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CircleDot[0, 1]]]]}]}, "Substitution Lemma 4" -> { Tooltip -> Column[{"Substitution Lemma 4", CircleDot[1, CircleDot[0, 1]] == CircleDot[0, 1]}]}, "Axiom 1" -> { Tooltip -> Column[{"Axiom 1", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x1, $CellContext`x2], \ $CellContext`x3]}]}, "Critical Pair Lemma 1" -> { Tooltip -> Column[{"Critical Pair Lemma 1", CircleDot[0, CircleDot[0, $CellContext`x1]] == CircleDot[ CircleDot[0, 1], $CellContext`x1]}]}, "Substitution Lemma 2" -> { Tooltip -> Column[{"Substitution Lemma 2", CircleDot[0, CircleDot[0, $CellContext`x1]] == CircleDot[0, CircleDot[1, $CellContext`x1]]}]}, "Substitution Lemma 3" -> { Tooltip -> Column[{"Substitution Lemma 3", CircleDot[1, CircleDot[0, 1]] == CircleDot[0, 0]}]}, "Axiom 3" -> { Tooltip -> Column[{"Axiom 3", CircleDot[0, 1] == CircleDot[1, CircleDot[0, 0]]}]}, "Axiom 2" -> { Tooltip -> Column[{"Axiom 2", CircleDot[0, 0] == CircleDot[1, CircleDot[0, 0]]}]}, "Substitution Lemma 7" -> { Tooltip -> Column[{"Substitution Lemma 7", CircleDot[0, CircleDot[0, 0]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]}]}, "Conclusion 2" -> {Tooltip -> Column[{"Conclusion 2", True}]}, "Hypothesis 2" -> {Tooltip -> Column[{"Hypothesis 2", CircleDot[0, CircleDot[0, 1]] == CircleDot[1, CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]}]}, "Substitution Lemma 8" -> { Tooltip -> Column[{"Substitution Lemma 8", CircleDot[0, CircleDot[0, 1]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]}]}, "Substitution Lemma 5" -> { Tooltip -> Column[{"Substitution Lemma 5", CircleDot[1, CircleDot[0, CircleDot[1, $CellContext`x1]]] == CircleDot[ CircleDot[0, 1], $CellContext`x1]}]}, "Conclusion 1" -> {Tooltip -> Column[{"Conclusion 1", True}]}, "Critical Pair Lemma 2" -> { Tooltip -> Column[{"Critical Pair Lemma 2", CircleDot[1, CircleDot[ CircleDot[0, 1], $CellContext`x1]] == CircleDot[ CircleDot[0, 1], $CellContext`x1]}]}, "Hypothesis 1" -> {Tooltip -> Column[{"Hypothesis 1", CircleDot[0, CircleDot[0, 0]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 0]]]]]}]}, "Substitution Lemma 9" -> { Tooltip -> Column[{"Substitution Lemma 9", CircleDot[0, CircleDot[0, 1]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 1]]]]}]}, "Substitution Lemma 13" -> { Tooltip -> Column[{"Substitution Lemma 13", CircleDot[0, CircleDot[0, 1]] == CircleDot[1, CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 1]]]]]}]}, "Substitution Lemma 11" -> { Tooltip -> Column[{"Substitution Lemma 11", CircleDot[0, CircleDot[1, 1]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[1, 1]]]]}]}, "Substitution Lemma 15" -> { Tooltip -> Column[{"Substitution Lemma 15", CircleDot[0, CircleDot[1, 1]] == CircleDot[1, CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[1, 1]]]]]}]}, "Substitution Lemma 6" -> { Tooltip -> Column[{"Substitution Lemma 6", CircleDot[1, CircleDot[0, CircleDot[1, $CellContext`x1]]] == CircleDot[0, CircleDot[1, $CellContext`x1]]}]}, "Substitution Lemma 14" -> { Tooltip -> Column[{"Substitution Lemma 14", CircleDot[0, CircleDot[1, 1]] == CircleDot[1, CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[0, 1]]]]]}]}, "Substitution Lemma 1" -> { Tooltip -> Column[{ "Substitution Lemma 1", CircleDot[0, 1] == CircleDot[0, 0]}]}, "Substitution Lemma 16" -> { Tooltip -> Column[{"Substitution Lemma 16", CircleDot[0, CircleDot[1, 1]] == CircleDot[1, CircleDot[1, CircleDot[0, CircleDot[1, 1]]]]}]}}, EdgeStyle -> { DirectedEdge["Substitution Lemma 11", "Substitution Lemma 12"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Critical Pair Lemma 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 6", "Substitution Lemma 17"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 9"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 9", "Substitution Lemma 10"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 2", "Substitution Lemma 15"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 1", "Substitution Lemma 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 2", "Substitution Lemma 10"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 6", "Conclusion 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 16", "Substitution Lemma 17"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Substitution Lemma 7"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 4", "Critical Pair Lemma 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 10", "Substitution Lemma 11"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 6", "Substitution Lemma 16"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 15", "Substitution Lemma 16"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 6", "Conclusion 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 2", "Substitution Lemma 5"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 2", "Substitution Lemma 14"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 14", "Substitution Lemma 15"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Substitution Lemma 5"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 13", "Substitution Lemma 14"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 6"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 13"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 6", "Substitution Lemma 12"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 3"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 3"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Hypothesis 1", "Substitution Lemma 7"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 8", "Substitution Lemma 9"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 1", "Substitution Lemma 2"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 3", "Substitution Lemma 4"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 2", "Substitution Lemma 11"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 3", "Substitution Lemma 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 17", "Conclusion 2"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Substitution Lemma 6"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Critical Pair Lemma 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 8"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 4"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 12", "Conclusion 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Critical Pair