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»)} : Mathlib.Tactic.Ring.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»)} : Mathlib.Tactic.Ring.ExProd sα a → Bool

True if this represents an atomic expression.

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

True if this represents an atomic expression.

Equations

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

inductive Mathlib.Tactic.RingNF.RingMode : Type

The normalization style for ring_nf.

• SOP: Mathlib.Tactic.RingNF.RingMode

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

• raw: Mathlib.Tactic.RingNF.RingMode

  Raw form: the representation ring uses internally.

instance Mathlib.Tactic.RingNF.instInhabitedRingMode : Inhabited Mathlib.Tactic.RingNF.RingMode

Equations

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

instance Mathlib.Tactic.RingNF.instBEqRingMode : BEq Mathlib.Tactic.RingNF.RingMode

Equations

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

instance Mathlib.Tactic.RingNF.instReprRingMode : Repr Mathlib.Tactic.RingNF.RingMode

Equations

• Mathlib.Tactic.RingNF.instReprRingMode = { reprPrec := Mathlib.Tactic.RingNF.reprRingMode✝ }

structure Mathlib.Tactic.RingNF.Config : Type

Configuration for ring_nf.

• red : Lean.Meta.TransparencyMode

  the reducibility setting to use when comparing atoms for defeq

• recursive : Bool

  if true, atoms inside ring expressions will be reduced recursively

• mode : Mathlib.Tactic.RingNF.RingMode

  The normalization style.

instance Mathlib.Tactic.RingNF.instInhabitedConfig : Inhabited Mathlib.Tactic.RingNF.Config

Equations

• Mathlib.Tactic.RingNF.instInhabitedConfig = { default := { red := default, recursive := default, mode := default } }

instance Mathlib.Tactic.RingNF.instBEqConfig : BEq Mathlib.Tactic.RingNF.Config

Equations

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

instance Mathlib.Tactic.RingNF.instReprConfig : Repr Mathlib.Tactic.RingNF.Config

Equations

• Mathlib.Tactic.RingNF.instReprConfig = { reprPrec := Mathlib.Tactic.RingNF.reprConfig✝ }

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

Function elaborating RingNF.Config.

Equations

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

structure Mathlib.Tactic.RingNF.Context : Type

The read-only state of the RingNF monad.

• ctx : Lean.Meta.Simp.Context

  A basically empty simp context, passed to the simp traversal in RingNF.rewrite.

• simp : Lean.Meta.Simp.Result → Lean.Meta.SimpM Lean.Meta.Simp.Result

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

@[reducible, inline]

abbrev Mathlib.Tactic.RingNF.M (α : Type) : Type

The monad for RingNF contains, in addition to the AtomM state, a simp context for the main traversal and a simp function (which has another simp context) to simplify normalized polynomials.

Equations

• Mathlib.Tactic.RingNF.M = ReaderT Mathlib.Tactic.RingNF.Context Mathlib.Tactic.AtomM

def Mathlib.Tactic.RingNF.rewrite (parent : Lean.Expr) (root : Bool := true) : Mathlib.Tactic.RingNF.M Lean.Meta.Simp.Result

A tactic in the RingNF.M monad which will simplify expression parent to a normal form.

• root: true if this is a direct call to the function. RingNF.M.run sets this to false in recursive mode.

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.rat_rawCast_pos {n d : ℕ} {R : Type u_2} [DivisionRing R] : Rat.rawCast (Int.ofNat 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.M.run {α : Type} (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RingNF.Config) (x : Mathlib.Tactic.RingNF.M α) : Lean.MetaM α

Runs a tactic in the RingNF.M monad, given initial data:

• s: a reference to the mutable state of ring, for persisting across calls. This ensures that atom ordering is used consistently.
• cfg: the configuration options
• x: the tactic to run

Equations

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

partial def Mathlib.Tactic.RingNF.M.run.rctx (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RingNF.Config) (nctx : Mathlib.Tactic.RingNF.Context) : Mathlib.Tactic.AtomM.Context

The recursive context.

partial def Mathlib.Tactic.RingNF.M.run.evalAtom (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RingNF.Config) (nctx : Mathlib.Tactic.RingNF.Context) (e : Lean.Expr) : Lean.MetaM Lean.Meta.Simp.Result

The atom evaluator calls either RingNF.rewrite recursively, or nothing depending on cfg.recursive.

def Mathlib.Tactic.RingNF.ringNFTarget (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RingNF.Config) : Lean.Elab.Tactic.TacticM Unit

Use ring_nf to rewrite the main goal.

Equations

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

def Mathlib.Tactic.RingNF.ringNFLocalDecl (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RingNF.Config) (fvarId : Lean.FVarId) : Lean.Elab.Tactic.TacticM Unit

Use ring_nf to rewrite hypothesis h.

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 !)
  • 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 !)
  • 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 !)
  • 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 !)
  • 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)