lean2/library/data/examples/notencodable.lean

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