The Result type for norm_num

We set up predicates IsNat, IsInt, and IsRat, stating that an element of a ring is equal to the "normal form" of a natural number, integer, or rational number coerced into that ring.

We then define Result e, which contains a proof that a typed expression e : Q($α) is equal to the coercion of an explicit natural number, integer, or rational number, or is either true or false.

@[implicit_reducible]

def Mathlib.Meta.NormNum.instAddMonoidWithOne' {α : Type u} [Semiring α] : AddMonoidWithOne α

A shortcut (non)instance for AddMonoidWithOne α from Semiring α to shrink generated proofs.

@[implicit_reducible]

def Mathlib.Meta.NormNum.instAddMonoidWithOne {α : Type u} [Ring α] : AddMonoidWithOne α

A shortcut (non)instance for AddMonoidWithOne α from Ring α to shrink generated proofs.

theorem Mathlib.Meta.NormNum.instAtLeastTwo (n : ℕ) : (n + 2).AtLeastTwo

A shortcut (non)instance for Nat.AtLeastTwo (n + 2) to shrink generated proofs.

def Mathlib.Meta.NormNum.inferAddMonoidWithOne {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(AddMonoidWithOne «$α»)

Helper function to synthesize a typed AddMonoidWithOne α expression.

def Mathlib.Meta.NormNum.inferSemiring {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(Semiring «$α»)

Helper function to synthesize a typed Semiring α expression.

def Mathlib.Meta.NormNum.inferRing {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(Ring «$α»)

Helper function to synthesize a typed Ring α expression.

def Mathlib.Meta.NormNum.mkRawIntLit (n : ℤ) : Q(ℤ)

Represent an integer as a "raw" typed expression.

This uses .lit (.natVal n) internally to represent a natural number, rather than the preferred OfNat.ofNat form. We use this internally to avoid unnecessary typeclass searches.

This function is the inverse of Expr.intLit!.

def Mathlib.Meta.NormNum.mkRawRatLit (q : ℚ) : Q(ℚ)

Represent an integer as a "raw" typed expression.

This .lit (.natVal n) internally to represent a natural number, rather than the preferred OfNat.ofNat form. We use this internally to avoid unnecessary typeclass searches.

def Mathlib.Meta.NormNum.rawIntLitNatAbs (n : Q(ℤ)) : (m : Q(ℕ)) × Q(«$n».natAbs = «$m»)

Extract the raw natlit representing the absolute value of a raw integer literal (of the type produced by Mathlib.Meta.NormNum.mkRawIntLit) along with an equality proof.

def Mathlib.Meta.NormNum.mkOfNat {u : Lean.Level} (α : Q(Type u)) (_sα : Q(AddMonoidWithOne «$α»)) (lit : Q(ℕ)) : Lean.MetaM ((a' : Q(«$α»)) × Q(↑«$lit» = «$a'»))

Constructs an ofNat application a' with the canonical instance, together with a proof that the instance is equal to the result of Nat.cast on the given AddMonoidWithOne instance.

This function is performance-critical, as many higher level tactics have to construct numerals. So rather than using typeclass search we hardcode the (relatively small) set of solutions to the typeclass problem.

structure Mathlib.Meta.NormNum.IsNat {α : Type u} [AddMonoidWithOne α] (a : α) (n : ℕ) : Prop

Assert that an element of a semiring is equal to the coercion of some natural number.

• out : a = ↑n

  The element is equal to the coercion of the natural number.

theorem Mathlib.Meta.NormNum.IsNat.raw_refl (n : ℕ) : IsNat n n

def Nat.rawCast {α : Type u} [AddMonoidWithOne α] (n : ℕ) : α

A "raw nat cast" is an expression of the form (Nat.rawCast lit : α) where lit is a raw natural number literal. These expressions are used by tactics like ring to decrease the number of typeclass arguments required in each use of a number literal at type α.

