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28 lines
653 B
Plaintext
28 lines
653 B
Plaintext
2 years ago
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import NNG.MyNat.Multiplication
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-- this is one of *three* routes to
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-- canonically_ordered_comm_semiring
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namespace MyNat
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def le (a b : MyNat) := ∃ (c : MyNat), b = a + c
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-- Another choice is to define it recursively:
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-- | le 0 _
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-- | le (succ a) (succ b) = le ab
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-- notation
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instance : LE MyNat := ⟨MyNat.le⟩
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theorem le_def' : MyNat.le = (.≤.) := rfl
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theorem le_iff_exists_add (a b : MyNat) : a ≤ b ↔ ∃ (c : MyNat), b = a + c := Iff.rfl
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def lt_myNat (a b : MyNat) := a ≤ b ∧ ¬ (b ≤ a)
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instance : LT MyNat := ⟨lt_myNat⟩
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theorem lt : ∀ (a b : MyNat), a < b ↔ a ≤ b ∧ ¬b ≤ a := fun _ _ => Iff.rfl
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end MyNat
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