ring tactic

A tactic for solving equations in commutative (semi)rings, where the exponents can also contain variables. Based on http://www.cs.ru.nl/~freek/courses/tt-2014/read/10.1.1.61.3041.pdf .

More precisely, expressions of the following form are supported:

• constants (non-negative integers)
• variables
• coefficients (any rational number, embedded into the (semi)ring)
• addition of expressions
• multiplication of expressions (a * b)
• scalar multiplication of expressions (n • a; the multiplier must have type ℕ)
• exponentiation of expressions (the exponent must have type ℕ)
• subtraction and negation of expressions (if the base is a full ring)

The extension to exponents means that something like 2 * 2^n * b = b * 2^(n+1) can be proved, even though it is not strictly speaking an equation in the language of commutative rings.

Implementation notes

The basic approach to prove equalities is to normalise both sides and check for equality. The normalisation is guided by building a value in the type ExSum at the meta level, together with a proof (at the base level) that the original value is equal to the normalised version.

The outline of the file:

• Define a mutual inductive family of types ExSum, ExProd, ExBase, which can represent expressions with +, *, ^ and rational numerals. The mutual induction ensures that associativity and distributivity are applied, by restricting which kinds of subexpressions appear as arguments to the various operators.
• Represent addition, multiplication and exponentiation in the ExSum type, thus allowing us to map expressions to ExSum (the eval function drives this). We apply associativity and distributivity of the operators here (helped by Ex* types) and commutativity as well (by sorting the subterms; unfortunately not helped by anything). Any expression not of the above formats is treated as an atom (the same as a variable).

There are some details we glossed over which make the plan more complicated:

• The order on atoms is not initially obvious. We construct a list containing them in order of initial appearance in the expression, then use the index into the list as a key to order on.
• For pow, the exponent must be a natural number, while the base can be any semiring α. We swap out operations for the base ring α with those for the exponent ring ℕ as soon as we deal with exponents.

Caveats and future work

The normalized form of an expression is the one that is useful for the tactic, but not as nice to read. To remedy this, the user-facing normalization calls ringNFCore.

Subtraction cancels out identical terms, but division does not. That is: a - a = 0 := by ring solves the goal, but a / a := 1 by ring doesn't. Note that 0 / 0 is generally defined to be 0, so division cancelling out is not true in general.

Multiplication of powers can be simplified a little bit further: 2 ^ n * 2 ^ n = 4 ^ n := by ring could be implemented in a similar way that 2 * a + 2 * a = 4 * a := by ring already works. This feature wasn't needed yet, so it's not implemented yet.

Tags

ring, semiring, exponent, power

def Mathlib.Tactic.Ring.instCommSemiringNat : CommSemiring ℕ

A shortcut instance for CommSemiring ℕ used by ring.

Equations

• Mathlib.Tactic.Ring.instCommSemiringNat = inferInstance

def Mathlib.Tactic.Ring.instCommSemiringInt : CommSemiring ℤ

A shortcut instance for CommSemiring ℤ used by ring.

Equations

• Mathlib.Tactic.Ring.instCommSemiringInt = inferInstance

def Mathlib.Tactic.Ring.sℕ : Q(CommSemiring ℕ)

A typed expression of type CommSemiring ℕ used when we are working on ring subexpressions of type ℕ.

Equations

• Mathlib.Tactic.Ring.sℕ = q(Mathlib.Tactic.Ring.instCommSemiringNat)

def Mathlib.Tactic.Ring.sℤ : Q(CommSemiring ℤ)

A typed expression of type CommSemiring ℤ used when we are working on ring subexpressions of type ℤ.

Equations

• Mathlib.Tactic.Ring.sℤ = q(Mathlib.Tactic.Ring.instCommSemiringInt)

inductive Mathlib.Tactic.Ring.ExBase {u : Lean.Level} {α : Q(Type u)} : Q(CommSemiring «$α») → (e : Q(«$α»)) → Type

The base e of a normalized exponent expression.

• atom {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} (id : ℕ) : ExBase sα e

  An atomic expression e with id id.

  Atomic expressions are those which ring cannot parse any further. For instance, a + (a % b) has a and (a % b) as atoms. The ring1 tactic does not normalize the subexpressions in atoms, but ring_nf does.

  Atoms in fact represent equivalence classes of expressions, modulo definitional equality. The field index : ℕ should be a unique number for each class, while value : expr contains a representative of this class. The function resolve_atom determines the appropriate atom for a given expression.

• sum {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} : ExSum sα e → ExBase sα e

  A sum of monomials.

inductive Mathlib.Tactic.Ring.ExProd {u : Lean.Level} {α : Q(Type u)} : Q(CommSemiring «$α») → (e : Q(«$α»)) → Type

A monomial, which is a product of powers of ExBase expressions, terminated by a (nonzero) constant coefficient.

• const {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} (value : ℚ) (hyp : Option Lean.Expr := none) : ExProd sα e

  A coefficient value, which must not be 0. e is a raw rat cast. If value is not an integer, then hyp should be a proof of (value.den : α) ≠ 0.

• mul {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {x : Q(«$α»)} {e : Q(ℕ)} {b : Q(«$α»)} : ExBase sα x → ExProd sℕ e → ExProd sα b → ExProd sα q(«$x» ^ «$e» * «$b»)

  A product x ^ e * b is a monomial if b is a monomial. Here x is an ExBase and e is an ExProd representing a monomial expression in ℕ (it is a monomial instead of a polynomial because we eagerly normalize x ^ (a + b) = x ^ a * x ^ b.)

inductive Mathlib.Tactic.Ring.ExSum {u : Lean.Level} {α : Q(Type u)} : Q(CommSemiring «$α») → (e : Q(«$α»)) → Type

A polynomial expression, which is a sum of monomials.

• zero {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} : ExSum sα q(0)

  Zero is a polynomial. e is the expression 0.

• add {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExProd sα a → ExSum sα b → ExSum sα q(«$a» + «$b»)

  A sum a + b is a polynomial if a is a monomial and b is another polynomial.

def Mathlib.Tactic.Ring.ExBase.eq {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExBase sα a → ExBase sα b → Bool

Equality test for expressions. This is not a BEq instance because it is heterogeneous.

Equations

• (Mathlib.Tactic.Ring.ExBase.atom i).eq (Mathlib.Tactic.Ring.ExBase.atom j) = (i == j)
• (Mathlib.Tactic.Ring.ExBase.sum a_1).eq (Mathlib.Tactic.Ring.ExBase.sum b_1) = a_1.eq b_1
• x✝¹.eq x✝ = false

def Mathlib.Tactic.Ring.ExProd.eq {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExProd sα a → ExProd sα b → Bool

Equality test for expressions. This is not a BEq instance because it is heterogeneous.