theorem Mathlib.Meta.NormNum.IsNat.to_eq {α : Type u} [AddMonoidWithOne α] {n : ℕ} {a a' : α} : IsNat a n → ↑n = a' → a = a'

theorem Mathlib.Meta.NormNum.IsNat.to_raw_eq {α : Type u} {a : α} {n : ℕ} [AddMonoidWithOne α] : IsNat a n → a = n.rawCast

theorem Mathlib.Meta.NormNum.IsNat.of_raw (α : Type u_1) [AddMonoidWithOne α] (n : ℕ) : IsNat n.rawCast n

theorem Mathlib.Meta.NormNum.isNat.natElim {p : ℕ → Prop} {n n' : ℕ} : IsNat n n' → p n' → p n

structure Mathlib.Meta.NormNum.IsInt {α : Type u} [Ring α] (a : α) (n : ℤ) : Prop

Assert that an element of a ring is equal to the coercion of some integer.

• out : a = ↑n

  The element is equal to the coercion of the integer.

def Int.rawCast {α : Type u} [Ring α] (n : ℤ) : α

A "raw int cast" is an expression of the form:

• (Nat.rawCast lit : α) where lit is a raw natural number literal
• (Int.rawCast (Int.negOfNat lit) : α) where lit is a nonzero raw natural number literal

(That is, we only actually use this function for negative integers.) This representation is used by tactics like ring to decrease the number of typeclass arguments required in each use of a number literal at type α.

theorem Mathlib.Meta.NormNum.IsInt.to_isNat {α : Type u_1} [Ring α] {a : α} {n : ℕ} : IsInt a (Int.ofNat n) → IsNat a n

theorem Mathlib.Meta.NormNum.IsNat.to_isInt {α : Type u_1} [Ring α] {a : α} {n : ℕ} : IsNat a n → IsInt a (Int.ofNat n)

theorem Mathlib.Meta.NormNum.IsInt.to_raw_eq {α : Type u} {a : α} {n : ℤ} [Ring α] : IsInt a n → a = n.rawCast

theorem Mathlib.Meta.NormNum.IsInt.of_raw (α : Type u_1) [Ring α] (n : ℤ) : IsInt n.rawCast n

theorem Mathlib.Meta.NormNum.IsInt.neg_to_eq {α : Type u_1} [Ring α] {n : ℕ} {a a' : α} : IsInt a (Int.negOfNat n) → ↑n = a' → a = -a'

theorem Mathlib.Meta.NormNum.IsInt.nonneg_to_eq {α : Type u_1} [Ring α] {n : ℕ} {a a' : α} (h : IsInt a (Int.ofNat n)) (e : ↑n = a') : a = a'

inductive Mathlib.Meta.NormNum.IsRat {α : Type u} [Ring α] (a : α) (num : ℤ) (denom : ℕ) : Prop

Assert that an element of a ring is equal to num / denom (and denom is invertible so that this makes sense). We will usually also have num and denom coprime, although this is not part of the definition.

• mk {α : Type u} [Ring α] {a : α} {num : ℤ} {denom : ℕ} (inv : Invertible ↑denom) (eq : a = ↑num * ⅟↑denom) : IsRat a num denom

inductive Mathlib.Meta.NormNum.IsNNRat {α : Type u} [Semiring α] (a : α) (num denom : ℕ) : Prop

Assert that an element of a semiring is equal to num / denom (and denom is invertible so that this makes sense). We will usually also have num and denom coprime, although this is not part of the definition.

• mk {α : Type u} [Semiring α] {a : α} {num denom : ℕ} (inv : Invertible ↑denom) (eq : a = ↑num * ⅟↑denom) : IsNNRat a num denom

def NNRat.rawCast {α : Type u} [DivisionSemiring α] (n d : ℕ) : α

A "raw nnrat cast" is an expression of the form:

• (Nat.rawCast lit : α) where lit is a raw natural number literal
• (NNRat.rawCast n d : α) where n is a raw nat literal, d is a raw nat literal, and d is not 1 or 0.

This representation is used by tactics like ring to decrease the number of typeclass arguments required in each use of a number literal at type α.

def Rat.rawCast {α : Type u} [DivisionRing α] (n : ℤ) (d : ℕ) : α

A "raw rat cast" is an expression of the form:

• (Nat.rawCast lit : α) where lit is a raw natural number literal
• (Int.rawCast (Int.negOfNat lit) : α) where lit is a nonzero raw natural number literal
• (NNRat.rawCast n d : α) where n is a raw nat literal, d is a raw nat literal, and d is not 1 or 0.
• (Rat.rawCast (Int.negOfNat n) d : α) where n is a raw nat literal, d is a raw nat literal, n is not 0, and d is not 1 or 0.

