begin_problem(Scenario1_Piece2).

list_of_descriptions.
name({*Qualitative Reasoning About Containers - Case 1, Piece 2*}).
author({*Professor Ernest Davis, Angelica Chen, Casey McGinley*}).
status(unsatisfiable).
description({*November 11, 2014*}).
end_of_list.

list_of_symbols.
functions[(RUnion,2), (Singleton,1), (Place,2), (Slice,2), (HPlace,1), (Ox1,0), (Ob1,0), (Ta1,0), (Tb1,0), (Rc1,0), (Hc1,0)].
predicates[(Ordered,2), (Time,1), (Region,1), (History,1), (Object,1), (ObjectSet,1), (Lt,2), (Leq,2), (Leq3,3), (P,2), (Cavity,2), (OSPlace,3), (ClosedContainer,3), (CContained,3), (RigidObject,1), (RigidHistory,1), (O,2), (EC,2), (DR,2), (Outside,2), (NoExitCavity,4), (PersistentCavity,4), (NoEntranceCavity,4)].
end_of_list.

list_of_formulae(axioms).
% Time Ordering
% Sortal axioms
% TISA
formula(forall([x,y], implies(Lt(x,y), and(Time(x), Time(y))))).
% TISB
formula(forall([x,y], implies(Leq(x,y), and(Time(x), Time(y))))).
% TISC
formula(forall([x,y], implies(Ordered(x,y), and(Time(x), Time(y))))).
% TISD
formula(forall([x,y,z], implies(Leq3(x,y,z), and(Time(x), Time(y), Time(z))))).

% Space
% Basic spatial relations
% Sortal axioms
% SBSA
formula(forall([u,v], implies(or(P(u,v), O(u,v), EC(u,v), DR(u,v)), and(Region(u), Region(v))))).
% SBSB
formula(forall([u,v], implies(and(Region(u), Region(v)), Region(RUnion(u,v))))).

% Spatial Containment
% Sortal axioms
% SCSA
formula(forall([u,v], implies(or(Cavity(u,v), Outside(u,v)), and(Region(u), Region(v))))).

% Objects
% Object Sets
% Sortal axioms
% OSSA
formula(forall([x], implies(Object(x), ObjectSet(Singleton(x))))).

% Objects and Object Sets: Spatio-Temporal
% Sortal axioms
% OTSA
formula(forall([t,o], implies(and(Time(t), Object(o)), Region(Place(t,o))))).
% OTSB
formula(forall([t,s,r], implies(OSPlace(t,s,r), and(Time(t), ObjectSet(s), Region(r))))).

% Objects containing regions
% Sortal axiom
% ORSA
formula(forall([t,s,r], implies(ClosedContainer(t,s,r), and(Time(t), ObjectSet(s), Region(r))))).

% Object Containment
% Sortal axiom
% OCSA
formula(forall([t,ox,s], implies(CContained(t,ox,s), and(Time(t), Object(ox), ObjectSet(s))))).

% Histories
% Sortal axioms
% HISA
formula(forall([t,h], implies(and(Time(t), History(h)), Region(Slice(t,h))))).
% HISC
formula(forall([o], implies(Object(o), History(HPlace(o))))).

% Evolving cavities
% Sortal axioms
% HCSA
formula(forall([t1,t2,hc,ho], implies(or(NoExitCavity(t1,t2,hc,ho), NoEntranceCavity(t1,t2,hc,ho), PersistentCavity(t1,t2,hc,ho)), and(Time(t1), Time(t2), History(hc), History(ho))))).

% Rigid Objects
% Sortal axioms
% RGSA
formula(forall([o], implies(RigidObject(o), Object(o)))).
% RGSB
formula(forall([h], implies(RigidHistory(h), History(h)))).

% P.2.1 - implicit from ROA1 in Part1
formula(RigidHistory(HPlace(Ob1))).
% P.2.2 - proved by Part1
formula(Cavity(Rc1,Slice(Ta1,HPlace(Ob1)))).
% HCD1
formula(forall([t1,t2,hc,hb], equiv(PersistentCavity(t1,t2,hc,hb), and(NoExitCavity(t1,t2,hc,hb), NoEntranceCavity(t1,t2,hc,hb))))).
% C1A3 - given
formula(Lt(Ta1,Tb1)).
% TID1
formula(forall([x,y], equiv(Leq(x,y), or(Lt(x,y), equal(x,y))))).
% TID3
formula(forall([x,y,z], equiv(Leq3(x,y,z), and(Leq(x,y), Leq(y,z))))).
% ROA2
formula(forall([h,t1,tm,t2,r], implies(and(RigidHistory(h), Leq3(t1,tm,t2), Cavity(r,Slice(tm,h))), exists([hc], and(RigidHistory(hc), PersistentCavity(t1,t2,hc,h), equal(r, Slice(tm,hc))))))).

end_of_list.

list_of_formulae(conjectures).
formula(exists([hc], and(equal(Rc1, Slice(Ta1,hc)), NoExitCavity(Ta1,Tb1,hc,HPlace(Ob1))))).
end_of_list.

list_of_settings(SPASS).
{*set_flag(DocProof,1).*}
end_of_list.

end_problem.