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(* Copyright 2007 Milad Niqui                                           *)
(* This file is distributed under the terms of the                      *)
(* GNU Lesser General Public License Version 2.1                        *)
(* A copy of the license can be found at                                *)
(*                  <http://www.gnu.org/licenses/lgpl.txt>              *)
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Require OrderedTypeEx.
Require FSetList.
Module NatSet := FSetList.Make(OrderedTypeEx.Nat_as_OT).
Infix "++" := add (at level 60, right associativity).
Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
Require FSetFacts FSetProperties.
Require Extraction.
Module GeneralProperties := FSetProperties.Properties NatSet.
Section problem_knows_not_refl.
Variable town: t.
Variable n:nat.
Variable cardinality: cardinal town = 2*n+1.
Variable knows: elt -> elt -> Prop.
Variable knows_sym: forall m n, knows m n ->  knows n m.
Variable knows_extensional:forall m n p, E.eq n p -> knows m n-> knows m p.
Variable property: forall B, Subset B town -> cardinal B = n ->
{d:elt | In d (diff town B)/\(forall b, In b B -> knows d b)}.

(* Specifications: *)
Lemma extendible_by_one:forall B', cardinal B' <= (cardinal town)-1 -> {d:elt| In d town /\ ~(In d B')}.
Lemma extendible_to_n:forall B', Subset B' town -> cardinal B' <= n ->
{B:t| cardinal B = n /\  Subset B' B /\ Subset B town}.
(** The following property, proven by induction, is the heart of the solution, and is due to Tonny Hurkens. *)
Lemma inductive_invariant:forall m, m<= n ->
{B':t| Subset B' town /\ cardinal B' = m /\ forall b'0 b'1, In b'0 B' -> In b'1 B' -> ~(E.eq b'0 b'1) -> knows b'0 b'1}.
Theorem AMM11262: {e:elt | In e town /\ forall u, In u town /\ ~(E.eq u e)-> knows e u}.
End problem_knows_not_refl.