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-> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]]], { "CriticalPairLemma", 3} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x4]], "Proof" -> Association[ "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Input" -> {"Axiom", 4}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], { "SubstitutionLemma", 3} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 2]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"Hypothesis", 2}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 2]] == CircleTimes[1, 1, 1]]], {"Conclusion", 1} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "OutputExpression" -> True]], {"CriticalPairLemma", 4} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 5} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 6} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"SubstitutionLemma", 2}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 7} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, 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Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], { "CriticalPairLemma", 8} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 4}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 9} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 8}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 6}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 10} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x4], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 9}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "Side" -> 2, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 9}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 11} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x4, $CellContext`x5]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 10}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 12} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 5} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 4}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]], { "SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> {1, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> {1, 2, 2}, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {1, 2, 2, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 13} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 12}, "Position" -> {1, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 14} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 13}, "Position" -> {1, 2, 2, 2, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 15} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 14}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 16} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 15}, "Position" -> {1, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 17} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 16}, "Position" -> {1, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 12}, "Orientation" -> 1, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]] -> CircleTimes[1, 1, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 18} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 17}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 19} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 18}, "Position" -> {1, 1}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 20} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 19}, "Position" -> {1, 1}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 21} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 20}, "Position" -> {1, 1}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 22} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 21}, "Position" -> {1, 1}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]] == CircleTimes[1, 1, 1]]], {"Conclusion", 2} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 22}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> True]]}], Editable->False, SelectWithContents->True, Selectable->False]], "Output", TaggingRules->{}, CellChangeTimes->{3.805005981533059*^9, 3.805289297338427*^9}, CellLabel->"Out[391]=", CellID->1410837312] }, Open ]], Cell["Show the abstract proof network:", "Text", TaggingRules->{}, CellChangeTimes->{{3.805005981627852*^9, 3.805005988320613*^9}}, CellID->907113134], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"proof", "[", "\"\\"", "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.805005992247055*^9, 3.805005994932311*^9}}, CellLabel->"In[392]:=", CellID->229399979], Cell[BoxData[ GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{ "Axiom 1", "Axiom 2", "Axiom 3", "Axiom 4", "Hypothesis 1", "Hypothesis 2", "Critical Pair Lemma 1", "Critical Pair Lemma 2", "Substitution Lemma 1", "Critical Pair Lemma 3", "Substitution Lemma 2", "Substitution Lemma 3", "Conclusion 1", "Critical Pair Lemma 4", "Critical Pair Lemma 5", "Critical Pair Lemma 6", "Critical Pair Lemma 7", "Substitution Lemma 4", "Critical Pair Lemma 8", "Critical Pair Lemma 9", "Critical Pair Lemma 10", "Critical Pair Lemma 11", "Critical Pair Lemma 12", "Substitution Lemma 5", "Substitution Lemma 6", "Substitution Lemma 7", "Substitution Lemma 8", "Substitution Lemma 9", "Substitution Lemma 10", "Substitution Lemma 11", "Substitution Lemma 12", "Substitution Lemma 13", "Substitution Lemma 14", "Substitution Lemma 15", "Substitution Lemma 16", "Substitution Lemma 17", "Substitution Lemma 18", "Substitution Lemma 19", "Substitution Lemma 20", "Substitution Lemma 21", "Substitution Lemma 22", "Conclusion 2"}, {{{2, 7}, {1, 7}, {2, 8}, {1, 8}, {8, 9}, {2, 9}, {3, 10}, {1, 10}, {4, 11}, {3, 11}, {6, 12}, {3, 12}, {12, 13}, {11, 13}, {11, 14}, {1, 14}, {7, 15}, {1, 15}, {10, 16}, {11, 16}, {11, 17}, {16, 17}, {17, 18}, {3, 18}, {18, 19}, {18, 19}, {19, 20}, {16, 20}, {20, 21}, {20, 21}, {10, 22}, {21, 22}, {22, 23}, {3, 23}, {14, 24}, {22, 24}, {5, 25}, {9, 25}, {25, 26}, {24, 26}, {26, 27}, {24, 27}, {27, 28}, {9, 28}, {28, 29}, {24, 29}, {29, 30}, {24, 30}, {30, 31}, {24, 31}, {31, 32}, {15, 32}, {32, 33}, {24, 