43 lines
1.1 KiB
Text
43 lines
1.1 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Small example showing that (nat → nat) is not encodable.
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-/
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import data.encodable
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open nat encodable option
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section
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hypothesis nat_nat_encodable : encodable (nat → nat)
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private definition decode_fun (n : nat) : option (nat → nat) :=
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@decode (nat → nat) nat_nat_encodable n
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private definition encode_fun (f : nat → nat) : nat :=
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@encode (nat → nat) nat_nat_encodable f
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private lemma encodek_fun : ∀ f : nat → nat, decode_fun (encode_fun f) = some f :=
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λ f, !encodek
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private definition f (n : nat) : nat :=
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match decode_fun n with
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| some g := succ (g n)
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| none := 0
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end
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private definition v : nat := encode_fun f
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private lemma f_eq : succ (f v) = f v :=
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begin
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change succ (f v) =
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match decode_fun (encode_fun f) with
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| some g := succ (g v)
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| none := 0
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end,
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rewrite encodek_fun
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end
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end
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theorem not_encodable_nat_arrow_nat : (encodable (nat → nat)) → false :=
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assume h, absurd (f_eq h) succ_ne_self
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