Lemma 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Hypothesis 2", "Substitution Lemma 13"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 7", "Substitution Lemma 8"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]}, GraphLayout -> "LayeredDigraphEmbedding", VertexLabels -> {None}, VertexShapeFunction -> { "Substitution Lemma 4" -> "Circle", "Substitution Lemma 5" -> "Circle", "Substitution Lemma 1" -> "Circle", "Substitution Lemma 10" -> "Circle", "Substitution Lemma 6" -> "Circle", "Substitution Lemma 2" -> "Circle", "Axiom 1" -> "FiveDown", "Hypothesis 1" -> "Diamond", "Substitution Lemma 8" -> "Circle", "Conclusion 2" -> "Square", "Substitution Lemma 14" -> "Circle", "Substitution Lemma 7" -> "Circle", "Substitution Lemma 3" -> "Circle", "Critical Pair Lemma 1" -> "Triangle", "Substitution Lemma 16" -> "Circle", "Substitution Lemma 11" -> "Circle", "Substitution Lemma 12" -> "Circle", "Substitution Lemma 17" -> "Circle", "Axiom 2" -> "FiveDown", "Conclusion 1" -> "Square", "Critical Pair Lemma 2" -> "Triangle", "Axiom 3" -> "FiveDown", "Substitution Lemma 15" -> "Circle", "Substitution Lemma 9" -> "Circle", "Substitution Lemma 13" -> "Circle", "Hypothesis 2" -> "Diamond"}, VertexSize -> {{"Scaled", 0.021611613513818406`}}, VertexStyle -> {"Hypothesis 2" -> Directive[ RGBColor[ Rational[146, 255], Rational[10, 17], 0], EdgeForm[]], "Critical Pair Lemma 1" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Axiom 3" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 17" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 9" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 11" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Conclusion 1" -> Directive[ RGBColor[ Rational[13, 15], Rational[1, 15], 0], EdgeForm[]], "Substitution Lemma 14" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 6" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Hypothesis 1" -> Directive[ RGBColor[ Rational[146, 255], Rational[10, 17], 0], EdgeForm[]], "Substitution Lemma 2" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 12" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Conclusion 2" -> Directive[ RGBColor[ Rational[13, 15], Rational[1, 15], 0], EdgeForm[]], "Substitution Lemma 5" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 8" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 13" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 10" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 2" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 15" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 3" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 1" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Critical Pair Lemma 2" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 7" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 4" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 1" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 16" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]]}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.025789280468781373`], 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-> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic]}, "SummaryPanel"], DynamicModuleValues:>{}], "]"}], ProofObject["ListLogic", CircleDot[0, CircleDot[0, CircleDot[1, CircleDot[0, 0]]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], And[0 == CircleDot[0, CircleDot[1, 0]], CircleDot[0, 0] == 1, ForAll[{\[FormalA], \[FormalB], \[FormalC]}, CircleDot[\[FormalA], CircleDot[\[FormalB], \[FormalC]]] == CircleDot[ CircleDot[\[FormalA], \[FormalB]], \[FormalC]]]], {{"Axiom", 1} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x1, $CellContext`x2], $CellContext`x3], "Proof" -> Association[]], {"Axiom", 2} -> Association["Statement" -> CircleDot[0, CircleDot[1, 0]] == 0, "Proof" -> Association[]], {"Axiom", 3} -> Association[ "Statement" -> CircleDot[0, 0] == 1, "Proof" -> Association[]], { "Hypothesis", 1} -> Association["Statement" -> CircleDot[0, CircleDot[0, CircleDot[1, CircleDot[0, 0]]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[]], {"CriticalPairLemma", 1} -> Association["Statement" -> CircleDot[0, CircleDot[0, $CellContext`x1]] == CircleDot[1, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[0, 0] -> 1), "MatchingSide" -> 1]], { "CriticalPairLemma", 2} -> Association[ "Statement" -> CircleDot[1, 0] == CircleDot[0, 1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[0, Pattern[$CellContext`x1, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[0, 0] -> 1), "MatchingSide" -> 1]], { "CriticalPairLemma", 3} -> Association["Statement" -> CircleDot[1, 1] == CircleDot[0, CircleDot[1, 0]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[0, Pattern[$CellContext`x1, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 2}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "MatchingSide" -> 1]], {"SubstitutionLemma", 1} -> Association[ "Statement" -> CircleDot[1, 1] == 0, "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 3}, "Position" -> 2, "Construct" -> {"Axiom", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[1, 0]] -> 0), "OutputExpression" -> CircleDot[1, 1] == 0]], {"CriticalPairLemma", 4} -> Association["Statement" -> CircleDot[1, CircleDot[1, $CellContext`x1]] == CircleDot[0, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[1, 1] -> 0), "MatchingSide" -> 1]], { "SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[0, CircleDot[0, CircleDot[1, 1]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> {1, 2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[0, CircleDot[0, CircleDot[1, 1]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 3} -> Association["Statement" -> CircleDot[1, CircleDot[1, 1]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 2}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "OutputExpression" -> CircleDot[1, CircleDot[1, 1]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 4} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> {1, 2}, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[1, 1] -> 0), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 5} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, CircleDot[1, 1]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 4}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, CircleDot[1, 