Equations

• (Mathlib.Tactic.Ring.ExProd.const i hyp).eq (Mathlib.Tactic.Ring.ExProd.const j hyp_1) = (i == j)
• (Mathlib.Tactic.Ring.ExProd.mul a₁ a₂ a₃).eq (Mathlib.Tactic.Ring.ExProd.mul b₁ b₂ b₃) = (a₁.eq b₁ && a₂.eq b₂ && a₃.eq b₃)
• x✝¹.eq x✝ = false

def Mathlib.Tactic.Ring.ExSum.eq {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExSum sα a → ExSum sα b → Bool

Equality test for expressions. This is not a BEq instance because it is heterogeneous.

Equations

• Mathlib.Tactic.Ring.ExSum.zero.eq Mathlib.Tactic.Ring.ExSum.zero = true
• (Mathlib.Tactic.Ring.ExSum.add a₁ a₂).eq (Mathlib.Tactic.Ring.ExSum.add b₁ b₂) = (a₁.eq b₁ && a₂.eq b₂)
• x✝¹.eq x✝ = false

def Mathlib.Tactic.Ring.ExBase.cmp {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExBase sα a → ExBase sα b → Ordering

A total order on normalized expressions. This is not an Ord instance because it is heterogeneous.

Equations

• (Mathlib.Tactic.Ring.ExBase.atom i).cmp (Mathlib.Tactic.Ring.ExBase.atom j) = compare i j
• (Mathlib.Tactic.Ring.ExBase.sum a_1).cmp (Mathlib.Tactic.Ring.ExBase.sum b_1) = a_1.cmp b_1
• (Mathlib.Tactic.Ring.ExBase.atom id).cmp (Mathlib.Tactic.Ring.ExBase.sum x_2) = Ordering.lt
• (Mathlib.Tactic.Ring.ExBase.sum x_2).cmp (Mathlib.Tactic.Ring.ExBase.atom id) = Ordering.gt

def Mathlib.Tactic.Ring.ExProd.cmp {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExProd sα a → ExProd sα b → Ordering

A total order on normalized expressions. This is not an Ord instance because it is heterogeneous.

Equations

• (Mathlib.Tactic.Ring.ExProd.const i hyp).cmp (Mathlib.Tactic.Ring.ExProd.const j hyp_1) = compare i j
• (Mathlib.Tactic.Ring.ExProd.mul a₁ a₂ a₃).cmp (Mathlib.Tactic.Ring.ExProd.mul b₁ b₂ b₃) = ((a₁.cmp b₁).then (a₂.cmp b₂)).then (a₃.cmp b₃)
• (Mathlib.Tactic.Ring.ExProd.const value hyp).cmp (Mathlib.Tactic.Ring.ExProd.mul a_3 a_4 a_5) = Ordering.lt
• (Mathlib.Tactic.Ring.ExProd.mul a_2 a_3 a_4).cmp (Mathlib.Tactic.Ring.ExProd.const value hyp) = Ordering.gt

def Mathlib.Tactic.Ring.ExSum.cmp {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a b : Q(«$α»)} : ExSum sα a → ExSum sα b → Ordering

A total order on normalized expressions. This is not an Ord instance because it is heterogeneous.

Equations

• Mathlib.Tactic.Ring.ExSum.zero.cmp Mathlib.Tactic.Ring.ExSum.zero = Ordering.eq
• (Mathlib.Tactic.Ring.ExSum.add a₁ a₂).cmp (Mathlib.Tactic.Ring.ExSum.add b₁ b₂) = (a₁.cmp b₁).then (a₂.cmp b₂)
• Mathlib.Tactic.Ring.ExSum.zero.cmp (Mathlib.Tactic.Ring.ExSum.add a_3 a_4) = Ordering.lt
• (Mathlib.Tactic.Ring.ExSum.add a_3 a_4).cmp Mathlib.Tactic.Ring.ExSum.zero = Ordering.gt

instance Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExBase {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} : Inhabited ((e : Q(«$α»)) × ExBase sα e)

Equations

• Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExBase = { default := ⟨default, Mathlib.Tactic.Ring.ExBase.atom 0⟩ }

instance Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExSum {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} : Inhabited ((e : Q(«$α»)) × ExSum sα e)

Equations

• Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExSum = { default := ⟨q(0), Mathlib.Tactic.Ring.ExSum.zero⟩ }

instance Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExProd {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} : Inhabited ((e : Q(«$α»)) × ExProd sα e)

Equations

• Mathlib.Tactic.Ring.instInhabitedSigmaQuotedExProd = { default := ⟨default, Mathlib.Tactic.Ring.ExProd.const 0⟩ }

def Mathlib.Tactic.Ring.ExBase.cast {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring «$β»)} {a : Q(«$α»)} : ExBase sα a → (a : Q(«$β»)) × ExBase sβ a

Converts ExBase sα to ExBase sβ, assuming sα and sβ are defeq.

Equations

• (Mathlib.Tactic.Ring.ExBase.atom i).cast = ⟨a, Mathlib.Tactic.Ring.ExBase.atom i⟩
• (Mathlib.Tactic.Ring.ExBase.sum a_1).cast = match a_1.cast with | ⟨fst, vb⟩ => ⟨fst, Mathlib.Tactic.Ring.ExBase.sum vb⟩

def Mathlib.Tactic.Ring.ExProd.cast {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring «$β»)} {a : Q(«$α»)} : ExProd sα a → (a : Q(«$β»)) × ExProd sβ a

Converts ExProd sα to ExProd sβ, assuming sα and sβ are defeq.

Equations

• (Mathlib.Tactic.Ring.ExProd.const i h).cast = ⟨a, Mathlib.Tactic.Ring.ExProd.const i h⟩
• (Mathlib.Tactic.Ring.ExProd.mul a₁ a₂ a₃).cast = ⟨q(unknown_1 ^ «$e» * unknown_2), Mathlib.Tactic.Ring.ExProd.mul a₁.cast.snd a₂ a₃.cast.snd⟩

def Mathlib.Tactic.Ring.ExSum.cast {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring «$β»)} {a : Q(«$α»)} : ExSum sα a → (a : Q(«$β»)) × ExSum sβ a

Converts ExSum sα to ExSum sβ, assuming sα and sβ are defeq.

Equations

• Mathlib.Tactic.Ring.ExSum.zero.cast = ⟨q(0), Mathlib.Tactic.Ring.ExSum.zero⟩
• (Mathlib.Tactic.Ring.ExSum.add a₁ a₂).cast = ⟨q(unknown_1 + unknown_2), Mathlib.Tactic.Ring.ExSum.add a₁.cast.snd a₂.cast.snd⟩

structure Mathlib.Tactic.Ring.Result {u : Lean.Level} {α : Q(Type u)} (E : Q(«$α») → Type) (e : Q(«$α»)) : Type

The result of evaluating an (unnormalized) expression e into the type family E (one of ExSum, ExProd, ExBase) is a (normalized) element e' and a representation E e' for it, and a proof of e = e'.

• expr : Q(«$α»)

  The normalized result.

• val : E self.expr

  The data associated to the normalization.

