@prefix skolem: <https://eyereasoner.github.io/.well-known/genid/8b98b360-9a70-4845-b52c-c675af60ad01#>.
@prefix r: <http://www.w3.org/2000/10/swap/reason#>.
@prefix : <http://www.agfa.com/w3c/euler/graph.axiom#>.
@prefix n3: <http://www.w3.org/2004/06/rei#>.
@prefix owl: <http://www.w3.org/2002/07/owl#>.
@prefix var: <http://www.w3.org/2000/10/swap/var#>.
@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>.

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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#nantes"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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 <https://eyereasoner.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 <https://eyereasoner.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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#angers"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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 <https://eyereasoner.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 <https://eyereasoner.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 <https://eyereasoner.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 <https://eyereasoner.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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#chartres"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#lemans"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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 <https://eyereasoner.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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#chartres"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#oneway"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#path"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo [ n3:uri "http://www.agfa.com/w3c/euler/graph.axiom#paris"]];
    r:binding [ r:variable [ n3:uri "http://www.w3.org/2000/10/swap/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 <https://eyereasoner.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 <https://eyereasoner.github.io/eye/reasoning/graph/graph.axiom.n3>].

