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.

When Logic.Equiv.Basic and Order.Hom.Basic have been ported some additional testing is required.

ink

def Mathlib.Tactic.applyFunHyp (f : Lean.TSyntax `term) (using? : Option Lean.Expr) (h : Lean.FVarId) (g : Lean.MVarId) : Lean.Elab.Tactic.TacticM (List Lean.MVarId)

Apply a function to a hypothesis.

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ink

def Mathlib.Tactic.applyFunTargetFailure (f : Lean.Expr) : Lean.MetaM (List Lean.MVarId)

Failure message for applyFunTarget.

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def Mathlib.Tactic.applyFunTarget (f : Lean.Expr) (using? : Option Lean.Expr) (g : Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

Apply a function to the main goal.

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def Mathlib.Tactic.applyFun : Lean.ParserDescr

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 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

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