26 lines
879 B
Text
26 lines
879 B
Text
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import data.list data.nat
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open nat list eq.ops
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theorem nat.le_of_eq {x y : ℕ} (H : x = y) : x ≤ y := H ▸ !le.refl
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section
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variable {Q : Type}
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definition f : list Q → ℕ -- default if l is empty, else max l
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| [] := 0
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| (h :: t) := f t + 1
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theorem f_foo : ∀{l : list Q}, ∀{q : Q}, q ∈ l → f l ≥ 1
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| [] := take q, assume Hq, absurd Hq !not_mem_nil
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| [h] := take q, assume Hq, nat.le_of_eq !rfl
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| (h :: (h' :: t)) := take q, assume Hq,
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have Hor : q = h ∨ q ∈ (h' :: t), from iff.mp !mem_cons_iff Hq,
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have H : f (h' :: t) ≥ 1, from f_foo (mem_cons h' t),
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have H1 : 1 + 1 ≤ f (h' :: t) + 1, from nat.add_le_add_right H 1,
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calc
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f (h :: h' :: t) = f (h' :: t) + 1 : rfl
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... ≥ 1 + 1 : H1
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... = 1 : sorry
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end
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