• proof : Q(«$e» = unknown_1)

  A proof that the original expression is equal to the normalized result.

instance Mathlib.Tactic.Ring.instInhabitedResultOfSigmaQuoted {u : Lean.Level} {α : Q(Type u)} {E : Q(«$α») → Type} {e : Q(«$α»)} [Inhabited ((e : Q(«$α»)) × E e)] : Inhabited (Result E e)

Equations

• Mathlib.Tactic.Ring.instInhabitedResultOfSigmaQuoted = match default with | ⟨e', v⟩ => { default := { expr := e', val := v, proof := default } }

def Mathlib.Tactic.Ring.ExProd.mkNat {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (n : ℕ) : (e : Q(«$α»)) × ExProd sα e

Constructs the expression corresponding to .const n. (The .const constructor does not check that the expression is correct.)

def Mathlib.Tactic.Ring.ExProd.mkNegNat {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) : Q(Ring «$α») → (n : ℕ) → (e : Q(«$α»)) × ExProd sα e

Constructs the expression corresponding to .const (-n). (The .const constructor does not check that the expression is correct.)

def Mathlib.Tactic.Ring.ExProd.mkNNRat {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) : Q(DivisionSemiring «$α») → (q : ℚ) → (n d : Q(ℕ)) → (h : Lean.Expr) → (e : Q(«$α»)) × ExProd sα e

Constructs the expression corresponding to .const q h for q = n / d and h a proof that (d : α) ≠ 0. (The .const constructor does not check that the expression is correct.)

Equations

• Mathlib.Tactic.Ring.ExProd.mkNNRat sα x✝ q n d h = ⟨q(NNRat.rawCast «$n» «$d»), Mathlib.Tactic.Ring.ExProd.const q (some h)⟩

def Mathlib.Tactic.Ring.ExProd.mkNegNNRat {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) : Q(DivisionRing «$α») → (q : ℚ) → (n d : Q(ℕ)) → (h : Lean.Expr) → (e : Q(«$α»)) × ExProd sα e

Constructs the expression corresponding to .const q h for q = -(n / d) and h a proof that (d : α) ≠ 0. (The .const constructor does not check that the expression is correct.)

Equations

• Mathlib.Tactic.Ring.ExProd.mkNegNNRat sα x✝ q n d h = ⟨q(Rat.rawCast (Int.negOfNat «$n») «$d»), Mathlib.Tactic.Ring.ExProd.const q (some h)⟩

def Mathlib.Tactic.Ring.ExBase.toProd {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {a : Q(«$α»)} {b : Q(ℕ)} (va : ExBase sα a) (vb : ExProd sℕ b) : ExProd sα q(«$a» ^ «$b» * Nat.rawCast 1)

Embed an exponent (an ExBase, ExProd pair) as an ExProd by multiplying by 1.

Equations

• va.toProd vb = Mathlib.Tactic.Ring.ExProd.mul va vb (Mathlib.Tactic.Ring.ExProd.const 1)

def Mathlib.Tactic.Ring.ExProd.toSum {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} (v : ExProd sα e) : ExSum sα q(«$e» + 0)

Embed ExProd in ExSum by adding 0.

Equations

• v.toSum = Mathlib.Tactic.Ring.ExSum.add v Mathlib.Tactic.Ring.ExSum.zero

def Mathlib.Tactic.Ring.ExProd.coeff {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} : ExProd sα e → ℚ

Get the leading coefficient of an ExProd.

Equations

• (Mathlib.Tactic.Ring.ExProd.const i h).coeff = i
• (Mathlib.Tactic.Ring.ExProd.mul a₁ a₂ a₃).coeff = a₃.coeff

inductive Mathlib.Tactic.Ring.Overlap {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (e : Q(«$α»)) : Type

Two monomials are said to "overlap" if they differ by a constant factor, in which case the constants just add. When this happens, the constant may be either zero (if the monomials cancel) or nonzero (if they add up); the zero case is handled specially.

• zero {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} : Q(Meta.NormNum.IsNat «$e» 0) → Overlap sα e

  The expression e (the sum of monomials) is equal to 0.

• nonzero {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring «$α»)} {e : Q(«$α»)} : Result (ExProd sα) e → Overlap sα e

  The expression e (the sum of monomials) is equal to another monomial (with nonzero leading coefficient).

Addition

theorem Mathlib.Tactic.Ring.add_overlap_pf {R : Type u_1} [CommSemiring R] {a b c : R} (x : R) (e : ℕ) (pq_pf : a + b = c) : x ^ e * a + x ^ e * b = x ^ e * c

theorem Mathlib.Tactic.Ring.add_overlap_pf_zero {R : Type u_1} [CommSemiring R] {a b : R} (x : R) (e : ℕ) : Meta.NormNum.IsNat (a + b) 0 → Meta.NormNum.IsNat (x ^ e * a + x ^ e * b) 0

def Mathlib.Tactic.Ring.evalAddOverlap {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (va : ExProd sα a) (vb : ExProd sα b) : OptionT Lean.MetaM (Overlap sα q(«$a» + «$b»))

Given monomials va, vb, attempts to add them together to get another monomial. If the monomials are not compatible, returns none. For example, xy + 2xy = 3xy is a .nonzero overlap, while xy + xz returns none and xy + -xy = 0 is a .zero overlap.

Equations

• One or more equations did not get rendered due to their size.
• Mathlib.Tactic.Ring.evalAddOverlap sα va vb = do liftM (Lean.checkSystem `Mathlib.Tactic.Ring.evalAddOverlap.toString) OptionT.fail

theorem Mathlib.Tactic.Ring.add_pf_zero_add {R : Type u_1} [CommSemiring R] (b : R) : 0 + b = b

theorem Mathlib.Tactic.Ring.add_pf_add_zero {R : Type u_1} [CommSemiring R] (a : R) : a + 0 = a

theorem Mathlib.Tactic.Ring.add_pf_add_overlap {R : Type u_1} [CommSemiring R] {a₁ a₂ b₁ b₂ c₁ c₂ : R} : a₁ + b₁ = c₁ → a₂ + b₂ = c₂ → a₁ + a₂ + (b₁ + b₂) = c₁ + c₂

theorem Mathlib.Tactic.Ring.add_pf_add_overlap_zero {R : Type u_1} [CommSemiring R] {a₁ a₂ b₁ b₂ c : R} (h : Meta.NormNum.IsNat (a₁ + b₁) 0) (h₄ : a₂ + b₂ = c) : a₁ + a₂ + (b₁ + b₂) = c

theorem Mathlib.Tactic.Ring.add_pf_add_lt {R : Type u_1} [CommSemiring R] {a₂ b c : R} (a₁ : R) : a₂ + b = c → a₁ + a₂ + b = a₁ + c

theorem Mathlib.Tactic.Ring.add_pf_add_gt {R : Type u_1} [CommSemiring R] {a b₂ c : R} (b₁ : R) : a + b₂ = c → a + (b₁ + b₂) = b₁ + c

partial def Mathlib.Tactic.Ring.evalAdd {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (va : ExSum sα a) (vb : ExSum sα b) : Lean.MetaM (Result (ExSum sα) q(«$a» + «$b»))

Adds two polynomials va, vb together to get a normalized result polynomial.