33}, {33, 34}, {24, 34}, {34, 35}, {22, 35}, {35, 36}, {23, 36}, {36, 37}, {24, 37}, {37, 38}, {22, 38}, {38, 39}, {22, 39}, {39, 40}, {22, 40}, {40, 41}, {22, 41}, {41, 42}, {22, 42}}, Null}, { AnnotationRules -> { "Critical Pair Lemma 5" -> { Tooltip -> Column[{ "Critical Pair Lemma 5", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]]}]}, "Substitution Lemma 17" -> { Tooltip -> Column[{"Substitution Lemma 17", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 4" -> { Tooltip -> Column[{ "Critical Pair Lemma 4", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]}]}, "Substitution Lemma 10" -> { Tooltip -> Column[{"Substitution Lemma 10", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 12" -> { Tooltip -> Column[{"Substitution Lemma 12", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 8" -> { Tooltip -> Column[{ "Critical Pair Lemma 8", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x2]}]}, "Substitution Lemma 4" -> { Tooltip -> Column[{ "Substitution Lemma 4", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]}]}, "Axiom 1" -> { Tooltip -> Column[{ "Axiom 1", CircleDot[$CellContext`x1, $CellContext`x2] == CircleDot[$CellContext`x2, $CellContext`x1]}]}, "Critical Pair Lemma 1" -> { Tooltip -> Column[{ "Critical Pair Lemma 1", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x1, $CellContext`x2]]}]}, "Substitution Lemma 2" -> { Tooltip -> Column[{"Substitution Lemma 2", CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, \ $CellContext`x3]] == CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]}]}, "Substitution Lemma 3" -> { Tooltip -> Column[{"Substitution Lemma 3", CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 2]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 18" -> { Tooltip -> Column[{"Substitution Lemma 18", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1]}]}, "Axiom 3" -> {Tooltip -> Column[{"Axiom 3", CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, \ $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x4, \ $CellContext`x2]] == CircleTimes[$CellContext`x4, $CellContext`x3, \ $CellContext`x2]}]}, "Substitution Lemma 19" -> { Tooltip -> Column[{"Substitution Lemma 19", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]] == CircleTimes[1, 1, 1]}]}, "Axiom 2" -> { Tooltip -> Column[{"Axiom 2", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x1, $CellContext`x2], \ $CellContext`x3]}]}, "Critical Pair Lemma 9" -> { Tooltip -> Column[{ "Critical Pair Lemma 9", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]}]}, "Substitution Lemma 20" -> { Tooltip -> Column[{"Substitution Lemma 20", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 7" -> { Tooltip -> Column[{ "Critical Pair Lemma 7", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]}]}, "Critical Pair Lemma 3" -> { Tooltip -> Column[{ "Critical Pair Lemma 3", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x3, \ $CellContext`x4]]}]}, "Conclusion 2" -> {Tooltip -> Column[{"Conclusion 2", True}]}, "Critical Pair Lemma 11" -> { Tooltip -> Column[{ "Critical Pair Lemma 11", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x4, \ $CellContext`x5]]}]}, "Substitution Lemma 7" -> { Tooltip -> Column[{"Substitution Lemma 7", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 6" -> { Tooltip -> Column[{ "Critical Pair Lemma 6", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]}]}, "Substitution Lemma 5" -> { Tooltip -> Column[{ "Substitution Lemma 5", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x2]}]}, "Hypothesis 2" -> {Tooltip -> Column[{"Hypothesis 2", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleTimes[1, 1, 2]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 8" -> { Tooltip -> Column[{"Substitution Lemma 8", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Conclusion 1" -> {Tooltip -> Column[{"Conclusion 1", True}]}, "Critical Pair Lemma 10" -> { Tooltip -> Column[{ "Critical Pair Lemma 10", CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x4]}]}, "Substitution Lemma 21" -> { Tooltip -> Column[{"Substitution Lemma 21", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 2" -> { Tooltip -> Column[{ "Critical Pair Lemma 2", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x2, $CellContext`x1], \ $CellContext`x3]}]}, "Hypothesis 1" -> {Tooltip -> Column[{"Hypothesis 1", CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Critical Pair Lemma 12" -> { Tooltip -> Column[{ "Critical Pair Lemma 12", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x1]}]}, "Substitution Lemma 9" -> { Tooltip -> Column[{"Substitution Lemma 9", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 13" -> { Tooltip -> Column[{"Substitution Lemma 13", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 11" -> { Tooltip -> Column[{"Substitution Lemma 11", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 6" -> { Tooltip -> Column[{"Substitution Lemma 6", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 15" -> { Tooltip -> Column[{"Substitution Lemma 15", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}]}, "Axiom 4" -> {Tooltip -> Column[{"Axiom 4", CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, \ $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]}]}, "Substitution Lemma 14" -> { Tooltip -> Column[{"Substitution Lemma 14", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 22" -> { Tooltip -> Column[{"Substitution Lemma 22", CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 16" -> { Tooltip -> Column[{"Substitution Lemma 16", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1]}]}, "Substitution Lemma 1" -> { Tooltip -> Column[{ "Substitution Lemma 1", CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]}]}}, EdgeStyle -> { DirectedEdge["Critical Pair Lemma 11", "Conclusion 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 2", "Conclusion 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 6", "Critical Pair Lemma 7"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 2", "Substitution Lemma 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 