The abel tactic

Evaluate expressions in the language of additive, commutative monoids and groups.

def Mathlib.Tactic.Abel.abel : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

structure Mathlib.Tactic.Abel.Context : Type

The Context for a call to abel.

Stores a few options for this call, and caches some common subexpressions such as typeclass instances and 0 : α.

• α : Lean.Expr

  The type of the ambient additive commutative group or monoid.

• univ : Lean.Level

  The universe level for α.

• α0 : Lean.Expr

  The expression representing 0 : α.

• isGroup : Bool

  Specify whether we are in an additive commutative group or an additive commutative monoid.

• inst : Lean.Expr

  The AddCommGroup α or AddCommMonoid α expression.

def Mathlib.Tactic.Abel.mkContext (e : Lean.Expr) : Lean.MetaM Context

Populate a context object for evaluating e.

Equations

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

@[reducible, inline]

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

The monad for Abel contains, in addition to the AtomM state, some information about the current type we are working over, so that we can consistently use group lemmas or monoid lemmas as appropriate.

Equations

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

def Mathlib.Tactic.Abel.Context.app (c : Context) (n : Lean.Name) (inst : Lean.Expr) : Array Lean.Expr → Lean.Expr

Apply the function n : ∀ {α} [inst : AddWhatever α], _ to the implicit parameters in the context, and the given list of arguments.

Equations

• c.app n inst = Lean.mkAppN (((Lean.Expr.const n [c.univ]).app c.α).app inst)

def Mathlib.Tactic.Abel.Context.mkApp (c : Context) (n inst : Lean.Name) (l : Array Lean.Expr) : Lean.MetaM Lean.Expr

Apply the function n : ∀ {α} [inst α], _ to the implicit parameters in the context, and the given list of arguments.

Compared to context.app, this takes the name of the typeclass, rather than an inferred typeclass instance.

Equations

• c.mkApp n inst l = do let __do_lift ← Lean.Meta.synthInstance ((Lean.Expr.const inst [c.univ]).app c.α) pure (c.app n __do_lift l)

def Mathlib.Tactic.Abel.addG : Lean.Name → Lean.Name

Add the letter "g" to the end of the name, e.g. turning term into termg.

This is used to choose between declarations taking AddCommMonoid and those taking AddCommGroup instances.

Equations

• Mathlib.Tactic.Abel.addG (p.str s) = p.str (s ++ "g")
• Mathlib.Tactic.Abel.addG x✝ = x✝

def Mathlib.Tactic.Abel.iapp (n : Lean.Name) (xs : Array Lean.Expr) : M Lean.Expr

Apply the function n : ∀ {α} [AddComm{Monoid,Group} α] to the given list of arguments.

Will use the AddComm{Monoid,Group} instance that has been cached in the context.

Equations

• Mathlib.Tactic.Abel.iapp n xs = do let c ← read pure (c.app (if c.isGroup = true then Mathlib.Tactic.Abel.addG n else n) c.inst xs)

def Mathlib.Tactic.Abel.term {α : Type u_1} [AddCommMonoid α] (n : ℕ) (x a : α) : α

A type synonym used by abel to represent n • x + a in an additive commutative monoid.

Equations

• Mathlib.Tactic.Abel.term n x a = n • x + a

def Mathlib.Tactic.Abel.termg {α : Type u_1} [AddCommGroup α] (n : ℤ) (x a : α) : α

A type synonym used by abel to represent n • x + a in an additive commutative group.

Equations

• Mathlib.Tactic.Abel.termg n x a = n • x + a

def Mathlib.Tactic.Abel.mkTerm (n x a : Lean.Expr) : M Lean.Expr

Evaluate a term with coefficient n, atom x and successor terms a.

Equations

• Mathlib.Tactic.Abel.mkTerm n x a = Mathlib.Tactic.Abel.iapp `Mathlib.Tactic.Abel.term #[n, x, a]

def Mathlib.Tactic.Abel.intToExpr (n : ℤ) : M Lean.Expr

Interpret an integer as a coefficient to a term.