• 0 + b = b
• a + 0 = a
• a * x + a * y = a * (x + y) (for x, y coefficients; uses evalAddOverlap)
• (a₁ + a₂) + (b₁ + b₂) = a₁ + (a₂ + (b₁ + b₂)) (if a₁.lt b₁)
• (a₁ + a₂) + (b₁ + b₂) = b₁ + ((a₁ + a₂) + b₂) (if not a₁.lt b₁)

Multiplication

theorem Mathlib.Tactic.Ring.one_mul {R : Type u_1} [CommSemiring R] (a : R) : Nat.rawCast 1 * a = a

theorem Mathlib.Tactic.Ring.mul_one {R : Type u_1} [CommSemiring R] (a : R) : a * Nat.rawCast 1 = a

theorem Mathlib.Tactic.Ring.mul_pf_left {R : Type u_1} [CommSemiring R] {a₃ b c : R} (a₁ : R) (a₂ : ℕ) : a₃ * b = c → a₁ ^ a₂ * a₃ * b = a₁ ^ a₂ * c

theorem Mathlib.Tactic.Ring.mul_pf_right {R : Type u_1} [CommSemiring R] {a b₃ c : R} (b₁ : R) (b₂ : ℕ) : a * b₃ = c → a * (b₁ ^ b₂ * b₃) = b₁ ^ b₂ * c

theorem Mathlib.Tactic.Ring.mul_pp_pf_overlap {R : Type u_1} [CommSemiring R] {a₂ b₂ c : R} {ea eb e : ℕ} (x : R) : ea + eb = e → a₂ * b₂ = c → x ^ ea * a₂ * (x ^ eb * b₂) = x ^ e * c

partial def Mathlib.Tactic.Ring.evalMulProd {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (va : ExProd sα a) (vb : ExProd sα b) : Lean.MetaM (Result (ExProd sα) q(«$a» * «$b»))

Multiplies two monomials va, vb together to get a normalized result monomial.

• x * y = (x * y) (for x, y coefficients)
• x * (b₁ * b₂) = b₁ * (b₂ * x) (for x coefficient)
• (a₁ * a₂) * y = a₁ * (a₂ * y) (for y coefficient)
• (x ^ ea * a₂) * (x ^ eb * b₂) = x ^ (ea + eb) * (a₂ * b₂) (if ea and eb are identical except coefficient)
• (a₁ * a₂) * (b₁ * b₂) = a₁ * (a₂ * (b₁ * b₂)) (if a₁.lt b₁)
• (a₁ * a₂) * (b₁ * b₂) = b₁ * ((a₁ * a₂) * b₂) (if not a₁.lt b₁)

theorem Mathlib.Tactic.Ring.mul_zero {R : Type u_1} [CommSemiring R] (a : R) : a * 0 = 0

theorem Mathlib.Tactic.Ring.mul_add {R : Type u_1} [CommSemiring R] {a b₁ b₂ c₁ c₂ d : R} : a * b₁ = c₁ → a * b₂ = c₂ → c₁ + 0 + c₂ = d → a * (b₁ + b₂) = d

def Mathlib.Tactic.Ring.evalMul₁ {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (va : ExProd sα a) (vb : ExSum sα b) : Lean.MetaM (Result (ExSum sα) q(«$a» * «$b»))

Multiplies a monomial va to a polynomial vb to get a normalized result polynomial.

• a * 0 = 0
• a * (b₁ + b₂) = (a * b₁) + (a * b₂)

Equations

• One or more equations did not get rendered due to their size.
• Mathlib.Tactic.Ring.evalMul₁ sα va Mathlib.Tactic.Ring.ExSum.zero = pure { expr := q(0), val := Mathlib.Tactic.Ring.ExSum.zero, proof := q(⋯) }

theorem Mathlib.Tactic.Ring.zero_mul {R : Type u_1} [CommSemiring R] (b : R) : 0 * b = 0

theorem Mathlib.Tactic.Ring.add_mul {R : Type u_1} [CommSemiring R] {a₁ a₂ b c₁ c₂ d : R} : a₁ * b = c₁ → a₂ * b = c₂ → c₁ + c₂ = d → (a₁ + a₂) * b = d

def Mathlib.Tactic.Ring.evalMul {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (va : ExSum sα a) (vb : ExSum sα b) : Lean.MetaM (Result (ExSum sα) q(«$a» * «$b»))

Multiplies two polynomials va, vb together to get a normalized result polynomial.

• 0 * b = 0
• (a₁ + a₂) * b = (a₁ * b) + (a₂ * b)

Equations

• One or more equations did not get rendered due to their size.
• Mathlib.Tactic.Ring.evalMul sα Mathlib.Tactic.Ring.ExSum.zero vb = pure { expr := q(0), val := Mathlib.Tactic.Ring.ExSum.zero, proof := q(⋯) }

Scalar multiplication by ℕ

theorem Mathlib.Tactic.Ring.natCast_nat {R : Type u_1} [CommSemiring R] (n : ℕ) : ↑n.rawCast = n.rawCast

theorem Mathlib.Tactic.Ring.natCast_mul {R : Type u_1} [CommSemiring R] {b₁ b₃ : R} {a₁ a₃ : ℕ} (a₂ : ℕ) : ↑a₁ = b₁ → ↑a₃ = b₃ → ↑(a₁ ^ a₂ * a₃) = b₁ ^ a₂ * b₃

theorem Mathlib.Tactic.Ring.natCast_zero {R : Type u_1} [CommSemiring R] : ↑0 = 0

theorem Mathlib.Tactic.Ring.natCast_add {R : Type u_1} [CommSemiring R] {b₁ b₂ : R} {a₁ a₂ : ℕ} : ↑a₁ = b₁ → ↑a₂ = b₂ → ↑(a₁ + a₂) = b₁ + b₂

partial def Mathlib.Tactic.Ring.ExBase.evalNatCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℕ)} (va : ExBase sℕ a) : AtomM (Result (ExBase sα) q(↑«$a»))

Applies Nat.cast to a nat polynomial to produce a polynomial in α.

• An atom e causes ↑e to be allocated as a new atom.
• A sum delegates to ExSum.evalNatCast.

partial def Mathlib.Tactic.Ring.ExProd.evalNatCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℕ)} (va : ExProd sℕ a) : AtomM (Result (ExProd sα) q(↑«$a»))

Applies Nat.cast to a nat monomial to produce a monomial in α.

• ↑c = c if c is a numeric literal
• ↑(a ^ n * b) = ↑a ^ n * ↑b

partial def Mathlib.Tactic.Ring.ExSum.evalNatCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℕ)} (va : ExSum sℕ a) : AtomM (Result (ExSum sα) q(↑«$a»))

Applies Nat.cast to a nat polynomial to produce a polynomial in α.

• ↑0 = 0
• ↑(a + b) = ↑a + ↑b

theorem Mathlib.Tactic.Ring.smul_nat {a b c : ℕ} (h : a * b = c) : a • b = c

theorem Mathlib.Tactic.Ring.smul_eq_cast {R : Type u_1} [CommSemiring R] {a' b c : R} {a : ℕ} : ↑a = a' → a' * b = c → a • b = c

def Mathlib.Tactic.Ring.evalNSMul {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℕ)} {b : Q(«$α»)} (va : ExSum sℕ a) (vb : ExSum sα b) : AtomM (Result (ExSum sα) q(«$a» • «$b»))

Constructs the scalar multiplication n • a, where both n : ℕ and a : α are normalized polynomial expressions.