This representation is used by tactics like ring to decrease the number of typeclass arguments required in each use of a number literal at type α.

theorem Mathlib.Meta.NormNum.IsNNRat.to_isNat {α : Type u_1} [Semiring α] {a : α} {n : ℕ} : IsNNRat a n 1 → IsNat a n

theorem Mathlib.Meta.NormNum.IsRat.to_isNNRat {α : Type u_1} [Ring α] {a : α} {n d : ℕ} : IsRat a (Int.ofNat n) d → IsNNRat a n d

theorem Mathlib.Meta.NormNum.IsNat.to_isNNRat {α : Type u_1} [Semiring α] {a : α} {n : ℕ} : IsNat a n → IsNNRat a n 1

theorem Mathlib.Meta.NormNum.IsNNRat.to_isRat {α : Type u_1} [Ring α] {a : α} {n d : ℕ} : IsNNRat a n d → IsRat a (Int.ofNat n) d

theorem Mathlib.Meta.NormNum.IsRat.to_isInt {α : Type u_1} [Ring α] {a : α} {n : ℤ} : IsRat a n 1 → IsInt a n

theorem Mathlib.Meta.NormNum.IsInt.to_isRat {α : Type u_1} [Ring α] {a : α} {n : ℤ} : IsInt a n → IsRat a n 1

theorem Mathlib.Meta.NormNum.IsNNRat.to_raw_eq {α : Type u} {n d : ℕ} [DivisionSemiring α] {a : α} : IsNNRat a n d → a = NNRat.rawCast n d

theorem Mathlib.Meta.NormNum.IsRat.to_raw_eq {α : Type u} {n : ℤ} {d : ℕ} [DivisionRing α] {a : α} : IsRat a n d → a = Rat.rawCast n d

theorem Mathlib.Meta.NormNum.IsRat.neg_to_eq {α : Type u_1} [DivisionRing α] {n d : ℕ} {a n' d' : α} : IsRat a (Int.negOfNat n) d → ↑n = n' → ↑d = d' → a = -(n' / d')

theorem Mathlib.Meta.NormNum.IsNNRat.to_eq {α : Type u_1} [DivisionSemiring α] {n d : ℕ} {a n' d' : α} : IsNNRat a n d → ↑n = n' → ↑d = d' → a = n' / d'

theorem Mathlib.Meta.NormNum.IsNNRat.of_raw (α : Type u_1) [DivisionSemiring α] (n d : ℕ) (h : ↑d ≠ 0) : IsNNRat (NNRat.rawCast n d) n d

theorem Mathlib.Meta.NormNum.IsRat.of_raw (α : Type u_1) [DivisionRing α] (n : ℤ) (d : ℕ) (h : ↑d ≠ 0) : IsRat (Rat.rawCast n d) n d

theorem Mathlib.Meta.NormNum.IsNNRat.den_nz {α : Type u_1} [DivisionSemiring α] {a : α} {n d : ℕ} : IsNNRat a n d → ↑d ≠ 0

theorem Mathlib.Meta.NormNum.IsRat.den_nz {α : Type u_1} [DivisionRing α] {a : α} {n : ℤ} {d : ℕ} : IsRat a n d → ↑d ≠ 0

inductive Mathlib.Meta.NormNum.Result' : Type

The result of norm_num running on an expression x of type α. Untyped version of Result.

• isBool (val : Bool) (proof : Lean.Expr) : Result'

  Untyped version of Result.isBool.

• isNat (inst lit proof : Lean.Expr) : Result'

  Untyped version of Result.isNat.

• isNegNat (inst lit proof : Lean.Expr) : Result'

  Untyped version of Result.isNegNat.

• isNNRat (inst : Lean.Expr) (q : ℚ) (n d proof : Lean.Expr) : Result'

  Untyped version of Result.isNNRat.