Equations

• Mathlib.Tactic.Abel.intToExpr n = do let __do_lift ← read liftM ((Lean.mkConst (if __do_lift.isGroup = true then `Int else `Nat)).ofInt n)

inductive Mathlib.Tactic.Abel.NormalExpr : Type

A normal form for abel. Expressions are represented as a list of terms of the form e = n • x, where n : ℤ and x is an arbitrary element of the additive commutative monoid or group. We explicitly track the Expr forms of e and n, even though they could be reconstructed, for efficiency.

• zero (e : Lean.Expr) : NormalExpr
• nterm (e : Lean.Expr) (n : Lean.Expr × ℤ) (x : ℕ × Lean.Expr) (a : NormalExpr) : NormalExpr

instance Mathlib.Tactic.Abel.instInhabitedNormalExpr : Inhabited NormalExpr

Equations

• Mathlib.Tactic.Abel.instInhabitedNormalExpr = { default := Mathlib.Tactic.Abel.NormalExpr.zero default }

def Mathlib.Tactic.Abel.NormalExpr.e : NormalExpr → Lean.Expr

Extract the expression from a normal form.

Equations

• (Mathlib.Tactic.Abel.NormalExpr.zero e).e = e
• (Mathlib.Tactic.Abel.NormalExpr.nterm e n x_1 a).e = e

instance Mathlib.Tactic.Abel.instCoeNormalExprExpr : Coe NormalExpr Lean.Expr

Equations

• Mathlib.Tactic.Abel.instCoeNormalExprExpr = { coe := Mathlib.Tactic.Abel.NormalExpr.e }

def Mathlib.Tactic.Abel.NormalExpr.term' (n : Lean.Expr × ℤ) (x : ℕ × Lean.Expr) (a : NormalExpr) : M NormalExpr

Construct the normal form representing a single term.

Equations

• Mathlib.Tactic.Abel.NormalExpr.term' n x a = do let __do_lift ← Mathlib.Tactic.Abel.mkTerm n.1 x.2 a.e pure (Mathlib.Tactic.Abel.NormalExpr.nterm __do_lift n x a)

def Mathlib.Tactic.Abel.NormalExpr.zero' : M NormalExpr

Construct the normal form representing zero.

Equations

• Mathlib.Tactic.Abel.NormalExpr.zero' = do let __do_lift ← read pure (Mathlib.Tactic.Abel.NormalExpr.zero __do_lift.α0)

theorem Mathlib.Tactic.Abel.const_add_term {α : Type u_1} [AddCommMonoid α] (k : α) (n : ℕ) (x a a' : α) (h : k + a = a') : k + term n x a = term n x a'

theorem Mathlib.Tactic.Abel.const_add_termg {α : Type u_1} [AddCommGroup α] (k : α) (n : ℤ) (x a a' : α) (h : k + a = a') : k + termg n x a = termg n x a'

theorem Mathlib.Tactic.Abel.term_add_const {α : Type u_1} [AddCommMonoid α] (n : ℕ) (x a k a' : α) (h : a + k = a') : term n x a + k = term n x a'

theorem Mathlib.Tactic.Abel.term_add_constg {α : Type u_1} [AddCommGroup α] (n : ℤ) (x a k a' : α) (h : a + k = a') : termg n x a + k = termg n x a'

theorem Mathlib.Tactic.Abel.term_add_term {α : Type u_1} [AddCommMonoid α] (n₁ : ℕ) (x a₁ : α) (n₂ : ℕ) (a₂ : α) (n' : ℕ) (a' : α) (h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') : term n₁ x a₁ + term n₂ x a₂ = term n' x a'

theorem Mathlib.Tactic.Abel.term_add_termg {α : Type u_1} [AddCommGroup α] (n₁ : ℤ) (x a₁ : α) (n₂ : ℤ) (a₂ : α) (n' : ℤ) (a' : α) (h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') : termg n₁ x a₁ + termg n₂ x a₂ = termg n' x a'

theorem Mathlib.Tactic.Abel.zero_term {α : Type u_1} [AddCommMonoid α] (x a : α) : term 0 x a = a

theorem Mathlib.Tactic.Abel.zero_termg {α : Type u_1} [AddCommGroup α] (x a : α) : termg 0 x a = a

partial def Mathlib.Tactic.Abel.evalAdd : NormalExpr → NormalExpr → M (NormalExpr × Lean.Expr)