3", "Critical Pair Lemma 11"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 3", "Substitution Lemma 3"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 7"] -> Directive[ Dashing[{Small, Small}], 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["Axiom 3", "Substitution Lemma 4"] -> 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["Substitution Lemma 2", "Critical Pair Lemma 6"] -> 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["Critical Pair Lemma 11", "Critical Pair Lemma 12"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 4", "Critical Pair Lemma 8"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 5"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 10", "Critical Pair Lemma 11"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 16"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 10"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 11", "Substitution Lemma 12"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 18"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 1", "Critical Pair Lemma 4"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 4", "Substitution Lemma 2"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 12", "Substitution Lemma 17"] -> 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 1", "Substitution Lemma 9"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 4", "Substitution Lemma 5"] -> 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["Axiom 3", "Critical Pair Lemma 12"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 8", "Critical Pair Lemma 9"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 19"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 20"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 15"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 12", "Substitution Lemma 13"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 6", "Substitution Lemma 7"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 14"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 1", "Critical Pair Lemma 5"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 19", "Substitution Lemma 20"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Critical Pair Lemma 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 21", "Substitution Lemma 22"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 17", "Substitution Lemma 18"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 11"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 6"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 5", "Substitution Lemma 13"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 9", "Critical Pair Lemma 10"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 21"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 12"] -> 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 3", "Conclusion 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 3", "Critical Pair Lemma 3"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Critical Pair Lemma 2"] -> Directive[ Dashing[{Small, Small}], 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]], DirectedEdge["Substitution Lemma 14", "Substitution Lemma 15"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 3", "Substitution Lemma 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 2", "Critical Pair Lemma 4"] -> Directive[ Dashing[{Small, Small}], 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 5", "Substitution Lemma 8"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 18", "Substitution Lemma 19"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Critical Pair Lemma 3"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 6", "Critical Pair Lemma 9"] -> Directive[ Dashing[{Small, Small}], 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["Hypothesis 2", "Substitution Lemma 3"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 7", "Substitution Lemma 4"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 11", "Substitution Lemma 22"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Critical Pair Lemma 1", "Critical Pair Lemma 5"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 22", "Conclusion 2"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 2", "Critical Pair Lemma 7"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 20", "Substitution Lemma 21"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Critical Pair Lemma 3", "Critical Pair Lemma 6"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Hypothesis 1", "Substitution Lemma 6"] -> 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]]}, GraphLayout -> "LayeredDigraphEmbedding", VertexLabels -> {None}, VertexShapeFunction -> { "Critical Pair Lemma 4" -> "Triangle", "Hypothesis 2" -> "Diamond", "Substitution Lemma 17" -> "Circle", "Substitution Lemma 3" -> "Circle", "Critical Pair Lemma 7" -> "Triangle", "Substitution Lemma 20" -> "Circle", "Substitution Lemma 15" -> "Circle", "Conclusion 1" -> "Square", "Substitution Lemma 12" -> "Circle", "Substitution Lemma 10" -> "Circle", "Substitution Lemma 8" -> "Circle", "Critical Pair Lemma 2" -> "Triangle", "Axiom 1" -> "FiveDown", "Critical Pair Lemma 8" -> "Triangle", "Critical Pair Lemma 10" -> "Triangle", "Substitution Lemma 11" -> "Circle", "Hypothesis 1" -> "Diamond", "Conclusion 2" -> "Square", "Substitution Lemma 1" -> "Circle", "Critical Pair Lemma 9" -> "Triangle", "Substitution Lemma 2" -> "Circle", "Substitution Lemma 18" -> "Circle", "Axiom 2" -> "FiveDown", "Critical Pair Lemma 11" -> "Triangle", "Substitution Lemma 16" -> "Circle", "Axiom 3" -> "FiveDown", "Substitution Lemma 13" -> "Circle", "Substitution Lemma 7" -> "Circle", "Substitution Lemma 21" -> "Circle", "Critical Pair Lemma 6" -> "Triangle", "Axiom 4" -> "FiveDown", "Substitution Lemma 6" -> "Circle", "Critical Pair Lemma 5" -> "Triangle", "Critical Pair Lemma 1" -> "Triangle", "Substitution Lemma 19" -> "Circle", "Substitution Lemma 9" -> "Circle", "Substitution Lemma 22" -> "Circle", "Critical Pair Lemma 12" -> "Triangle", "Substitution Lemma 5" -> "Circle", "Substitution Lemma 4" -> "Circle", "Critical Pair Lemma 