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(* Copyright (c) 2009, Adam Chlipala
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* - Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* - Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
* - The names of contributors may not be used to endorse or promote products
* derived from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*)
Set Implicit Arguments.
Fixpoint name' (n : nat) : Type :=
match n with
| O => Empty_set
| S n' => option (name' n')
end.
Definition name'_eq_dec : forall n (x y : name' n), {x = y} + {x <> y}.
Hint Extern 1 (_ = _ -> False) => congruence.
induction n; simpl; intuition;
repeat match goal with
| [ x : Empty_set |- _ ] => destruct x
| [ x : option _ |- _ ] => destruct x
end; intuition;
match goal with
| [ IH : _, n1 : name' _, n2 : name' _ |- _ ] =>
destruct (IHn n1 n0); subst; intuition
end.
Qed.
Definition badName' n (P : name' n -> bool) :
{nm : _ | P nm = false} + {forall nm, P nm = true}.
Hint Constructors sig.
Hint Extern 1 (_ = true) =>
match goal with
| [ nm : option _ |- _ ] => destruct nm
end; auto.
induction n; simpl; intuition;
match goal with
| [ IH : forall P : _ -> _,_ |- _ ] =>
case_eq (P None);
destruct (IH (fun nm => P (Some nm))); firstorder eauto
end.
Qed.
Parameter numNames : nat.
Definition name := name' (S numNames).
Definition name_eq_dec : forall (x y : name), {x = y} + {x <> y} := @name'_eq_dec _.
Definition badName : forall P : name -> bool, {nm : _ | P nm = false} + {forall nm, P nm = true} := @badName' _.
Definition defaultName : name := None.
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