@prefix log: <http://www.w3.org/2000/10/swap/log#>.
@prefix owl: <http://www.w3.org/2002/07/owl#>.
@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>.
@prefix : <http://www.agfa.com/w3c/euler/graph.axiom#>.
@prefix math: <http://www.w3.org/2000/10/swap/math#>.
@prefix list: <http://www.w3.org/2000/10/swap/list#>.
@prefix xsd: <http://www.w3.org/2001/XMLSchema#>.
@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>.
@prefix ns1: <http://eulersharp.sourceforge.net/2003/03swap/rpo-rules#>.
@prefix r: <http://www.w3.org/2000/10/swap/reason#>.
@prefix var: <http://josd.github.io/var#>.
@prefix skolem: <http://josd.github.io/.well-known/genid/3917894951695475063#>.
@prefix n3: <http://www.w3.org/2004/06/rei#>.

skolem:proof a r:Proof, r:Conjunction;
  r:component skolem:lemma1;
  r:gives {
    :paris :path :nantes.
  }.

skolem:lemma1 a r:Inference;
  r:gives {
    :paris :path :nantes.
  };
  r:evidence (
    skolem:lemma2
  );
  r:rule skolem:lemma3.

skolem:lemma2 a r:Inference;
  r:gives {
    :paris :path :nantes.
  };
  r:evidence (
    skolem:lemma4
    skolem:lemma5
    skolem:lemma6
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#nantes"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#paris"]];
  r:rule skolem:lemma7.

skolem:lemma3 a r:Extraction;
  r:gives {
    {:paris :path :nantes} => {:paris :path :nantes}.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.filter.n3>].

skolem:lemma4 a r:Extraction;
  r:gives {
    :path a owl:TransitiveProperty.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

skolem:lemma5 a r:Inference;
  r:gives {
    :angers :path :nantes.
  };
  r:evidence (
    skolem:lemma8
    skolem:lemma9
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#nantes"]];
  r:rule skolem:lemma10.

skolem:lemma6 a r:Inference;
  r:gives {
    :paris :path :angers.
  };
  r:evidence (
    skolem:lemma4
    skolem:lemma11
    skolem:lemma12
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#paris"]];
  r:rule skolem:lemma7.

skolem:lemma7 a r:Extraction;
  r:gives {
    @forAll var:x_0, var:x_1, var:x_2, var:x_3. {var:x_0 a owl:TransitiveProperty.
     var:x_1 var:x_0 var:x_2.
     var:x_3 var:x_0 var:x_1} => {var:x_3 var:x_0 var:x_2}.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/rpo-rules.n3>].

skolem:lemma8 a r:Extraction;
  r:gives {
    :oneway rdfs:subPropertyOf :path.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

skolem:lemma9 a r:Extraction;
  r:gives {
    :angers :oneway :nantes.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

skolem:lemma10 a r:Extraction;
  r:gives {
    @forAll var:x_0, var:x_1, var:x_2, var:x_3. {var:x_0 rdfs:subPropertyOf var:x_1.
     var:x_2 var:x_0 var:x_3} => {var:x_2 var:x_1 var:x_3}.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/rpo-rules.n3>].

skolem:lemma11 a r:Inference;
  r:gives {
    :lemans :path :angers.
  };
  r:evidence (
    skolem:lemma8
    skolem:lemma13
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
  r:rule skolem:lemma10.

skolem:lemma12 a r:Inference;
  r:gives {
    :paris :path :lemans.
  };
  r:evidence (
    skolem:lemma4
    skolem:lemma14
    skolem:lemma15
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#chartres"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#paris"]];
  r:rule skolem:lemma7.

skolem:lemma13 a r:Extraction;
  r:gives {
    :lemans :oneway :angers.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

skolem:lemma14 a r:Inference;
  r:gives {
    :chartres :path :lemans.
  };
  r:evidence (
    skolem:lemma8
    skolem:lemma16
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#chartres"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
  r:rule skolem:lemma10.

skolem:lemma15 a r:Inference;
  r:gives {
    :paris :path :chartres.
  };
  r:evidence (
    skolem:lemma8
    skolem:lemma17
  );
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#paris"]];
  r:binding [ r:variable [ n3:uri "http://josd.github.io/var#x_3"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#chartres"]];
  r:rule skolem:lemma10.

skolem:lemma16 a r:Extraction;
  r:gives {
    :chartres :oneway :lemans.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

skolem:lemma17 a r:Extraction;
  r:gives {
    :paris :oneway :chartres.
  };
  r:because [ a r:Parsing; r:source <http://josd.github.io/eye/reasoning/graph/graph.axiom.n3>].

