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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** * MSetRBT : Implementation of MSetInterface via Red-Black trees *)
(** Initial author: Andrew W. Appel, 2011. Extra modifications by: Pierre Letouzey  The design decisions behind this implementation are described here:  - Efficient Verified Red-Black Trees, by Andrew W. Appel, September 2011. http://www.cs.princeton.edu/~appel/papers/redblack.pdf  Additional suggested reading:  - Red-Black Trees in a Functional Setting by Chris Okasaki. Journal of Functional Programming, 9(4):471-477, July 1999. http://www.eecs.usma.edu/webs/people/okasaki/jfp99redblack.pdf  - Red-black trees with types, by Stefan Kahrs. Journal of Functional Programming, 11(4), 425-432, 2001.  - Functors for Proofs and Programs, by J.-C. Filliatre and P. Letouzey. ESOP'04: European Symposium on Programming, pp. 370-384, 2004. http://www.lri.fr/~filliatr/ftp/publis/fpp.ps.gz *)
Require MSetGenTree.
Local Open Scope list_scope.
(* For nicer extraction, we create induction principles only when needed *)
Local Unset Elimination Schemes.
(** An extra function not (yet?) in MSetInterface.S *)
Module Type MSetRemoveMin (Import M:MSetInterface.S).
Parameter remove_min : t -> option (elt * t).

(* Specifications: *)
Axiom remove_min_spec1 : forall s k s',
remove_min s = Some (k,s') ->
min_elt s = Some k /\ remove k s [=] s'.
Axiom remove_min_spec2 : forall s, remove_min s = None -> Empty s.
End MSetRemoveMin.