Documentation

Init.Control.Lawful.Instances

ExceptT #

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theorem ExceptT.ext {ε : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_1} {x y : ExceptT ε m α} (h : x.run = y.run) :
x = y
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@[simp]
theorem ExceptT.run_pure {m : Type u_1 → Type u_2} {α ε : Type u_1} [Monad m] (x : α) :
(pure x).run = pure (Except.ok x)
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@[simp]
theorem ExceptT.run_lift {m : Type u → Type v} {α ε : Type u} [Monad m] (x : m α) :
(ExceptT.lift x).run = Except.ok <$> x
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@[simp]
theorem ExceptT.run_throw {m : Type u_1 → Type u_2} {ε β : Type u_1} {e : ε} [Monad m] :
(throw e).run = pure (Except.error e)
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@[simp]
theorem ExceptT.run_bind_lift {m : Type u_1 → Type u_2} {α ε β : Type u_1} [Monad m] [LawfulMonad m] (x : m α) (f : α → ExceptT ε m β) :
(ExceptT.lift x >>= f).run = do let a ← x (f a).run
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@[simp]
theorem ExceptT.bind_throw {m : Type u_1 → Type u_2} {α ε β : Type u_1} {e : ε} [Monad m] [LawfulMonad m] (f : α → ExceptT ε m β) :
throw e >>= f = throw e
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theorem ExceptT.run_bind {m : Type u_1 → Type u_2} {ε α β : Type u_1} {f : α → ExceptT ε m β} [Monad m] (x : ExceptT ε m α) :
(x >>= f).run = do let x ← x.run match x with | Except.ok x => (f x).run | Except.error e => pure (Except.error e)
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@[simp]
theorem ExceptT.lift_pure {m : Type u_1 → Type u_2} {α ε : Type u_1} [Monad m] [LawfulMonad m] (a : α) :
ExceptT.lift (pure a) = pure a
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@[simp]
theorem ExceptT.run_map {m : Type u_1 → Type u_2} {α β ε : Type u_1} [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α) :
(f <$> x).run = Except.map f <$> x.run
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theorem ExceptT.seq_eq {m : Type u → Type u_1} {α β ε : Type u} [Monad m] (mf : ExceptT ε m (α → β)) (x : ExceptT ε m α) :
mf <*> x = do let f ← mf f <$> x
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theorem ExceptT.bind_pure_comp {m : Type u_1 → Type u_2} {α β ε : Type u_1} [Monad m] (f : α → β) (x : ExceptT ε m α) :
x >>= pure ∘ f = f <$> x
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theorem ExceptT.seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) :
x <* y = Function.const β <$> x <*> y
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theorem ExceptT.seqRight_eq {m : Type u_1 → Type u_2} {ε α β : Type u_1} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) :
x *> y = Function.const α id <$> x <*> y
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theorem ExceptT.instLawfulMonad {m : Type u_1 → Type u_2} {ε : Type u_1} [Monad m] [LawfulMonad m] :
LawfulMonad (ExceptT ε m)
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@[simp]
theorem ExceptT.map_throw {m : Type u_1 → Type u_2} {ε : Type u_1} [Monad m] [LawfulMonad m] {α β : Type u_1} (f : α → β) (e : ε) :
f <$> throw e = throw e

Except #

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theorem instLawfulMonadExcept {ε : Type u_1} :
LawfulMonad (Except ε)
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theorem instLawfulApplicativeExcept {ε : Type u_1} :
LawfulApplicative (Except ε)
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theorem instLawfulFunctorExcept {ε : Type u_1} :
LawfulFunctor (Except ε)

