16 lines
792 B
Text
16 lines
792 B
Text
example (x₁ x₂ y₁ y₂ : Nat) : (x₁ + x₂) + (y₁ + y₂) = (x₁ + y₁) + (x₂ + y₂) := by
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calc (x₁ + x₂) + (y₁ + y₂)
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= x₁ + (x₂ + (y₁ + y₂)) := by rw [Nat.add_assoc]
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_ = x₁ + (y₁ + (x₂ + y₂)) := by rw [Nat.add_left_comm x₂ y₁ y₂]
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_ = (x₁ + y₁) + (x₂ + y₂) := by rw [Nat.add_assoc]
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example (n x₁ x₂ y₁ y₂ : Nat) : n = 0 → (x₁ + x₂) + (y₁ + y₂) = (x₁ + y₁) + (x₂ + y₂) := by
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intro h
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induction n with
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| zero =>
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calc (x₁ + x₂) + (y₁ + y₂)
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= x₁ + (x₂ + (y₁ + y₂)) := by rw [Nat.add_assoc]
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_ = x₁ + (y₁ + (x₂ + y₂)) := by rw [Nat.add_left_comm x₂ y₁ y₂]
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_ = (x₁ + y₁) + (x₂ + y₂) := by rw [Nat.add_assoc]
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| succ _ _ => contradiction
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