• a • b = a * b if α = ℕ
• a • b = ↑a * b otherwise

Equations

• One or more equations did not get rendered due to their size.

Scalar multiplication by ℤ

theorem Mathlib.Tactic.Ring.natCast_int {R : Type u_2} [CommRing R] (n : ℕ) : ↑n.rawCast = n.rawCast

theorem Mathlib.Tactic.Ring.intCast_negOfNat_Int {R : Type u_2} [CommRing R] (n : ℕ) : ↑(Int.negOfNat n).rawCast = (Int.negOfNat n).rawCast

theorem Mathlib.Tactic.Ring.intCast_mul {R : Type u_2} [CommRing R] {b₁ b₃ : R} {a₁ a₃ : ℤ} (a₂ : ℕ) : ↑a₁ = b₁ → ↑a₃ = b₃ → ↑(a₁ ^ a₂ * a₃) = b₁ ^ a₂ * b₃

theorem Mathlib.Tactic.Ring.intCast_zero {R : Type u_2} [CommRing R] : ↑0 = 0

theorem Mathlib.Tactic.Ring.intCast_add {R : Type u_2} [CommRing R] {b₁ b₂ : R} {a₁ a₂ : ℤ} : ↑a₁ = b₁ → ↑a₂ = b₂ → ↑(a₁ + a₂) = b₁ + b₂

partial def Mathlib.Tactic.Ring.ExBase.evalIntCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℤ)} (rα : Q(CommRing «$α»)) (va : ExBase sℤ a) : AtomM (Result (ExBase sα) q(↑«$a»))

Applies Int.cast to an int polynomial to produce a polynomial in α.

• An atom e causes ↑e to be allocated as a new atom.
• A sum delegates to ExSum.evalIntCast.

partial def Mathlib.Tactic.Ring.ExProd.evalIntCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℤ)} (rα : Q(CommRing «$α»)) (va : ExProd sℤ a) : AtomM (Result (ExProd sα) q(↑«$a»))

Applies Int.cast to an int monomial to produce a monomial in α.

• ↑c = c if c is a numeric literal
• ↑(a ^ n * b) = ↑a ^ n * ↑b

partial def Mathlib.Tactic.Ring.ExSum.evalIntCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℤ)} (rα : Q(CommRing «$α»)) (va : ExSum sℤ a) : AtomM (Result (ExSum sα) q(↑«$a»))

Applies Int.cast to an int polynomial to produce a polynomial in α.

• ↑0 = 0
• ↑(a + b) = ↑a + ↑b

theorem Mathlib.Tactic.Ring.smul_int {a b c : ℤ} (h : a * b = c) : a • b = c

theorem Mathlib.Tactic.Ring.smul_eq_intCast {R : Type u_2} [CommRing R] {a' b c : R} {a : ℤ} : ↑a = a' → a' * b = c → a • b = c

def Mathlib.Tactic.Ring.evalZSMul {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(ℤ)} {b : Q(«$α»)} (rα : Q(CommRing «$α»)) (va : ExSum sℤ a) (vb : ExSum sα b) : AtomM (Result (ExSum sα) q(«$a» • «$b»))

Constructs the scalar multiplication n • a, where both n : ℤ and a : α are normalized polynomial expressions.

• a • b = a * b if α = ℤ
• a • b = a' * b otherwise, where a' is ↑a with the coercion pushed as deep as possible.

Equations

• One or more equations did not get rendered due to their size.

Negation

theorem Mathlib.Tactic.Ring.neg_one_mul {R : Type u_2} [CommRing R] {a b : R} : (Int.negOfNat 1).rawCast * a = b → -a = b

theorem Mathlib.Tactic.Ring.neg_mul {R : Type u_2} [CommRing R] (a₁ : R) (a₂ : ℕ) {a₃ b : R} : -a₃ = b → -(a₁ ^ a₂ * a₃) = a₁ ^ a₂ * b

def Mathlib.Tactic.Ring.evalNegProd {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} (rα : Q(CommRing «$α»)) (va : ExProd sα a) : Lean.MetaM (Result (ExProd sα) q(-«$a»))

Negates a monomial va to get another monomial.

• -c = (-c) (for c coefficient)
• -(a₁ * a₂) = a₁ * -a₂

Equations

• One or more equations did not get rendered due to their size.

theorem Mathlib.Tactic.Ring.neg_zero {R : Type u_2} [CommRing R] : -0 = 0

theorem Mathlib.Tactic.Ring.neg_add {R : Type u_2} [CommRing R] {a₁ a₂ b₁ b₂ : R} : -a₁ = b₁ → -a₂ = b₂ → -(a₁ + a₂) = b₁ + b₂

def Mathlib.Tactic.Ring.evalNeg {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} (rα : Q(CommRing «$α»)) (va : ExSum sα a) : Lean.MetaM (Result (ExSum sα) q(-«$a»))

Negates a polynomial va to get another polynomial.

• -0 = 0 (for c coefficient)
• -(a₁ + a₂) = -a₁ + -a₂

Equations

• One or more equations did not get rendered due to their size.
• Mathlib.Tactic.Ring.evalNeg sα rα Mathlib.Tactic.Ring.ExSum.zero = pure { expr := q(0), val := Mathlib.Tactic.Ring.ExSum.zero, proof := q(⋯) }

Subtraction

theorem Mathlib.Tactic.Ring.sub_pf {R : Type u_2} [CommRing R] {a b c d : R} : -b = c → a + c = d → a - b = d

def Mathlib.Tactic.Ring.evalSub {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (rα : Q(CommRing «$α»)) (va : ExSum sα a) (vb : ExSum sα b) : Lean.MetaM (Result (ExSum sα) q(«$a» - «$b»))

Subtracts two polynomials va, vb to get a normalized result polynomial.

• a - b = a + -b

Equations

• One or more equations did not get rendered due to their size.

Exponentiation

theorem Mathlib.Tactic.Ring.pow_prod_atom {R : Type u_1} [CommSemiring R] (a : R) (b : ℕ) : a ^ b = (a + 0) ^ b * Nat.rawCast 1

def Mathlib.Tactic.Ring.evalPowProdAtom {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} {b : Q(ℕ)} (va : ExProd sα a) (vb : ExProd sℕ b) : Result (ExProd sα) q(«$a» ^ «$b»)

The fallback case for exponentiating polynomials is to use ExBase.toProd to just build an exponent expression. (This has a slightly different normalization than evalPowAtom because the input types are different.)

