37 lines
1.1 KiB
Text
37 lines
1.1 KiB
Text
class CommAddSemigroup (α : Type u) extends Add α where
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addComm : {a b : α} → a + b = b + a
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addAssoc : {a b c : α} → a + b + c = a + (b + c)
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open CommAddSemigroup
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theorem addComm3 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
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have h : b + c = c + b := addComm;
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have h' := congrArg (a + .) h;
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simp at h';
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rw [←addAssoc] at h';
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rw [←addAssoc (a := a)] at h';
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exact h';
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}
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theorem addComm4 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
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rw [addAssoc, addAssoc];
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rw [addComm (a := b)];
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}
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theorem addComm5 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
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have h : b + c = c + b := addComm;
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have h' := congrArg (a + .) h;
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simp at h';
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rw [←addAssoc] at h';
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rw [←@addAssoc (a := a)] at h';
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exact h';
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}
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theorem addComm6 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
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have h : b + c = c + b := addComm;
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have h' := congrArg (a + .) h;
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simp at h';
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rw [←addAssoc] at h';
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rw [←addAssoc] at h';
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exact h';
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}
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