ring_nf tactic

A tactic which uses ring to rewrite expressions. This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

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

True if this represents an atomic expression.

Equations

• (Mathlib.Tactic.Ring.ExBase.atom id).isAtom = true
• x✝.isAtom = false

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

True if this represents an atomic expression.

Equations

• One or more equations did not get rendered due to their size.
• x✝.isAtom = false

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

True if this represents an atomic expression.

Equations

• (Mathlib.Tactic.Ring.ExSum.add va₁ Mathlib.Tactic.Ring.ExSum.zero).isAtom = va₁.isAtom
• (Mathlib.Tactic.Ring.ExSum.add va₁ va₂).isAtom = false
• x✝.isAtom = false

inductive Mathlib.Tactic.RingNF.RingMode : Type

The normalization style for ring_nf.

• SOP : RingMode

  Sum-of-products form, like x + x * y * 2 + z ^ 2.

• raw : RingMode

  Raw form: the representation ring uses internally.

instance Mathlib.Tactic.RingNF.instInhabitedRingMode : Inhabited RingMode

Equations

• Mathlib.Tactic.RingNF.instInhabitedRingMode = { default := Mathlib.Tactic.RingNF.instInhabitedRingMode.default }

def Mathlib.Tactic.RingNF.instInhabitedRingMode.default : RingMode

Equations

• Mathlib.Tactic.RingNF.instInhabitedRingMode.default = Mathlib.Tactic.RingNF.RingMode.SOP

instance Mathlib.Tactic.RingNF.instBEqRingMode : BEq RingMode

Equations

• Mathlib.Tactic.RingNF.instBEqRingMode = { beq := Mathlib.Tactic.RingNF.instBEqRingMode.beq }

def Mathlib.Tactic.RingNF.instBEqRingMode.beq : RingMode → RingMode → Bool

Equations

• Mathlib.Tactic.RingNF.instBEqRingMode.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Mathlib.Tactic.RingNF.instReprRingMode : Repr RingMode

Equations

• Mathlib.Tactic.RingNF.instReprRingMode = { reprPrec := Mathlib.Tactic.RingNF.instReprRingMode.repr }

def Mathlib.Tactic.RingNF.instReprRingMode.repr : RingMode → ℕ → Std.Format

Equations

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

structure Mathlib.Tactic.RingNF.Configextends Mathlib.Tactic.AtomM.Recurse.Config : Type

Configuration for ring_nf.

• red : Lean.Meta.TransparencyMode
• zetaDelta : Bool
• failIfUnchanged : Bool

  if true, then fail if no progress is made

• mode : RingMode

  The normalization style.

instance Mathlib.Tactic.RingNF.instInhabitedConfig : Inhabited Config

Equations

• Mathlib.Tactic.RingNF.instInhabitedConfig = { default := Mathlib.Tactic.RingNF.instInhabitedConfig.default }

def Mathlib.Tactic.RingNF.instInhabitedConfig.default : Config

Equations

• Mathlib.Tactic.RingNF.instInhabitedConfig.default = { toConfig := default, failIfUnchanged := default, mode := default }

def Mathlib.Tactic.RingNF.instBEqConfig.beq : Config → Config → Bool

Equations

• Mathlib.Tactic.RingNF.instBEqConfig.beq { toConfig := a, failIfUnchanged := a_1, mode := a_2 } { toConfig := b, failIfUnchanged := b_1, mode := b_2 } = (a == b && (a_1 == b_1 && a_2 == b_2))
• Mathlib.Tactic.RingNF.instBEqConfig.beq x✝¹ x✝ = false

instance Mathlib.Tactic.RingNF.instBEqConfig : BEq Config

Equations

• Mathlib.Tactic.RingNF.instBEqConfig = { beq := Mathlib.Tactic.RingNF.instBEqConfig.beq }

def Mathlib.Tactic.RingNF.instReprConfig.repr : Config → ℕ → Std.Format

Equations

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

instance Mathlib.Tactic.RingNF.instReprConfig : Repr Config

Equations

• Mathlib.Tactic.RingNF.instReprConfig = { reprPrec := Mathlib.Tactic.RingNF.instReprConfig.repr }

def Mathlib.Tactic.RingNF.elabConfig : Lean.Syntax → Lean.Elab.Tactic.TacticM Config

Function elaborating RingNF.Config.

Equations

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

def Mathlib.Tactic.RingNF.evalExpr (e : Lean.Expr) : AtomM Lean.Meta.Simp.Result

Evaluates an expression e into a normalized representation as a polynomial.

This is a variant of Mathlib.Tactic.Ring.eval, the main driver of the ring tactic. It differs in

• operating on Expr (input) and Simp.Result (output), rather than typed Qq versions of these;
• throwing an error if the expression e is an atom for the ring tactic.

