Tactic fun_prop for proving function properties like Continuous f, Differentiable ℝ f, ...

def Mathlib.Meta.FunProp.synthesizeInstance (thmId : Mathlib.Meta.FunProp.Origin) (x type : Lean.Expr) : Lean.MetaM Bool

Synthesize instance of type type and

1. assign it to x if x is meta variable
2. check it is equal to x

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def Mathlib.Meta.FunProp.synthesizeArgs (thmId : Mathlib.Meta.FunProp.Origin) (xs : Array Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM Bool

Synthesize arguments xs either with typeclass synthesis, with fun_prop or with discharger.

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def Mathlib.Meta.FunProp.tryTheoremCore (xs : Array Lean.Expr) (val type e : Lean.Expr) (thmId : Mathlib.Meta.FunProp.Origin) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to apply theorem - core function

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def Mathlib.Meta.FunProp.tryTheoremWithHint? (e : Lean.Expr) (thmOrigin : Mathlib.Meta.FunProp.Origin) (hint : Array (ℕ × Lean.Expr)) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) (newMCtxDepth : Bool := false) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to apply a theorem provided some of the theorem arguments.

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def Mathlib.Meta.FunProp.tryTheorem? (e : Lean.Expr) (thmOrigin : Mathlib.Meta.FunProp.Origin) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) (newMCtxDepth : Bool := false) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to apply a theorem thmOrigin to the goal e.

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• Mathlib.Meta.FunProp.tryTheorem? e thmOrigin funProp newMCtxDepth = Mathlib.Meta.FunProp.tryTheoremWithHint? e thmOrigin #[] funProp newMCtxDepth

def Mathlib.Meta.FunProp.applyIdRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e using using identity lambda theorem.

For example, e = q(Continuous fun x => x) and funPropDecl is FunPropDecl for Continuous.

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def Mathlib.Meta.FunProp.applyConstRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e using using constant lambda theorem.

For example, e = q(Continuous fun x => y) and funPropDecl is FunPropDecl for Continuous.

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def Mathlib.Meta.FunProp.applyApplyRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e using using apply lambda theorem.

For example, e = q(Continuous fun f => f x) and funPropDecl is FunPropDecl for Continuous.

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def Mathlib.Meta.FunProp.applyCompRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e f g : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e using composition lambda theorem.

For example, e = q(Continuous fun x => f (g x)) and funPropDecl is FunPropDecl for Continuous

You also have to provide the functions f and g.

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def Mathlib.Meta.FunProp.applyPiRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e using pi lambda theorem.

For example, e = q(Continuous fun x y => f x y) and funPropDecl is FunPropDecl for Continuous

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def Mathlib.Meta.FunProp.letCase (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e f : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to prove e = q(P (fun x => let y := φ x; ψ x y).

For example,

• funPropDecl is FunPropDecl for Continuous
• e = q(Continuous fun x => let y := φ x; ψ x y)
• f = q(fun x => let y := φ x; ψ x y)

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• Mathlib.Meta.FunProp.letCase funPropDecl e f funProp = Lean.throwError (Lean.toMessageData "expected expression of the form `fun x => lam y := ..; ..`")

def Mathlib.Meta.FunProp.applyMorRules (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Prove function property of using morphism theorems.

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def Mathlib.Meta.FunProp.applyTransitionRules (e : Lean.Expr) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Prove function property of using transition theorems.

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def Mathlib.Meta.FunProp.removeArgRule (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to remove applied argument i.e. prove P (fun x => f x y) from P (fun x => f x).

For example

• funPropDecl is FunPropDecl for Continuous
• e = q(Continuous fun x => foo (bar x) y)
• fData contains info on fun x => foo (bar x) y This tries to prove Continuous fun x => foo (bar x) y from Continuous fun x => foo (bar x)

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def Mathlib.Meta.FunProp.bvarAppCase (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Prove function property of fun f => f x₁ ... xₙ.

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def Mathlib.Meta.FunProp.getDeclTheorems (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (funName : Lean.Name) (mainArgs : Array ℕ) (appliedArgs : ℕ) : Lean.MetaM (Array Mathlib.Meta.FunProp.FunctionTheorem)

Get candidate theorems from the environment for function property funPropDecl and function funName.

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def Mathlib.Meta.FunProp.getLocalTheorems (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (funOrigin : Mathlib.Meta.FunProp.Origin) (mainArgs : Array ℕ) (appliedArgs : ℕ) : Mathlib.Meta.FunProp.FunPropM (Array Mathlib.Meta.FunProp.FunctionTheorem)

Get candidate theorems from the local context for function property funPropDecl and function funName.

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def Mathlib.Meta.FunProp.tryTheorems (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (thms : Array Mathlib.Meta.FunProp.FunctionTheorem) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Try to apply function theorems thms to e.

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def Mathlib.Meta.FunProp.fvarAppCase (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Prove function property of fun x => f x₁ ... xₙ where f is free variable.

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def Mathlib.Meta.FunProp.constAppCase (funPropDecl : Mathlib.Meta.FunProp.FunPropDecl) (e : Lean.Expr) (fData : Mathlib.Meta.FunProp.FunctionData) (funProp : Lean.Expr → Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Prove function property of fun x => f x₁ ... xₙ where f is declared function.

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def Mathlib.Meta.FunProp.cacheResult (e : Lean.Expr) (r : Mathlib.Meta.FunProp.Result) : Mathlib.Meta.FunProp.FunPropM Mathlib.Meta.FunProp.Result

Cache result if it does not have any subgoals.

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def Mathlib.Meta.FunProp.cacheFailure (e : Lean.Expr) : Mathlib.Meta.FunProp.FunPropM Unit

Cache for failed goals such that fun_prop can fail fast next time.

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partial def Mathlib.Meta.FunProp.funProp (e : Lean.Expr) : Mathlib.Meta.FunProp.FunPropM (Option Mathlib.Meta.FunProp.Result)

Main funProp function. Returns proof of e.