ReaderT #

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theorem ReaderT.ext {ρ : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_1} {x y : ReaderT ρ m α} (h : ∀ (ctx : ρ), x.run ctx = y.run ctx) :
x = y
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@[simp]
theorem ReaderT.run_pure {m : Type u_1 → Type u_2} {α ρ : Type u_1} [Monad m] (a : α) (ctx : ρ) :
(pure a).run ctx = pure a
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@[simp]
theorem ReaderT.run_bind {m : Type u_1 → Type u_2} {ρ α β : Type u_1} [Monad m] (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) (ctx : ρ) :
(x >>= f).run ctx = do let a ← x.run ctx (f a).run ctx
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@[simp]
theorem ReaderT.run_mapConst {m : Type u_1 → Type u_2} {α ρ β : Type u_1} [Monad m] (a : α) (x : ReaderT ρ m β) (ctx : ρ) :
(Functor.mapConst a x).run ctx = Functor.mapConst a (x.run ctx)
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@[simp]
theorem ReaderT.run_map {m : Type u_1 → Type u_2} {α β ρ : Type u_1} [Monad m] (f : α → β) (x : ReaderT ρ m α) (ctx : ρ) :
(f <$> x).run ctx = f <$> x.run ctx
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@[simp]
theorem ReaderT.run_monadLift {n : Type u_1 → Type u_2} {m : Type u_1 → Type u_3} {α ρ : Type u_1} [MonadLiftT n m] (x : n α) (ctx : ρ) :
(monadLift x).run ctx = monadLift x
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@[simp]
theorem ReaderT.run_monadMap {n : Type u → Type u_1} {m : Type u → Type u_2} {ρ α : Type u} [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : ReaderT ρ m α) (ctx : ρ) :
(monadMap f x).run ctx = monadMap f (x.run ctx)
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@[simp]
theorem ReaderT.run_read {m : Type u_1 → Type u_2} {ρ : Type u_1} [Monad m] (ctx : ρ) :
ReaderT.read.run ctx = pure ctx
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@[simp]
theorem ReaderT.run_seq {m : Type u → Type u_1} {ρ α β : Type u} [Monad m] (f : ReaderT ρ m (α → β)) (x : ReaderT ρ m α) (ctx : ρ) :
(f <*> x).run ctx = f.run ctx <*> x.run ctx
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@[simp]
theorem ReaderT.run_seqRight {m : Type u_1 → Type u_2} {ρ α β : Type u_1} [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ) :
(x *> y).run ctx = x.run ctx *> y.run ctx
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@[simp]
theorem ReaderT.run_seqLeft {m : Type u_1 → Type u_2} {ρ α β : Type u_1} [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ) :
(x <* y).run ctx = x.run ctx <* y.run ctx
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theorem ReaderT.instLawfulFunctor {m : Type u_1 → Type u_2} {ρ : Type u_1} [Monad m] [LawfulFunctor m] :
LawfulFunctor (ReaderT ρ m)
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theorem ReaderT.instLawfulApplicative {m : Type u_1 → Type u_2} {ρ : Type u_1} [Monad m] [LawfulApplicative m] :
LawfulApplicative (ReaderT ρ m)
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theorem ReaderT.instLawfulMonad {m : Type u_1 → Type u_2} {ρ : Type u_1} [Monad m] [LawfulMonad m] :
LawfulMonad (ReaderT ρ m)

StateRefT #

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theorem instLawfulMonadStateRefT' {m : Type → Type} {ω σ : Type} [Monad m] [LawfulMonad m] :
LawfulMonad (StateRefT' ω σ m)

