@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://example.org/test#>.
@prefix n3: <https://www.w3.org/ns/n3#>.
@prefix log: <http://www.w3.org/2000/10/swap/log#>.
@prefix var: <http://www.w3.org/2000/10/swap/var#>.

skolem:proof a r:Proof, r:Conjunction;
    r:component skolem:lemma1;
    r:component skolem:lemma2;
    r:gives {
        :task1 a :NotDiscardedTask.
        :task2 a :NotDiscardedTask.
    }.

skolem:lemma1 a r:Inference;
    r:gives {
        :task1 a :NotDiscardedTask.
    };
    r:evidence (
        skolem:lemma3
        skolem:lemma4
    );
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ <http://www.w3.org/2004/06/rei#uri> "http://example.org/test#task1"]];
    r:rule skolem:lemma5.

skolem:lemma2 a r:Inference;
    r:gives {
        :task2 a :NotDiscardedTask.
    };
    r:evidence (
        skolem:lemma6
        skolem:lemma7
    );
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ <http://www.w3.org/2004/06/rei#uri> "http://example.org/test#task2"]];
    r:rule skolem:lemma5.

skolem:lemma3 a r:Extraction;
    r:gives {
        :task1 a :CompositeTask.
    };
    r:because [ a r:Parsing; r:source <https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3>].

skolem:lemma4 a r:Inference;
    r:gives {
        {
            :task1 a :DiscardedTask.
        } n3:notIn <https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3>.
    };
    r:evidence (
        [ a r:Fact; r:gives {<https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3> log:semantics {
            :task1 a :CompositeTask.
            :task2 a :CompositeTask.
        }}]
        [ a r:Fact; r:gives {{
            :task1 a :CompositeTask.
            :task2 a :CompositeTask.
        } log:notIncludes {
            :task1 a :DiscardedTask.
        }}]
    );
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ <http://www.w3.org/2004/06/rei#uri> "https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3"]];
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo {
        :task1 a :CompositeTask.
        :task2 a :CompositeTask.
    }];
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo {
        :task1 a :DiscardedTask.
    }];
    r:rule skolem:lemma8.

skolem:lemma5 a r:Extraction;
    r:gives {
        @forAll var:x_0. {
            var:x_0 a :CompositeTask.
            {
                var:x_0 a :DiscardedTask.
            } n3:notIn <https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3>.
        } => {
            var:x_0 a :NotDiscardedTask.
        }.
    };
    r:because [ a r:Parsing; r:source <https://eyereasoner.github.io/eye/reasoning/n3c/notIn_query.n3>].

skolem:lemma6 a r:Extraction;
    r:gives {
        :task2 a :CompositeTask.
    };
    r:because [ a r:Parsing; r:source <https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3>].

skolem:lemma7 a r:Inference;
    r:gives {
        {
            :task2 a :DiscardedTask.
        } n3:notIn <https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3>.
    };
    r:evidence (
        [ a r:Fact; r:gives {<https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3> log:semantics {
            :task1 a :CompositeTask.
            :task2 a :CompositeTask.
        }}]
        [ a r:Fact; r:gives {{
            :task1 a :CompositeTask.
            :task2 a :CompositeTask.
        } log:notIncludes {
            :task2 a :DiscardedTask.
        }}]
    );
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_0"]; r:boundTo [ <http://www.w3.org/2004/06/rei#uri> "https://eyereasoner.github.io/eye/reasoning/n3c/tasks.n3"]];
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_1"]; r:boundTo {
        :task1 a :CompositeTask.
        :task2 a :CompositeTask.
    }];
    r:binding [ r:variable [ <http://www.w3.org/2004/06/rei#uri> "http://www.w3.org/2000/10/swap/var#x_2"]; r:boundTo {
        :task2 a :DiscardedTask.
    }];
    r:rule skolem:lemma8.

skolem:lemma8 a r:Extraction;
    r:gives {
        @forAll var:x_0, var:x_1, var:x_2. {
            var:x_2 n3:notIn var:x_0.
        } <= {
            var:x_0 log:semantics var:x_1.
            var:x_1 log:notIncludes var:x_2.
        }.
    };
    r:because [ a r:Parsing; r:source <https://eyereasoner.github.io/eye/reasoning/n3c/components.n3>].