Equations

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

theorem Mathlib.Tactic.RingNF.add_assoc_rev {R : Type u_1} [CommSemiring R] (a b c : R) : a + (b + c) = a + b + c

theorem Mathlib.Tactic.RingNF.mul_assoc_rev {R : Type u_1} [CommSemiring R] (a b c : R) : a * (b * c) = a * b * c

theorem Mathlib.Tactic.RingNF.mul_neg {R : Type u_2} [Ring R] (a b : R) : a * -b = -(a * b)

theorem Mathlib.Tactic.RingNF.add_neg {R : Type u_2} [Ring R] (a b : R) : a + -b = a - b

theorem Mathlib.Tactic.RingNF.nat_rawCast_0 {R : Type u_1} [CommSemiring R] : Nat.rawCast 0 = 0

theorem Mathlib.Tactic.RingNF.nat_rawCast_1 {R : Type u_1} [CommSemiring R] : Nat.rawCast 1 = 1

theorem Mathlib.Tactic.RingNF.nat_rawCast_2 {R : Type u_1} [CommSemiring R] {n : ℕ} [n.AtLeastTwo] : n.rawCast = OfNat.ofNat n

theorem Mathlib.Tactic.RingNF.int_rawCast_neg {n : ℕ} {R : Type u_2} [Ring R] : (Int.negOfNat n).rawCast = -n.rawCast

theorem Mathlib.Tactic.RingNF.nnrat_rawCast {n d : ℕ} {R : Type u_2} [DivisionSemiring R] : NNRat.rawCast n d = n.rawCast / d.rawCast

theorem Mathlib.Tactic.RingNF.rat_rawCast_neg {n d : ℕ} {R : Type u_2} [DivisionRing R] : Rat.rawCast (Int.negOfNat n) d = (Int.negOfNat n).rawCast / d.rawCast

def Mathlib.Tactic.RingNF.cleanup (cfg : Config) (r : Lean.Meta.Simp.Result) : Lean.MetaM Lean.Meta.Simp.Result

A cleanup routine, which simplifies normalized polynomials to a more human-friendly format.

Equations

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

def Mathlib.Tactic.RingNF.ringNF : Lean.ParserDescr

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • zetaDelta: if true, local let variables can be unfolded (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

Equations

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

def Mathlib.Tactic.RingNF.tacticRing_nf!__ : Lean.ParserDescr

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • zetaDelta: if true, local let variables can be unfolded (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

Equations

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

def Mathlib.Tactic.RingNF.ringNFConv : Lean.ParserDescr

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • zetaDelta: if true, local let variables can be unfolded (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

Equations

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

def Mathlib.Tactic.RingNF.ring1NF : Lean.ParserDescr

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring1 uses ring_nf to simplify in atoms.
• The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.

Equations

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

def Mathlib.Tactic.RingNF.tacticRing1_nf!_ : Lean.ParserDescr

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring1 uses ring_nf to simplify in atoms.
• The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.

Equations

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

def Mathlib.Tactic.RingNF.elabRingNFConv : Lean.Elab.Tactic.Tactic

Elaborator for the ring_nf tactic.

Equations

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

def Mathlib.Tactic.RingNF.convRing_nf!_ : Lean.ParserDescr

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • zetaDelta: if true, local let variables can be unfolded (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

Equations

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

def Mathlib.Tactic.RingNF.ring : Lean.ParserDescr

Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent. If the goal is not appropriate for ring (e.g. not an equality) ring_nf will be suggested.

• ring! will use a more aggressive reducibility setting to determine equality of atoms.
• ring1 fails if the target is not an equality.

For example:

example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
example (x y : ℕ) : x + id y = y + id x := by ring!
example (x : ℕ) (h : x * 2 > 5): x + x > 5 := by ring; assumption -- suggests ring_nf

Equations

• Mathlib.Tactic.RingNF.ring = Lean.ParserDescr.node `Mathlib.Tactic.RingNF.ring 1024 (Lean.ParserDescr.nonReservedSymbol "ring" false)

def Mathlib.Tactic.RingNF.tacticRing! : Lean.ParserDescr

Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent. If the goal is not appropriate for ring (e.g. not an equality) ring_nf will be suggested.

• ring! will use a more aggressive reducibility setting to determine equality of atoms.
• ring1 fails if the target is not an equality.

For example:

example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
example (x y : ℕ) : x + id y = y + id x := by ring!
example (x : ℕ) (h : x * 2 > 5): x + x > 5 := by ring; assumption -- suggests ring_nf

Equations

• Mathlib.Tactic.RingNF.tacticRing! = Lean.ParserDescr.node `Mathlib.Tactic.RingNF.tacticRing! 1024 (Lean.ParserDescr.nonReservedSymbol "ring!" false)

def Mathlib.Tactic.RingNF.ringConv : Lean.ParserDescr

The tactic ring evaluates expressions in commutative (semi)rings. This is the conv tactic version, which rewrites a target which is a ring equality to True.

See also the ring tactic.

Equations

• Mathlib.Tactic.RingNF.ringConv = Lean.ParserDescr.node `Mathlib.Tactic.RingNF.ringConv 1024 (Lean.ParserDescr.nonReservedSymbol "ring" false)

def Mathlib.Tactic.RingNF.convRing! : Lean.ParserDescr

The tactic ring evaluates expressions in commutative (semi)rings. This is the conv tactic version, which rewrites a target which is a ring equality to True.

See also the ring tactic.

Equations

• Mathlib.Tactic.RingNF.convRing! = Lean.ParserDescr.node `Mathlib.Tactic.RingNF.convRing! 1024 (Lean.ParserDescr.nonReservedSymbol "ring!" false)

We register ring with the hint tactic.