# Documentation

Mathlib.Tactic.ApplyFun

# The apply_fun tactic. #

Apply a function to an equality or inequality in either a local hypothesis or the goal.

## Porting notes #

When the mono tactic has been ported we can attempt to automatically discharge Monotone f goals.

def Mathlib.Tactic.applyFunHyp (f : Lean.Term) (using? : ) (h : Lean.FVarId) (g : Lean.MVarId) :

Apply a function to a hypothesis.

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Failure message for applyFunTarget.

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Given a metavariable ginj of type Injective f, try to prove it. Returns whether it was successful.

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theorem Mathlib.Tactic.ApplyFun.le_of_le {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x : α} {y : α} :
e x e yx y

Alias of the forward direction of OrderIso.le_iff_le.

theorem Mathlib.Tactic.ApplyFun.lt_of_lt {α : Type u_2} {β : Type u_3} [] [] (e : α ≃o β) {x : α} {y : α} :
e x < e yx < y

Alias of the forward direction of OrderIso.lt_iff_lt.

Apply a function to the main goal.

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Apply a function to an equality or inequality in either a local hypothesis or the goal.

• If we have h : a = b, then apply_fun f at h will replace this with h : f a = f b.
• If we have h : a ≤ b, then apply_fun f at h will replace this with h : f a ≤ f b, and create a subsidiary goal Monotone f. apply_fun will automatically attempt to discharge this subsidiary goal using mono, or an explicit solution can be provided with apply_fun f at h using P, where P : Monotone f.
• If we have h : a < b, then apply_fun f at h will replace this with h : f a < f b, and create a subsidiary goal StrictMono f and behaves as in the previous case.
• If we have h : a ≠ b, then apply_fun f at h will replace this with h : f a ≠ f b, and create a subsidiary goal Injective f and behaves as in the previous two cases.
• If the goal is a ≠ b, apply_fun f will replace this with f a ≠ f b.
• If the goal is a = b, apply_fun f will replace this with f a = f b, and create a subsidiary goal injective f. apply_fun will automatically attempt to discharge this subsidiary goal using local hypotheses, or if f is actually an Equiv, or an explicit solution can be provided with apply_fun f using P, where P : Injective f.
• If the goal is a ≤ b (or similarly for a < b), and f is actually an OrderIso, apply_fun f will replace the goal with f a ≤ f b. If f is anything else (e.g. just a function, or an Equiv), apply_fun will fail.

Typical usage is:

open Function

example (X Y Z : Type) (f : X → Y) (g : Y → Z) (H : Injective <| g ∘ f) :
Injective f := by
intros x x' h
apply_fun g at h
exact H h


The function f is handled similarly to how it would be handled by refine in that f can contain placeholders. Named placeholders (like ?a or ?_) will produce new goals.

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