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protectedtheoremseq_eq {α β ε : Type u} [Monad m] (mf : ExceptT ε m (α → β)) (x : ExceptT ε m α) : mf <*> x = mf >>= fun f => f <$> x :=
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rfl
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-
protectedtheorembind_pure_comp [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α) : x >>= pure ∘ f = f <$> x := by
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protectedtheorembind_pure_comp [Monad m] (f : α → β) (x : ExceptT ε m α) : x >>= pure ∘ f = f <$> x := by
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intros; rfl
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protectedtheoremseqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x <* y = const β <$> x <*> y := by
@@ -200,11 +200,11 @@ theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
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show (f >>= fun g => g <$> x).run s = _
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simp
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-
@[simp]theoremrun_seqRight [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
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@[simp]theoremrun_seqRight [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
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show (x >>= fun _ => y).run s = _
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simp
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-
@[simp]theoremrun_seqLeft {α β σ : Type u} [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
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@[simp]theoremrun_seqLeft {α β σ : Type u} [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
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show (x >>= fun a => y >>= fun _ => pure a).run s = _
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