Documentation

Mathlib.Control.Lawful

Functor Laws, applicative laws, and monad Laws #

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def StateT.mk {σ : Type u} {m : Type u → Type v} {α : Type u} (f : σ → m (α × σ)) :
StateT σ m α
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    @[simp]
    theorem StateT.run_mk {σ : Type u} {m : Type u → Type v} {α : Type u} (f : σ → m (α × σ)) (st : σ) :
    (StateT.mk f).run st = f st
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    @[simp]
    theorem ExceptT.run_monadLift {α ε : Type u} {m : Type u → Type v} {n : Type u → Type u_1} [Monad m] [MonadLiftT n m] (x : n α) :
    (monadLift x).run = Except.ok <$> monadLift x
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    @[simp]
    theorem ExceptT.run_monadMap {α ε : Type u} {m : Type u → Type v} (x : ExceptT ε m α) {n : Type u → Type u_1} [MonadFunctorT n m] (f : {α : Type u} → n α → n α) :
    (monadMap f x).run = monadMap f x.run
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    def ReaderT.mk {m : Type u → Type v} {α σ : Type u} (f : σ → m α) :
    ReaderT σ m α
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    Instances For
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      @[simp]
      theorem ReaderT.run_mk {m : Type u → Type v} {α σ : Type u} (f : σ → m α) (r : σ) :
      (ReaderT.mk f).run r = f r
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      theorem OptionT.ext {α : Type u} {m : Type u → Type v} {x x' : OptionT m α} (h : x.run = x'.run) :
      x = x'
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      @[simp]
      theorem OptionT.run_mk {α : Type u} {m : Type u → Type v} (x : m (Option α)) :
      (OptionT.mk x).run = x
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      @[simp]
      theorem OptionT.run_pure {α : Type u} {m : Type u → Type v} [Monad m] (a : α) :
      (pure a).run = pure (some a)
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      @[simp]
      theorem OptionT.run_bind {α β : Type u} {m : Type u → Type v} (x : OptionT m α) [Monad m] (f : α → OptionT m β) :
      (x >>= f).run = do let x ← x.run match x with | some a => (f a).run | none => pure none
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      @[simp]
      theorem OptionT.run_map {α β : Type u} {m : Type u → Type v} (x : OptionT m α) [Monad m] (f : α → β) [LawfulMonad m] :
      (f <$> x).run = Option.map f <$> x.run
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      @[simp]
      theorem OptionT.run_monadLift {α : Type u} {m : Type u → Type v} [Monad m] {n : Type u → Type u_1} [MonadLiftT n m] (x : n α) :
      (monadLift x).run = do let a ← monadLift x pure (some a)
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      @[simp]
      theorem OptionT.run_monadMap {α : Type u} {m : Type u → Type v} (x : OptionT m α) {n : Type u → Type u_1} [MonadFunctorT n m] (f : {α : Type u} → n α → n α) :
      (monadMap f x).run = monadMap f x.run
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      instance instLawfulMonadOptionT_mathlib (m : Type u → Type v) [Monad m] [LawfulMonad m] :
      LawfulMonad (OptionT m)
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      • ⋯ = ⋯