Interpret the sum of two expressions in abel's normal form.

theorem Mathlib.Tactic.Abel.term_neg {α : Type u_1} [AddCommGroup α] (n : ℤ) (x a : α) (n' : ℤ) (a' : α) (h₁ : -n = n') (h₂ : -a = a') : -termg n x a = termg n' x a'

def Mathlib.Tactic.Abel.evalNeg : NormalExpr → M (NormalExpr × Lean.Expr)

Interpret a negated expression in abel's normal form.

Equations

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

def Mathlib.Tactic.Abel.smul {α : Type u_1} [AddCommMonoid α] (n : ℕ) (x : α) : α

A synonym for •, used internally in abel.

Equations

• Mathlib.Tactic.Abel.smul n x = n • x

def Mathlib.Tactic.Abel.smulg {α : Type u_1} [AddCommGroup α] (n : ℤ) (x : α) : α

A synonym for •, used internally in abel.

Equations

• Mathlib.Tactic.Abel.smulg n x = n • x

theorem Mathlib.Tactic.Abel.zero_smul {α : Type u_1} [AddCommMonoid α] (c : ℕ) : smul c 0 = 0

theorem Mathlib.Tactic.Abel.zero_smulg {α : Type u_1} [AddCommGroup α] (c : ℤ) : smulg c 0 = 0

theorem Mathlib.Tactic.Abel.term_smul {α : Type u_1} [AddCommMonoid α] (c n : ℕ) (x a : α) (n' : ℕ) (a' : α) (h₁ : c * n = n') (h₂ : smul c a = a') : smul c (term n x a) = term n' x a'

theorem Mathlib.Tactic.Abel.term_smulg {α : Type u_1} [AddCommGroup α] (c n : ℤ) (x a : α) (n' : ℤ) (a' : α) (h₁ : c * n = n') (h₂ : smulg c a = a') : smulg c (termg n x a) = termg n' x a'

def Mathlib.Tactic.Abel.evalSMul (k : Lean.Expr × ℤ) : NormalExpr → M (NormalExpr × Lean.Expr)

Auxiliary function for evalSMul'.

Equations

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

theorem Mathlib.Tactic.Abel.term_atom {α : Type u_1} [AddCommMonoid α] (x : α) : x = term 1 x 0

theorem Mathlib.Tactic.Abel.term_atomg {α : Type u_1} [AddCommGroup α] (x : α) : x = termg 1 x 0

theorem Mathlib.Tactic.Abel.term_atom_pf {α : Type u_1} [AddCommMonoid α] (x x' : α) (h : x = x') : x = term 1 x' 0

theorem Mathlib.Tactic.Abel.term_atom_pfg {α : Type u_1} [AddCommGroup α] (x x' : α) (h : x = x') : x = termg 1 x' 0

def Mathlib.Tactic.Abel.evalAtom (e : Lean.Expr) : M (NormalExpr × Lean.Expr)

Interpret an expression as an atom for abel's normal form.

Equations

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

theorem Mathlib.Tactic.Abel.unfold_sub {α : Type u_1} [SubtractionMonoid α] (a b c : α) (h : a + -b = c) : a - b = c

theorem Mathlib.Tactic.Abel.unfold_smul {α : Type u_1} [AddCommMonoid α] (n : ℕ) (x y : α) (h : smul n x = y) : n • x = y

theorem Mathlib.Tactic.Abel.unfold_smulg {α : Type u_1} [AddCommGroup α] (n : ℕ) (x y : α) (h : smulg (Int.ofNat n) x = y) : ↑n • x = y

theorem Mathlib.Tactic.Abel.unfold_zsmul {α : Type u_1} [AddCommGroup α] (n : ℤ) (x y : α) (h : smulg n x = y) : n • x = y

