {v : α → ℕ | (fun (i : Fin2 n) => f i v) ∈ S}

theorem Dioph.diophFn_comp {α : Type} {n : ℕ} {f : Vector3 ℕ n → ℕ} (df : DiophFn f) (g : Vector3 ((α → ℕ) → ℕ) n) (dg : VectorAllP DiophFn g) : DiophFn fun (v : α → ℕ) => f fun (i : Fin2 n) => g i v

def Dioph.«term_D∧_» : Lean.TrailingParserDescr

Diophantine sets are closed under intersection.

def Dioph.«term_D∨_» : Lean.TrailingParserDescr

Diophantine sets are closed under union.

def Dioph.«termD∃» : Lean.ParserDescr

Deleting the first component preserves the Diophantine property.

def Dioph.«term&_» : Lean.ParserDescr

Local abbreviation for Fin2.ofNat'

theorem Dioph.proj_dioph_of_nat {n : ℕ} (m : ℕ) [Fin2.IsLT m n] : DiophFn fun (v : Vector3 ℕ n) => v (Fin2.ofNat' m)

def Dioph.«termD&_» : Lean.ParserDescr

Projection preserves Diophantine functions.

theorem Dioph.const_dioph {α : Type} (n : ℕ) : DiophFn (Function.const (α → ℕ) n)

def Dioph.«termD._» : Lean.ParserDescr

The constant function is Diophantine.

theorem Dioph.dioph_comp2 {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) {S : ℕ → ℕ → Prop} (d : Dioph fun (v : Vector3 ℕ 2) => S (v (Fin2.ofNat' 0)) (v (Fin2.ofNat' 1))) : Dioph fun (v : α → ℕ) => S (f v) (g v)

theorem Dioph.diophFn_comp2 {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) {h : ℕ → ℕ → ℕ} (d : DiophFn fun (v : Vector3 ℕ 2) => h (v (Fin2.ofNat' 0)) (v (Fin2.ofNat' 1))) : DiophFn fun (v : α → ℕ) => h (f v) (g v)

theorem Dioph.eq_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : Dioph fun (v : α → ℕ) => f v = g v

The set of places where two Diophantine functions are equal is Diophantine.

def Dioph.«term_D=_» : Lean.TrailingParserDescr

The set of places where two Diophantine functions are equal is Diophantine.

theorem Dioph.add_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v + g v

Diophantine functions are closed under addition.

def Dioph.«term_D+_» : Lean.TrailingParserDescr

Diophantine functions are closed under addition.

theorem Dioph.mul_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v * g v

Diophantine functions are closed under multiplication.

def Dioph.«term_D*_» : Lean.TrailingParserDescr

Diophantine functions are closed under multiplication.

theorem Dioph.le_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : Dioph {v : α → ℕ | f v ≤ g v}

The set of places where one Diophantine function is at most another is Diophantine.

def Dioph.«term_D≤_» : Lean.TrailingParserDescr

The set of places where one Diophantine function is at most another is Diophantine.

theorem Dioph.lt_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : Dioph {v : α → ℕ | f v < g v}

The set of places where one Diophantine function is less than another is Diophantine.

def Dioph.«term_D<_» : Lean.TrailingParserDescr

The set of places where one Diophantine function is less than another is Diophantine.

theorem Dioph.ne_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : Dioph {v : α → ℕ | f v ≠ g v}

The set of places where two Diophantine functions are unequal is Diophantine.

def Dioph.«term_D≠_» : Lean.TrailingParserDescr

The set of places where two Diophantine functions are unequal is Diophantine.

theorem Dioph.sub_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v - g v

Diophantine functions are closed under subtraction.

def Dioph.«term_D-_» : Lean.TrailingParserDescr

Diophantine functions are closed under subtraction.

theorem Dioph.dvd_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : Dioph fun (v : α → ℕ) => f v ∣ g v

The set of places where one Diophantine function divides another is Diophantine.

def Dioph.«term_D∣_» : Lean.TrailingParserDescr

The set of places where one Diophantine function divides another is Diophantine.

theorem Dioph.mod_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v % g v

Diophantine functions are closed under the modulo operation.

def Dioph.«term_D%_» : Lean.TrailingParserDescr

Diophantine functions are closed under the modulo operation.

theorem Dioph.modEq_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) {h : (α → ℕ) → ℕ} (dh : DiophFn h) : Dioph fun (v : α → ℕ) => f v ≡ g v [MOD h v]

The set of places where two Diophantine functions are congruent modulo a third is Diophantine.

def Dioph.«termD≡» : Lean.ParserDescr

The set of places where two Diophantine functions are congruent modulo a third is Diophantine.

theorem Dioph.div_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v / g v

Diophantine functions are closed under integer division.

def Dioph.«term_D/_» : Lean.TrailingParserDescr

Diophantine functions are closed under integer division.

theorem Dioph.pell_dioph : Dioph fun (v : Vector3 ℕ 4) => ∃ (h : 1 < v (Fin2.ofNat' 0)), Pell.xn h (v (Fin2.ofNat' 1)) = v (Fin2.ofNat' 2) ∧ Pell.yn h (v (Fin2.ofNat' 1)) = v (Fin2.ofNat' 3)

theorem Dioph.xn_dioph : DiophPFun fun (v : Vector3 ℕ 2) => { Dom := 1 < v (Fin2.ofNat' 0), get := fun (h : 1 < v (Fin2.ofNat' 0)) => Pell.xn h (v (Fin2.ofNat' 1)) }

theorem Dioph.pow_dioph {α : Type} {f g : (α → ℕ) → ℕ} (df : DiophFn f) (dg : DiophFn g) : DiophFn fun (v : α → ℕ) => f v ^ g v

A version of Matiyasevic's theorem