• isNegNNRat (inst : Lean.Expr) (q : ℚ) (n d proof : Lean.Expr) : Result'

  Untyped version of Result.isNegNNRat.

def Mathlib.Meta.NormNum.instInhabitedResult'.default : Result'

@[implicit_reducible]

instance Mathlib.Meta.NormNum.instInhabitedResult' : Inhabited Result'

def Mathlib.Meta.NormNum.Result {u : Lean.Level} {α : Q(Type u)} (x : Q(«$α»)) : Type

The result of norm_num running on an expression x of type α.

@[implicit_reducible]

instance Mathlib.Meta.NormNum.instInhabitedResult {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} : Inhabited (Result x)

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isTrue {x : Q(Prop)} (proof : Q(«$x»)) : Result q(«$x»)

The result is proof : x, where x is a (true) proposition.

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isFalse {x : Q(Prop)} (proof : Q(¬«$x»)) : Result q(«$x»)

The result is proof : ¬x, where x is a (false) proposition.

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isNat {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(AddMonoidWithOne «$α») := by assumption) (lit : Q(ℕ)) (proof : Q(IsNat «$x» «$lit»)) : Result x

The result is lit : ℕ (a raw nat literal) and proof : isNat x lit.

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isNegNat {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(Ring «$α») := by assumption) (lit : Q(ℕ)) (proof : Q(IsInt «$x» (Int.negOfNat «$lit»))) : Result x

The result is -lit where lit is a raw nat literal and proof : isInt x (.negOfNat lit).

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isNNRat {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(DivisionSemiring «$α») := by assumption) (q : ℚ) (n d : Q(ℕ)) (proof : Q(IsNNRat «$x» «$n» «$d»)) : Result x

The result is proof : IsNNRat x n d, where n a raw nat literal, d is a raw nat literal (not 0 or 1), n and d are coprime, and q is the value of n / d.

@[match_pattern, inline]

def Mathlib.Meta.NormNum.Result.isNegNNRat {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(DivisionRing «$α») := by assumption) (q : ℚ) (n d : Q(ℕ)) (proof : Q(IsRat «$x» (Int.negOfNat «$n») «$d»)) : Result x

The result is proof : IsRat x n d, where n is .negOfNat lit with lit a raw nat literal, d is a raw nat literal (not 0 or 1), n and d are coprime, and q is the value of n / d.

def Mathlib.Meta.NormNum.Result.isInt {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(Ring «$α») := by assumption) (z : Q(ℤ)) (n : ℤ) (proof : Q(IsInt «$x» «$z»)) : Result x

The result is z : ℤ and proof : isNat x z.

def Mathlib.Meta.NormNum.Result.isNNRat' {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(DivisionSemiring «$α») := by assumption) (q : ℚ) (n d : Q(ℕ)) (proof : Q(IsNNRat «$x» «$n» «$d»)) : Result x

The result is q : NNRat and proof : isNNRat x q.

def Mathlib.Meta.NormNum.Result.isRat {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} (inst : Q(DivisionRing «$α») := by assumption) (q : ℚ) (n : Q(ℤ)) (d : Q(ℕ)) (proof : Q(IsRat «$x» «$n» «$d»)) : Result x

The result is q : ℚ and proof : isRat x q.

@[implicit_reducible]

instance Mathlib.Meta.NormNum.instToMessageDataResult {u : Lean.Level} {α : Q(Type u)} {x : Q(«$α»)} : Lean.ToMessageData (Result x)

def Mathlib.Meta.NormNum.Result.toRat {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} : Result e → Option ℚ

Returns the rational number that is the result of norm_num evaluation.

def Mathlib.Meta.NormNum.Result.toRatNZ {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} : Result e → Option (ℚ × Option Lean.Expr)

Returns the rational number that is the result of norm_num evaluation, along with a proof that the denominator is nonzero in the isRat case.

def Mathlib.Meta.NormNum.Result.toInt {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} (_i : Q(Ring «$α») := by with_reducible assumption) : Result e → Option (ℤ × (lit : Q(ℤ)) × Q(IsInt «$e» «$lit»))