• x ^ e = (x + 0) ^ e * 1

Equations

• Mathlib.Tactic.Ring.evalPowProdAtom sα va vb = { expr := q((«$a» + 0) ^ «$b» * Nat.rawCast 1), val := (Mathlib.Tactic.Ring.ExBase.sum va.toSum).toProd vb, proof := q(⋯) }

theorem Mathlib.Tactic.Ring.pow_atom {R : Type u_1} [CommSemiring R] (a : R) (b : ℕ) : a ^ b = a ^ b * Nat.rawCast 1 + 0

def Mathlib.Tactic.Ring.evalPowAtom {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} {b : Q(ℕ)} (va : ExBase sα a) (vb : ExProd sℕ b) : Result (ExSum sα) q(«$a» ^ «$b»)

The fallback case for exponentiating polynomials is to use ExBase.toProd to just build an exponent expression.

• x ^ e = x ^ e * 1 + 0

Equations

• Mathlib.Tactic.Ring.evalPowAtom sα va vb = { expr := q(«$a» ^ «$b» * Nat.rawCast 1 + 0), val := (va.toProd vb).toSum, proof := q(⋯) }

theorem Mathlib.Tactic.Ring.const_pos (n : ℕ) (h : Nat.ble 1 n = true) : 0 < n.rawCast

theorem Mathlib.Tactic.Ring.mul_exp_pos {a₁ a₂ : ℕ} (n : ℕ) (h₁ : 0 < a₁) (h₂ : 0 < a₂) : 0 < a₁ ^ n * a₂

theorem Mathlib.Tactic.Ring.add_pos_left {a₁ : ℕ} (a₂ : ℕ) (h : 0 < a₁) : 0 < a₁ + a₂

theorem Mathlib.Tactic.Ring.add_pos_right {a₂ : ℕ} (a₁ : ℕ) (h : 0 < a₂) : 0 < a₁ + a₂

partial def Mathlib.Tactic.Ring.ExBase.evalPos {a : Q(ℕ)} (va : ExBase sℕ a) : Option Q(0 < «$a»)

Attempts to prove that a polynomial expression in ℕ is positive.

• Atoms are not (necessarily) positive
• Sums defer to ExSum.evalPos

partial def Mathlib.Tactic.Ring.ExProd.evalPos {a : Q(ℕ)} (va : ExProd sℕ a) : Option Q(0 < «$a»)

Attempts to prove that a monomial expression in ℕ is positive.

• 0 < c (where c is a numeral) is true by the normalization invariant (c is not zero)
• 0 < x ^ e * b if 0 < x and 0 < b

partial def Mathlib.Tactic.Ring.ExSum.evalPos {a : Q(ℕ)} (va : ExSum sℕ a) : Option Q(0 < «$a»)

Attempts to prove that a polynomial expression in ℕ is positive.

• 0 < 0 fails
• 0 < a + b if 0 < a or 0 < b

theorem Mathlib.Tactic.Ring.pow_one {R : Type u_1} [CommSemiring R] (a : R) : a ^ 1 = a

theorem Mathlib.Tactic.Ring.pow_bit0 {R : Type u_1} [CommSemiring R] {a b c : R} {k : ℕ} : a ^ k = b → b * b = c → a ^ Nat.mul 2 k = c

theorem Mathlib.Tactic.Ring.pow_bit1 {R : Type u_1} [CommSemiring R] {a b c : R} {k : ℕ} {d : R} : a ^ k = b → b * b = c → c * a = d → a ^ (Nat.mul 2 k).add 1 = d

partial def Mathlib.Tactic.Ring.evalPowNat {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} (va : ExSum sα a) (n : Q(ℕ)) : Lean.MetaM (Result (ExSum sα) q(«$a» ^ «$n»))

The main case of exponentiation of ring expressions is when va is a polynomial and n is a nonzero literal expression, like (x + y)^5. In this case we work out the polynomial completely into a sum of monomials.

• x ^ 1 = x
• x ^ (2*n) = x ^ n * x ^ n
• x ^ (2*n+1) = x ^ n * x ^ n * x

theorem Mathlib.Tactic.Ring.one_pow {R : Type u_1} [CommSemiring R] (b : ℕ) : Nat.rawCast 1 ^ b = Nat.rawCast 1

theorem Mathlib.Tactic.Ring.mul_pow {R : Type u_1} [CommSemiring R] {a₂ c₂ : R} {ea₁ b c₁ : ℕ} {xa₁ : R} : ea₁ * b = c₁ → a₂ ^ b = c₂ → (xa₁ ^ ea₁ * a₂) ^ b = xa₁ ^ c₁ * c₂

def Mathlib.Tactic.Ring.evalPowProd {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} {b : Q(ℕ)} (va : ExProd sα a) (vb : ExProd sℕ b) : Lean.MetaM (Result (ExProd sα) q(«$a» ^ «$b»))

There are several special cases when exponentiating monomials:

• 1 ^ n = 1
• x ^ y = (x ^ y) when x and y are constants
• (a * b) ^ e = a ^ e * b ^ e

In all other cases we use evalPowProdAtom.

Equations

• One or more equations did not get rendered due to their size.

structure Mathlib.Tactic.Ring.ExtractCoeff (e : Q(ℕ)) : Type

The result of extractCoeff is a numeral and a proof that the original expression factors by this numeral.

• k : Q(ℕ)

  A raw natural number literal.

• e' : Q(ℕ)

  The result of extracting the coefficient is a monic monomial.

• ve' : ExProd sℕ self.e'

  e' is a monomial.

• p : Q(«$e» = unknown_1 * unknown_2)

  The proof that e splits into the coefficient k and the monic monomial e'.

theorem Mathlib.Tactic.Ring.coeff_one (k : ℕ) : k.rawCast = Nat.rawCast 1 * k

theorem Mathlib.Tactic.Ring.coeff_mul {a₃ c₂ k : ℕ} (a₁ a₂ : ℕ) : a₃ = c₂ * k → a₁ ^ a₂ * a₃ = a₁ ^ a₂ * c₂ * k

def Mathlib.Tactic.Ring.extractCoeff {a : Q(ℕ)} (va : ExProd sℕ a) : ExtractCoeff a

Given a monomial expression va, splits off the leading coefficient k and the remainder e', stored in the ExtractCoeff structure.

• c = 1 * c (if c is a constant)
• a * b = (a * b') * k if b = b' * k

theorem Mathlib.Tactic.Ring.pow_one_cast {R : Type u_1} [CommSemiring R] (a : R) : a ^ Nat.rawCast 1 = a

theorem Mathlib.Tactic.Ring.zero_pow {R : Type u_1} [CommSemiring R] {b : ℕ} : 0 < b → 0 ^ b = 0

theorem Mathlib.Tactic.Ring.single_pow {R : Type u_1} [CommSemiring R] {a c : R} {b : ℕ} : a ^ b = c → (a + 0) ^ b = c + 0

theorem Mathlib.Tactic.Ring.pow_nat {R : Type u_1} [CommSemiring R] {a : R} {b c k : ℕ} {d e : R} : b = c * k → a ^ c = d → d ^ k = e → a ^ b = e

partial def Mathlib.Tactic.Ring.evalPow₁ {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} {b : Q(ℕ)} (va : ExSum sα a) (vb : ExProd sℕ b) : Lean.MetaM (Result (ExSum sα) q(«$a» ^ «$b»))

Exponentiates a polynomial va by a monomial vb, including several special cases.