theorem Mathlib.Tactic.Abel.subst_into_smul {α : Type u_1} [AddCommMonoid α] (l : ℕ) (r : α) (tl : ℕ) (tr t : α) (prl : l = tl) (prr : r = tr) (prt : smul tl tr = t) : smul l r = t

theorem Mathlib.Tactic.Abel.subst_into_smulg {α : Type u_1} [AddCommGroup α] (l : ℤ) (r : α) (tl : ℤ) (tr t : α) (prl : l = tl) (prr : r = tr) (prt : smulg tl tr = t) : smulg l r = t

theorem Mathlib.Tactic.Abel.subst_into_smul_upcast {α : Type u_1} [AddCommGroup α] (l : ℕ) (r : α) (tl : ℕ) (zl : ℤ) (tr t : α) (prl₁ : l = tl) (prl₂ : ↑tl = zl) (prr : r = tr) (prt : smulg zl tr = t) : smul l r = t

theorem Mathlib.Tactic.Abel.subst_into_add {α : Type u_1} [AddCommMonoid α] (l r tl tr t : α) (prl : l = tl) (prr : r = tr) (prt : tl + tr = t) : l + r = t

theorem Mathlib.Tactic.Abel.subst_into_addg {α : Type u_1} [AddCommGroup α] (l r tl tr t : α) (prl : l = tl) (prr : r = tr) (prt : tl + tr = t) : l + r = t

theorem Mathlib.Tactic.Abel.subst_into_negg {α : Type u_1} [AddCommGroup α] (a ta t : α) (pra : a = ta) (prt : -ta = t) : -a = t

def Mathlib.Tactic.Abel.evalSMul' (eval : Lean.Expr → M (NormalExpr × Lean.Expr)) (is_smulg : Bool) (orig e₁ e₂ : Lean.Expr) : M (NormalExpr × Lean.Expr)

Normalize a term orig of the form smul e₁ e₂ or smulg e₁ e₂. Normalized terms use smul for monoids and smulg for groups, so there are actually four cases to handle:

• Using smul in a monoid just simplifies the pieces using subst_into_smul
• Using smulg in a group just simplifies the pieces using subst_into_smulg
• Using smul a b in a group requires converting a from a nat to an int and then simplifying smulg ↑a b using subst_into_smul_upcast
• Using smulg in a monoid is impossible (or at least out of scope), because you need a group argument to write a smulg term

Equations

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

partial def Mathlib.Tactic.Abel.eval (e : Lean.Expr) : M (NormalExpr × Lean.Expr)

Evaluate an expression into its abel normal form, by recursing into subexpressions.

def Mathlib.Tactic.Abel.abel1 : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

def Mathlib.Tactic.Abel.abel1! : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

• Mathlib.Tactic.Abel.abel1! = Lean.ParserDescr.node `Mathlib.Tactic.Abel.abel1! 1024 (Lean.ParserDescr.nonReservedSymbol "abel1!" false)

theorem Mathlib.Tactic.Abel.term_eq {α : Type u_1} [AddCommMonoid α] (n : ℕ) (x a : α) : term n x a = n • x + a

theorem Mathlib.Tactic.Abel.termg_eq {α : Type u_1} [AddCommGroup α] (n : ℤ) (x a : α) : termg n x a = n • x + a

A type synonym used by abel to represent n • x + a in an additive commutative group.

def Mathlib.Tactic.Abel.NormalExpr.isAtom : NormalExpr → Bool

True if this represents an atomic expression.

Equations

• (Mathlib.Tactic.Abel.NormalExpr.nterm e (fst, 1) x_1 (Mathlib.Tactic.Abel.NormalExpr.zero e_1)).isAtom = true
• x✝.isAtom = false

inductive Mathlib.Tactic.Abel.AbelMode : Type

The normalization style for abel_nf.

• term : AbelMode

  The default form

• raw : AbelMode

  Raw form: the representation abel uses internally.

structure Mathlib.Tactic.Abel.AbelNF.Config : Type

Configuration for abel_nf.