Extract from a Result the integer value (as both a term and an expression), and the proof that the original expression is equal to this integer.

def Mathlib.Meta.NormNum.Result.toNNRat' {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} (_i : Q(DivisionSemiring «$α») := by with_reducible assumption) : Result e → Option (ℚ × (n : Q(ℕ)) × (d : Q(ℕ)) × Q(IsNNRat «$e» «$n» «$d»))

Extract from a Result the rational value (as both a term and an expression), and the proof that the original expression is equal to this rational number.

def Mathlib.Meta.NormNum.Result.toRat' {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} (_i : Q(DivisionRing «$α») := by with_reducible assumption) : Result e → Option (ℚ × (n : Q(ℤ)) × (d : Q(ℕ)) × Q(IsRat «$e» «$n» «$d»))

Extract from a Result the rational value (as both a term and an expression), and the proof that the original expression is equal to this rational number.

def Mathlib.Meta.NormNum.Result.toRawEq {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} : Result e → (e' : Q(«$α»)) × Q(«$e» = «$e'»)

Given a NormNum.Result e (which uses IsNat, IsInt, IsRat to express equality to a rational numeral), converts it to an equality e = Nat.rawCast n, e = Int.rawCast n, or e = Rat.rawCast n d to a raw cast expression, so it can be used for rewriting.

def Mathlib.Meta.NormNum.Result.toRawIntEq {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} : Result e → Option (ℤ × (e' : Q(«$α»)) × Q(«$e» = «$e'»))

Result.toRawEq but providing an integer. Given a NormNum.Result e for something known to be an integer (which uses IsNat or IsInt to express equality to an integer numeral), converts it to an equality e = Nat.rawCast n or e = Int.rawCast n to a raw cast expression, so it can be used for rewriting. Gives none if not an integer.

def Mathlib.Meta.NormNum.Result.ofRawNat {u : Lean.Level} {α : Q(Type u)} (e : Q(«$α»)) : Result e

Constructs a Result out of a raw nat cast. Assumes e is a raw nat cast expression.

def Mathlib.Meta.NormNum.Result.ofRawInt {u : Lean.Level} {α : Q(Type u)} (n : ℤ) (e : Q(«$α»)) : Result e

Constructs a Result out of a raw int cast. Assumes e is a raw int cast expression denoting n.

def Mathlib.Meta.NormNum.Result.ofRawNNRat {u : Lean.Level} {α : Q(Type u)} (q : ℚ) (e : Q(«$α»)) (hyp : Option Lean.Expr := none) : Result e

Constructs a Result out of a raw rat cast. Assumes e is a raw rat cast expression denoting n.

def Mathlib.Meta.NormNum.Result.ofRawRat {u : Lean.Level} {α : Q(Type u)} (q : ℚ) (e : Q(«$α»)) (hyp : Option Lean.Expr := none) : Result e

Constructs a Result out of a raw rat cast. Assumes e is a raw rat cast expression denoting n.

def Mathlib.Meta.NormNum.Result.toSimpResult {u : Lean.Level} {α : Q(Type u)} {e : Q(«$α»)} : Result e → Lean.MetaM Lean.Meta.Simp.Result

Convert a Result to a Simp.Result.

@[reducible, inline]

abbrev Mathlib.Meta.NormNum.BoolResult (p : Q(Prop)) (b : Bool) : Type

Given Mathlib.Meta.NormNum.Result.isBool p b, this is the type of p. Note that BoolResult p b is definitionally equal to Expr, and if you write match b with ..., then in the true branch BoolResult p true is reducibly equal to Q($p) and in the false branch it is reducibly equal to Q(¬ $p).

def Mathlib.Meta.NormNum.Result.ofBoolResult {p : Q(Prop)} {b : Bool} (prf : BoolResult p b) : Result q(Prop)

Obtain a Result from a BoolResult.

def Mathlib.Meta.NormNum.Result.eqTrans {u : Lean.Level} {α : Q(Type u)} {a b : Q(«$α»)} (eq : Q(«$a» = «$b»)) : Result b → Result a

If a = b and we can evaluate b, then we can evaluate a.