StateT #

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theorem StateT.ext {σ : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_1} {x y : StateT σ m α} (h : ∀ (s : σ), x.run s = y.run s) :
x = y
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@[simp]
theorem StateT.run'_eq {m : Type u_1 → Type u_2} {σ α : Type u_1} [Monad m] (x : StateT σ m α) (s : σ) :
x.run' s = (fun (x : α × σ) => x.fst) <$> x.run s
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@[simp]
theorem StateT.run_pure {m : Type u_1 → Type u_2} {α σ : Type u_1} [Monad m] (a : α) (s : σ) :
(pure a).run s = pure (a, s)
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@[simp]
theorem StateT.run_bind {m : Type u_1 → Type u_2} {σ α β : Type u_1} [Monad m] (x : StateT σ m α) (f : α → StateT σ m β) (s : σ) :
(x >>= f).run s = do let p ← x.run s (f p.fst).run p.snd
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@[simp]
theorem StateT.run_map {m : Type u → Type u_1} {α β σ : Type u} [Monad m] [LawfulMonad m] (f : α → β) (x : StateT σ m α) (s : σ) :
(f <$> x).run s = (fun (p : α × σ) => (f p.fst, p.snd)) <$> x.run s
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@[simp]
theorem StateT.run_get {m : Type u_1 → Type u_2} {σ : Type u_1} [Monad m] (s : σ) :
get.run s = pure (s, s)
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@[simp]
theorem StateT.run_set {m : Type u_1 → Type u_2} {σ : Type u_1} [Monad m] (s s' : σ) :
(set s').run s = pure (PUnit.unit, s')
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@[simp]
theorem StateT.run_modify {m : Type u_1 → Type u_2} {σ : Type u_1} [Monad m] (f : σ → σ) (s : σ) :
(modify f).run s = pure (PUnit.unit, f s)
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@[simp]
theorem StateT.run_modifyGet {m : Type u_1 → Type u_2} {σ α : Type u_1} [Monad m] (f : σ → α × σ) (s : σ) :
(modifyGet f).run s = pure ((f s).fst, (f s).snd)
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@[simp]
theorem StateT.run_lift {m : Type u → Type u_1} {α σ : Type u} [Monad m] (x : m α) (s : σ) :
(StateT.lift x).run s = do let a ← x pure (a, s)
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theorem StateT.run_bind_lift {m : Type u → Type u_1} {β α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f : α → StateT σ m β) (s : σ) :
(StateT.lift x >>= f).run s = do let a ← x (f a).run s
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@[simp]
theorem StateT.run_monadLift {m : Type u → Type u_1} {n : Type u → Type u_2} {α σ : Type u} [Monad m] [MonadLiftT n m] (x : n α) (s : σ) :
(monadLift x).run s = do let a ← monadLift x pure (a, s)
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@[simp]
theorem StateT.run_monadMap {n : Type u → Type u_1} {m : Type u → Type u_2} {σ α : Type u} [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : StateT σ m α) (s : σ) :
(monadMap f x).run s = monadMap f (x.run s)
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@[simp]
theorem StateT.run_seq {m : Type u → Type u_1} {α β σ : Type u} [Monad m] [LawfulMonad m] (f : StateT σ m (α → β)) (x : StateT σ m α) (s : σ) :
(f <*> x).run s = do let fs ← f.run s (fun (p : α × σ) => (fs.fst p.fst, p.snd)) <$> x.run fs.snd
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@[simp]
theorem StateT.run_seqRight {m : Type u_1 → Type u_2} {σ α β : Type u_1} [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) :
(x *> y).run s = do let p ← x.run s y.run p.snd
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@[simp]
theorem StateT.run_seqLeft {m : Type u → Type u_1} {α β σ : Type u} [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) :
(x <* y).run s = do let p ← x.run s let p' ← y.run p.snd pure (p.fst, p'.snd)
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theorem StateT.seqRight_eq {m : Type u_1 → Type u_2} {σ α β : Type u_1} [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) :
x *> y = Function.const α id <$> x <*> y
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theorem StateT.seqLeft_eq {m : Type u_1 → Type u_2} {σ α β : Type u_1} [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) :
x <* y = Function.const β <$> x <*> y
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theorem StateT.instLawfulMonad {m : Type u_1 → Type u_2} {σ : Type u_1} [Monad m] [LawfulMonad m] :
LawfulMonad (StateT σ m)

EStateM #

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theorem instLawfulMonadEStateM {ε σ : Type u_1} :
LawfulMonad (EStateM ε σ)