• a ^ 1 = a
• 0 ^ e = 0 if 0 < e
• (a + 0) ^ b = a ^ b computed using evalPowProd
• a ^ b = (a ^ b') ^ k if b = b' * k and k > 1

Otherwise a ^ b is just encoded as a ^ b * 1 + 0 using evalPowAtom.

theorem Mathlib.Tactic.Ring.pow_zero {R : Type u_1} [CommSemiring R] (a : R) : a ^ 0 = Nat.rawCast 1 + 0

theorem Mathlib.Tactic.Ring.pow_add {R : Type u_1} [CommSemiring R] {a c₁ c₂ : R} {b₁ b₂ : ℕ} {d : R} : a ^ b₁ = c₁ → a ^ b₂ = c₂ → c₁ * c₂ = d → a ^ (b₁ + b₂) = d

def Mathlib.Tactic.Ring.evalPow {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a : Q(«$α»)} {b : Q(ℕ)} (va : ExSum sα a) (vb : ExSum sℕ b) : Lean.MetaM (Result (ExSum sα) q(«$a» ^ «$b»))

Exponentiates two polynomials va, vb.

• a ^ 0 = 1
• a ^ (b₁ + b₂) = a ^ b₁ * a ^ b₂

Equations

• One or more equations did not get rendered due to their size.
• Mathlib.Tactic.Ring.evalPow sα va Mathlib.Tactic.Ring.ExSum.zero = pure { expr := q(Nat.rawCast 1 + 0), val := (Mathlib.Tactic.Ring.ExProd.mkNat sα 1).snd.toSum, proof := q(⋯) }

structure Mathlib.Tactic.Ring.Cache {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) : Type

This cache contains data required by the ring tactic during execution.

• rα : Option Q(CommRing «$α»)

  A ring instance on α, if available.

• dsα : Option Q(Semifield «$α»)

  A division semiring instance on α, if available.

• czα : Option Q(CharZero «$α»)

  A characteristic zero ring instance on α, if available.

def Mathlib.Tactic.Ring.mkCache {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) : Lean.MetaM (Cache sα)

Create a new cache for α by doing the necessary instance searches.

Equations

• One or more equations did not get rendered due to their size.

theorem Mathlib.Tactic.Ring.cast_pos {R : Type u_1} [CommSemiring R] {a : R} {n : ℕ} : Meta.NormNum.IsNat a n → a = n.rawCast + 0

theorem Mathlib.Tactic.Ring.cast_zero {R : Type u_1} [CommSemiring R] {a : R} : Meta.NormNum.IsNat a 0 → a = 0

theorem Mathlib.Tactic.Ring.cast_neg {n : ℕ} {R : Type u_2} [Ring R] {a : R} : Meta.NormNum.IsInt a (Int.negOfNat n) → a = (Int.negOfNat n).rawCast + 0

theorem Mathlib.Tactic.Ring.cast_nnrat {n d : ℕ} {R : Type u_2} [DivisionSemiring R] {a : R} : Meta.NormNum.IsNNRat a n d → a = NNRat.rawCast n d + 0

theorem Mathlib.Tactic.Ring.cast_rat {n : ℤ} {d : ℕ} {R : Type u_2} [DivisionRing R] {a : R} : Meta.NormNum.IsRat a n d → a = Rat.rawCast n d + 0

def Mathlib.Tactic.Ring.evalCast {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {e : Q(«$α»)} : Meta.NormNum.Result e → Option (Result (ExSum sα) e)

Converts a proof by norm_num that e is a numeral, into a normalization as a monomial:

• e = 0 if norm_num returns IsNat e 0
• e = Nat.rawCast n + 0 if norm_num returns IsNat e n
• e = Int.rawCast n + 0 if norm_num returns IsInt e n
• e = NNRat.rawCast n d + 0 if norm_num returns IsNNRat e n d
• e = Rat.rawCast n d + 0 if norm_num returns IsRat e n d

theorem Mathlib.Tactic.Ring.toProd_pf {R : Type u_1} [CommSemiring R] {a a' : R} (p : a = a') : a = a' ^ Nat.rawCast 1 * Nat.rawCast 1

theorem Mathlib.Tactic.Ring.atom_pf {R : Type u_1} [CommSemiring R] (a : R) : a = a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0

theorem Mathlib.Tactic.Ring.atom_pf' {R : Type u_1} [CommSemiring R] {a a' : R} (p : a = a') : a = a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0

def Mathlib.Tactic.Ring.evalAtom {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (e : Q(«$α»)) : AtomM (Result (ExSum sα) e)

Evaluates an atom, an expression where ring can find no additional structure.

• a = a ^ 1 * 1 + 0

Equations

• One or more equations did not get rendered due to their size.

theorem Mathlib.Tactic.Ring.inv_mul {R : Type u_2} [Semifield R] {a₁ : R} {a₂ : ℕ} {a₃ b₁ b₃ c : R} : a₁⁻¹ = b₁ → a₃⁻¹ = b₃ → b₃ * (b₁ ^ a₂ * Nat.rawCast 1) = c → (a₁ ^ a₂ * a₃)⁻¹ = c

theorem Mathlib.Tactic.Ring.inv_zero {R : Type u_2} [Semifield R] : 0⁻¹ = 0

theorem Mathlib.Tactic.Ring.inv_single {R : Type u_2} [Semifield R] {a b : R} : a⁻¹ = b → (a + 0)⁻¹ = b + 0

theorem Mathlib.Tactic.Ring.inv_add {R : Type u_1} [CommSemiring R] {b₁ b₂ : R} {a₁ a₂ : ℕ} : ↑a₁ = b₁ → ↑a₂ = b₂ → ↑(a₁ + a₂) = b₁ + b₂

def Mathlib.Tactic.Ring.evalInvAtom {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (dsα : Q(Semifield «$α»)) (a : Q(«$α»)) : AtomM (Result (ExBase sα) q(«$a»⁻¹))

Applies ⁻¹ to a polynomial to get an atom.

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def Mathlib.Tactic.Ring.ExProd.evalInv {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (dsα : Q(Semifield «$α»)) {a : Q(«$α»)} (czα : Option Q(CharZero «$α»)) (va : ExProd sα a) : AtomM (Result (ExProd sα) q(«$a»⁻¹))

Inverts a polynomial va to get a normalized result polynomial.

• c⁻¹ = (c⁻¹) if c is a constant
• (a ^ b * c)⁻¹ = a⁻¹ ^ b * c⁻¹

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def Mathlib.Tactic.Ring.ExSum.evalInv {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (dsα : Q(Semifield «$α»)) {a : Q(«$α»)} (czα : Option Q(CharZero «$α»)) (va : ExSum sα a) : AtomM (Result (ExSum sα) q(«$a»⁻¹))

Inverts a polynomial va to get a normalized result polynomial.