3" -> "Triangle", "Substitution Lemma 14" -> "Circle"}, VertexSize -> {{"Scaled", 0.01743033499620919}}, VertexStyle -> {"Substitution Lemma 1" -> 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[]], "Substitution Lemma 20" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 15" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Critical Pair Lemma 9" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Critical Pair Lemma 12" -> 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[]], "Critical Pair Lemma 4" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Critical Pair Lemma 2" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 6" -> 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 10" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 14" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Critical Pair Lemma 10" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 5" -> 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[]], "Hypothesis 2" -> 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 16" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Critical Pair Lemma 3" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Critical Pair Lemma 6" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Critical Pair Lemma 8" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 13" -> 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[]], "Critical Pair Lemma 1" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 8" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 4" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 19" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Hypothesis 1" -> Directive[ RGBColor[ Rational[146, 255], Rational[10, 17], 0], EdgeForm[]], "Critical Pair Lemma 5" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 12" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Critical Pair Lemma 11" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Substitution Lemma 21" -> 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 4" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 22" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 3" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 18" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Critical Pair Lemma 7" -> Directive[ RGBColor[ Rational[47, 51], Rational[98, 255], Rational[53, 255]], EdgeForm[]], "Conclusion 1" -> Directive[ RGBColor[ Rational[13, 15], Rational[1, 15], 0], 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[]]}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.016045141714733657`], {RGBColor[ NCache[ Rational[167, 255], 0.6549019607843137], NCache[ Rational[167, 255], 0.6549019607843137], NCache[ Rational[167, 255], 0.6549019607843137]], Dashing[{Small, Small}], ArrowBox[BezierCurveBox[CompressedData[" 1:eJxV1GtMUwcUB/DSx711wCYTEXyFosBQMMht7wOVHRVEZhGBuQo4HioPRfEx F2BDxegGpuAUJejQzjEZBQItKMYp2DOQVSeBriritI5RdQo6InageJFp92n/ 5OTkl/y/niNbtzUmVSgQCCLezNv9/wSDU2a8b0vGRcNDfiR6+XMOesPjY/pX dBqkTcNn+qwcFJaLNNHWHsP6D3jvrt84aBcEfOw1oc/w8hOp1rmFA36O++z0 uEeGKzAtWFPJQZqyLSR73aDB/EB+N1/Nwf6gKTF+Jc8NbstWH2zK4iDrgn9N 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Annotation[#, Column[{"Critical Pair Lemma 12", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{2.9269579598944233, 17.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{ "x1", "\[CircleTimes]", "x2", "\[CircleTimes]", "x3"}], "\[Equal]", RowBox[{ "x1", "\[CircleTimes]", "x1", "\[CircleTimes]", "x2"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]}], "Tooltip"]& ], 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RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "2", "\[CircleTimes]", "4"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{1.0000000161259095, 13.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "3", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "2", "\[CircleTimes]", "4"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 10", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{2.000000016609249, 12.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 11\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "3"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "2", "\[CircleTimes]", "4"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 11", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{2.376477403581987, 11.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 12\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "2", "\[CircleTimes]", "4"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 12", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{2.3764774040866996, 10.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 13\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "2", "\[CircleTimes]", "4"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 13", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{4.000000028946374, 9.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 14\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 14", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{6.00000003544875, 8.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 15\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "4", "\[CircleTimes]", "2"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"2", "\[CircleTimes]", "2", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 15", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{6.000000026775183, 7.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 16\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"4", "\[CircleTimes]", "2", "\[CircleTimes]", "2"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 16", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{6.687394489687108, 6.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 17\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "4"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 17", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{6.68739448909389, 5.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 18\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 18", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{9.000000035742119, 4.