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[Merged by Bors] - chore(data/sum): Add trivial simp lemmas #5112

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[Merged by Bors] - chore(data/sum): Add trivial simp lemmas #5112

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chore(data/sum): Add trivial simp lemmas
eric-wieser Nov 25, 2020
49d7b22
Update sum.lean
eric-wieser Nov 25, 2020
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18 changes: 18 additions & 0 deletions src/data/sum.lean
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Expand Up @@ -93,6 +93,24 @@ protected def elim {α β γ : Sort*} (f : α → γ) (g : β → γ) : α ⊕
@[simp] lemma elim_inr {α β γ : Sort*} (f : α → γ) (g : β → γ) (x : β) :
sum.elim f g (inr x) = g x := rfl

@[simp] lemma elim_comp_inl {α β γ : Sort*} (f : α → γ) (g : β → γ) :
sum.elim f g ∘ inl = f := rfl

@[simp] lemma elim_comp_inr {α β γ : Sort*} (f : α → γ) (g : β → γ) :
sum.elim f g ∘ inr = g := rfl

@[simp] lemma elim_inl_inr {α β : Sort*} :
@sum.elim α β _ inl inr = id :=
funext $ λ x, sum.cases_on x (λ _, rfl) (λ _, rfl)

lemma comp_elim {α β γ δ : Sort*} (f : γ → δ) (g : α → γ) (h : β → γ):
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This would also be fine as a simp lemma, but I'm not sure if we actually want this to always simplify to the right-hand side.

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Yeah, I left this one out because the decision of which way to simplify felt subjective.

f ∘ sum.elim g h = sum.elim (f ∘ g) (f ∘ h) :=
funext $ λ x, sum.cases_on x (λ _, rfl) (λ _, rfl)

@[simp] lemma elim_comp_inl_inr {α β γ : Sort*} (f : α ⊕ β → γ) :
sum.elim (f ∘ inl) (f ∘ inr) = f :=
funext $ λ x, sum.cases_on x (λ _, rfl) (λ _, rfl)

lemma elim_injective {α β γ : Sort*} {f : α → γ} {g : β → γ}
(hf : function.injective f) (hg : function.injective g)
(hfg : ∀ a b, f a ≠ g b) : function.injective (sum.elim f g) :=
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