• red : Lean.Meta.TransparencyMode

  the reducibility setting to use when comparing atoms for defeq

• zetaDelta : Bool

  if true, local let variables can be unfolded

• recursive : Bool

  if true, atoms inside ring expressions will be reduced recursively

• mode : AbelMode

  The normalization style.

def Mathlib.Tactic.Abel.elabAbelNFConfig : Lean.Syntax → Lean.Elab.Tactic.TacticM AbelNF.Config

Function elaborating AbelNF.Config.

Equations

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

def Mathlib.Tactic.Abel.abelNFCore (s : IO.Ref AtomM.State) (cfg : AbelNF.Config) (e : Lean.Expr) : Lean.MetaM Lean.Meta.Simp.Result

The core of abel_nf, which rewrites the expression e into abel normal form.

• s: a reference to the mutable state of abel, for persisting across calls. This ensures that atom ordering is used consistently.
• cfg: the configuration options
• e: the expression to rewrite

Equations

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

partial def Mathlib.Tactic.Abel.abelNFCore.go (s : IO.Ref AtomM.State) (cfg : AbelNF.Config) (simp : Lean.Meta.Simp.Result → Lean.MetaM Lean.Meta.Simp.Result) (ctx : Lean.Meta.Simp.Context) (root : Bool) (parent : Lean.Expr) : Lean.MetaM Lean.Meta.Simp.Result

The recursive case of abelNF.

• root: true when the function is called directly from abelNFCore and false when called by evalAtom in recursive mode.
• parent: The input expression to simplify. In pre we make use of both parent and e to determine if we are at the top level in order to prevent a loop go -> eval -> evalAtom -> go which makes no progress.

partial def Mathlib.Tactic.Abel.abelNFCore.evalAtom (s : IO.Ref AtomM.State) (cfg : AbelNF.Config) (simp : Lean.Meta.Simp.Result → Lean.MetaM Lean.Meta.Simp.Result) (ctx : Lean.Meta.Simp.Context) : Lean.Expr → Lean.MetaM Lean.Meta.Simp.Result

The evalAtom implementation passed to eval calls go if cfg.recursive is true, and does nothing otherwise.

def Mathlib.Tactic.Abel.abelNFTarget (s : IO.Ref AtomM.State) (cfg : AbelNF.Config) : Lean.Elab.Tactic.TacticM Unit

Use abel_nf to rewrite the main goal.

Equations

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

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

Use abel_nf to rewrite hypothesis h.

Equations

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

def Mathlib.Tactic.Abel.abelNF : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

def Mathlib.Tactic.Abel.tacticAbel_nf!__ : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

def Mathlib.Tactic.Abel.abelNFConv : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

def Mathlib.Tactic.Abel.elabAbelNFConv : Lean.Elab.Tactic.Tactic

Elaborator for the abel_nf tactic.

Equations

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

def Mathlib.Tactic.Abel.convAbel_nf!_ : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

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

def Mathlib.Tactic.Abel.tacticAbel! : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

• Mathlib.Tactic.Abel.tacticAbel! = Lean.ParserDescr.node `Mathlib.Tactic.Abel.tacticAbel! 1024 (Lean.ParserDescr.nonReservedSymbol "abel!" false)

def Mathlib.Tactic.Abel.abelConv : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

• Mathlib.Tactic.Abel.abelConv = Lean.ParserDescr.node `Mathlib.Tactic.Abel.abelConv 1024 (Lean.ParserDescr.nonReservedSymbol "abel" false)

def Mathlib.Tactic.Abel.convAbel! : Lean.ParserDescr

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

• abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
• abel_nf rewrites all group expressions into a normal form.
  • In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
  • abel_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, abel_nf will also recurse into atoms
• abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work

• In mathlib 3, abel accepted additional optional arguments:

  syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
  

  It is undecided whether these features should be restored eventually.

Equations

• Mathlib.Tactic.Abel.convAbel! = Lean.ParserDescr.node `Mathlib.Tactic.Abel.convAbel! 1024 (Lean.ParserDescr.nonReservedSymbol "abel!" false)

We register abel with the hint tactic.