adamc@620: (* Copyright (c) 2009, Adam Chlipala
adamc@620:  * All rights reserved.
adamc@620:  *
adamc@620:  * Redistribution and use in source and binary forms, with or without
adamc@620:  * modification, are permitted provided that the following conditions are met:
adamc@620:  *
adamc@620:  * - Redistributions of source code must retain the above copyright notice,
adamc@620:  *   this list of conditions and the following disclaimer.
adamc@620:  * - Redistributions in binary form must reproduce the above copyright notice,
adamc@620:  *   this list of conditions and the following disclaimer in the documentation
adamc@620:  *   and/or other materials provided with the distribution.
adamc@620:  * - The names of contributors may not be used to endorse or promote products
adamc@620:  *   derived from this software without specific prior written permission.
adamc@620:  *
adamc@620:  * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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adamc@620:  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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adamc@620:  * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 
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adamc@620:  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
adamc@620:  * POSSIBILITY OF SUCH DAMAGE.
adamc@620:  *)
adamc@620: 
adamc@620: Set Implicit Arguments.
adamc@620: 
adamc@620: 
adamc@620: Fixpoint name' (n : nat) : Type :=
adamc@620:   match n with
adamc@620:     | O => Empty_set
adamc@620:     | S n' => option (name' n')
adamc@620:   end.
adamc@620: 
adamc@620: Definition name'_eq_dec : forall n (x y : name' n), {x = y} + {x <> y}.
adamc@620:   Hint Extern 1 (_ = _ -> False) => congruence.
adamc@620: 
adamc@620:   induction n; simpl; intuition;
adamc@620:     repeat match goal with
adamc@620:              | [ x : Empty_set |- _ ] => destruct x
adamc@620:              | [ x : option _ |- _ ] => destruct x
adamc@620:            end; intuition;
adamc@620:     match goal with
adamc@620:       | [ IH : _, n1 : name' _, n2 : name' _ |- _ ] =>
adamc@620:         destruct (IHn n1 n0); subst; intuition
adamc@620:     end.
adamc@620: Qed.
adamc@620: 
adamc@620: Definition badName' n (P : name' n -> bool) :
adamc@620:   {nm : _ | P nm = false} + {forall nm, P nm = true}.
adamc@620:   Hint Constructors sig.
adamc@620:   Hint Extern 1 (_ = true) =>
adamc@620:     match goal with
adamc@620:       | [ nm : option _ |- _ ] => destruct nm
adamc@620:     end; auto.
adamc@620: 
adamc@620:   induction n; simpl; intuition;
adamc@620:     match goal with
adamc@620:       | [ IH : forall P : _ -> _,_ |- _ ] =>
adamc@620:         case_eq (P None);
adamc@620:         destruct (IH (fun nm => P (Some nm))); firstorder eauto
adamc@620:     end.
adamc@620: Qed.
adamc@620: 
adamc@620: Parameter numNames : nat.
adamc@620: Definition name := name' (S numNames).
adamc@620: Definition name_eq_dec : forall (x y : name), {x = y} + {x <> y} := @name'_eq_dec _.
adamc@620: Definition badName : forall P : name -> bool, {nm : _ | P nm = false} + {forall nm, P nm = true} := @badName' _.
adamc@620: Definition defaultName : name := None.