• 0⁻¹ = 0
• a⁻¹ = (a⁻¹) if a is a nontrivial sum

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• Mathlib.Tactic.Ring.ExSum.evalInv sα dsα czα Mathlib.Tactic.Ring.ExSum.zero = pure { expr := q(0), val := Mathlib.Tactic.Ring.ExSum.zero, proof := q(⋯) }

theorem Mathlib.Tactic.Ring.div_pf {R : Type u_2} [Semifield R] {a b c d : R} : b⁻¹ = c → a * c = d → a / b = d

def Mathlib.Tactic.Ring.evalDiv {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) {a b : Q(«$α»)} (rα : Q(Semifield «$α»)) (czα : Option Q(CharZero «$α»)) (va : ExSum sα a) (vb : ExSum sα b) : AtomM (Result (ExSum sα) q(«$a» / «$b»))

Divides two polynomials va, vb to get a normalized result polynomial.

• a / b = a * b⁻¹

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theorem Mathlib.Tactic.Ring.add_congr {R : Type u_1} [CommSemiring R] {a a' b b' c : R} : a = a' → b = b' → a' + b' = c → a + b = c

theorem Mathlib.Tactic.Ring.mul_congr {R : Type u_1} [CommSemiring R] {a a' b b' c : R} : a = a' → b = b' → a' * b' = c → a * b = c

theorem Mathlib.Tactic.Ring.nsmul_congr {R : Type u_1} [CommSemiring R] {b b' c : R} {a a' : ℕ} : a = a' → b = b' → a' • b' = c → a • b = c

theorem Mathlib.Tactic.Ring.zsmul_congr {R : Type u_2} [CommRing R] {b b' c : R} {a a' : ℤ} : a = a' → b = b' → a' • b' = c → a • b = c

theorem Mathlib.Tactic.Ring.pow_congr {R : Type u_1} [CommSemiring R] {a a' c : R} {b b' : ℕ} : a = a' → b = b' → a' ^ b' = c → a ^ b = c

theorem Mathlib.Tactic.Ring.neg_congr {R : Type u_2} [CommRing R] {a a' b : R} : a = a' → -a' = b → -a = b

theorem Mathlib.Tactic.Ring.sub_congr {R : Type u_2} [CommRing R] {a a' b b' c : R} : a = a' → b = b' → a' - b' = c → a - b = c

theorem Mathlib.Tactic.Ring.inv_congr {R : Type u_2} [Semifield R] {a a' b : R} : a = a' → a'⁻¹ = b → a⁻¹ = b

theorem Mathlib.Tactic.Ring.div_congr {R : Type u_2} [Semifield R] {a a' b b' c : R} : a = a' → b = b' → a' / b' = c → a / b = c

def Mathlib.Tactic.Ring.Cache.nat : Cache sℕ

A precomputed Cache for ℕ.

Equations

• Mathlib.Tactic.Ring.Cache.nat = { rα := none, dsα := none, czα := some q(⋯) }

def Mathlib.Tactic.Ring.Cache.int : Cache sℤ

A precomputed Cache for ℤ.

Equations

• Mathlib.Tactic.Ring.Cache.int = { rα := some q(inferInstance), dsα := none, czα := some q(⋯) }

def Mathlib.Tactic.Ring.isAtomOrDerivable {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (c : Cache sα) (e : Q(«$α»)) : AtomM (Option (Option (Result (ExSum sα) e)))

Checks whether e would be processed by eval as a ring expression, or otherwise if it is an atom or something simplifiable via norm_num.

We use this in ring_nf to avoid rewriting atoms unnecessarily.

Returns:

• none if eval would process e as an algebraic ring expression
• some none if eval would treat e as an atom.
• some (some r) if eval would not process e as an algebraic ring expression, but NormNum.derive can nevertheless simplify e, with result r.

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partial def Mathlib.Tactic.Ring.eval {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)) (c : Cache sα) (e : Q(«$α»)) : AtomM (Result (ExSum sα) e)

Evaluates expression e of type α into a normalized representation as a polynomial. This is the main driver of ring, which calls out to evalAdd, evalMul etc.

class Mathlib.Tactic.Ring.CSLift (α : Type u) (β : outParam (Type u)) : Type u

CSLift α β is a typeclass used by ring for lifting operations from α (which is not a commutative semiring) into a commutative semiring β by using an injective map lift : α → β.

• lift : α → β

  lift is the "canonical injection" from α to β

• inj : Function.Injective lift

  lift is an injective function

class Mathlib.Tactic.Ring.CSLiftVal {α : Type u} {β : outParam (Type u)} [CSLift α β] (a : α) (b : outParam β) : Prop

CSLiftVal a b means that b = lift a. This is used by ring to construct an expression b from the input expression a, and then run the usual ring algorithm on b.

• eq : b = CSLift.lift a

  The output value b is equal to the lift of a. This can be supplied by the default instance which sets b := lift a, but ring will treat this as an atom so it is more useful when there are other instances which distribute addition or multiplication.

@[instance 100]

instance Mathlib.Tactic.Ring.instCSLiftValLift {α β : Type u_2} [CSLift α β] (a : α) : CSLiftVal a (CSLift.lift a)

theorem Mathlib.Tactic.Ring.of_lift {α β : Type u_2} [inst : CSLift α β] {a b : α} {a' b' : β} [h1 : CSLiftVal a a'] [h2 : CSLiftVal b b'] (h : a' = b') : a = b

theorem Mathlib.Tactic.Ring.of_eq {α : Sort u_2} {a b c : α} : a = c → b = c → a = b

opaque Mathlib.Tactic.Ring.ringCleanupRef : IO.Ref (Lean.Expr → Lean.MetaM Lean.Expr)

This is a routine which is used to clean up the unsolved subgoal of a failed ring1 application. It is overridden in Mathlib/Tactic/Ring/RingNF.lean to apply the ring_nf simp set to the goal.

def Mathlib.Tactic.Ring.proveEq (g : Lean.MVarId) : AtomM Unit

Frontend of ring1: attempt to close a goal g, assuming it is an equation of semirings.

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def Mathlib.Tactic.Ring.proveEq.ringCore {v : Lean.Level} {α : Q(Type v)} (sα : Q(CommSemiring «$α»)) (e₁ e₂ : Q(«$α»)) : AtomM Q(«$e₁» = «$e₂»)

The core of proveEq takes expressions e₁ e₂ : α where α is a CommSemiring, and returns a proof that they are equal (or fails).

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def Mathlib.Tactic.Ring.ring1 : Lean.ParserDescr

ring1 solves the goal when it is an equality in commutative (semi)rings, allowing variables in the exponent.

This version of ring fails if the target is not an equality.

• ring1! uses a more aggressive reducibility setting to determine equality of atoms.

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• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.Ring.tacticRing1! : Lean.ParserDescr

ring1 solves the goal when it is an equality in commutative (semi)rings, allowing variables in the exponent.

This version of ring fails if the target is not an equality.

• ring1! uses a more aggressive reducibility setting to determine equality of atoms.

Equations

• Mathlib.Tactic.Ring.tacticRing1! = Lean.ParserDescr.node `Mathlib.Tactic.Ring.tacticRing1! 1024 (Lean.ParserDescr.nonReservedSymbol "ring1!" false)