1]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]]]], {"SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]]]], {"SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]]]], {"SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]], {"SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]], {"SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]], {"SubstitutionLemma", 13} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 12}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]], {"SubstitutionLemma", 14} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 13}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]], {"SubstitutionLemma", 15} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 14}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]], {"SubstitutionLemma", 16} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 15}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]], {"SubstitutionLemma", 17} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 16}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]], {"SubstitutionLemma", 18} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 17}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]], {"SubstitutionLemma", 19} -> Association["Statement" -> CircleDot[1, 0] == 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Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[1, 1] -> 0), "MatchingSide" -> 1]], {"SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[0, CircleDot[0, CircleDot[1, 1]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> {1, 2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[0, CircleDot[0, CircleDot[1, 1]]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 3} -> Association["Statement" -> CircleDot[1, CircleDot[1, 1]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 2}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "OutputExpression" -> CircleDot[1, CircleDot[1, 1]] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 4} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> {1, 2}, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[1, 1] -> 0), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 5} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, CircleDot[1, 1]]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 4}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, CircleDot[1, 1]]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> 1, "Rule" -> (CircleDot[0, CircleDot[0, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[1, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, CircleDot[1, 1]]]]]]]]]]]]]]]]]], { "SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]]]], {"SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]]]], {"SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]]]], {"SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]]]], {"SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]]]], {"SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]]]], {"SubstitutionLemma", 13} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 12}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]]]], {"SubstitutionLemma", 14} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 13}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]]]], {"SubstitutionLemma", 15} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 14}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]]]], {"SubstitutionLemma", 16} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 15}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]]]], {"SubstitutionLemma", 17} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 16}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]]]], {"SubstitutionLemma", 18} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 17}, "Position" -> {2, 2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]]]], {"SubstitutionLemma", 19} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 18}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]]]], {"SubstitutionLemma", 20} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 19}, "Position" -> {2, 2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]]]], {"SubstitutionLemma", 21} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 20}, "Position" -> {2, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]]]], {"SubstitutionLemma", 22} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 21}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]]]], {"SubstitutionLemma", 23} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 22}, "Position" -> {2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]]]], {"SubstitutionLemma", 24} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 23}, "Position" -> {2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, CircleDot[0, 0]]]]]], {"SubstitutionLemma", 25} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, 1]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 24}, "Position" -> {2, 2, 1}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[0, 1]]]]], {"SubstitutionLemma", 26} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[1, 0]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 25}, "Position" -> {2, 2, 2}, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[1, CircleDot[1, 0]]]]], {"SubstitutionLemma", 27} -> Association["Statement" -> CircleDot[1, 0] == CircleDot[0, CircleDot[0, 0]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 26}, "Position" -> {2, 2}, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[1, CircleDot[1, Pattern[$CellContext`x1, Blank[]]]] -> CircleDot[0, $CellContext`x1]), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, CircleDot[0, 0]]]], {"SubstitutionLemma", 28} -> Association[ "Statement" -> CircleDot[1, 0] == CircleDot[0, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 27}, "Position" -> {2, 2}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[0, 0] -> 1), "OutputExpression" -> CircleDot[1, 0] == CircleDot[0, 1]]], {"Conclusion", 1} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 28}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[0, 1] -> CircleDot[1, 0]), "OutputExpression" -> True]]}], Editable->False, SelectWithContents->True, Selectable->False]], "Output", TaggingRules->{}, CellChangeTimes->{3.8050025193664007`*^9, 3.805288922113833*^9}, CellLabel->"Out[377]=", CellID->1827171523] }, Open ]], 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