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 19\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 19", CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]] == CircleTimes[1, 1, 1]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{10.00000003754809, 3.}, 0.26606251670568215]}, TagBox[ GridBox[{{"\"Substitution Lemma 20\""}, { RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}], "\[CircleDot]", RowBox[{"(", RowBox[{"1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}], ")"}]}], ")"}]}], ")"}]}], "\[Equal]", RowBox[{ "1", "\[CircleTimes]", "1", "\[CircleTimes]", "1"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 20", CircleDot[ 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Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]], \ $CellContext`x4]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$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[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 4}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]]], {"CriticalPairLemma", 5} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 3} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 5}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x3]]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]]], \ {"CriticalPairLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 7} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]]] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x3, CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x4, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x4]]]] == CircleDot[$CellContext`x3, CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 7}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 11} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 7}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 11}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]]], \ {"CriticalPairLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 10}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"CriticalPairLemma", 13} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 8}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 14} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 13}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 15} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 14}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 16} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x4, Blank[]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x4, Blank[]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 3}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 17} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 16}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 18} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 17}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 7}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 19} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 17}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 20} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 18}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"CriticalPairLemma", 21} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 19}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 15}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 22} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 21}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 23} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 9}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x4, Blank[]]]]]] -> CircleDot[$CellContext`x3, CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]), "MatchingSide" -> 1]], {"SubstitutionLemma", 5} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 23}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]], { "CriticalPairLemma", 24} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 25} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 24}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 20}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 26} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 25}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 12}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 27} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x2, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 26}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 20}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 28} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 25}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 28}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x2]]]]], {"SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], { "CriticalPairLemma", 29} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 22}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 29}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]]]], {"SubstitutionLemma", 9} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 10}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]]]], {"CriticalPairLemma", 30} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 27}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 30}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x2]]]]], {"SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], { "CriticalPairLemma", 31} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 9}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 32} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 31}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "MatchingSide" -> 1]], {"SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 32}, "Position" -> 2, "Construct" -> {"SubstitutionLemma", 11}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], { "CriticalPairLemma", 33} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 12}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 7}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 34} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 33}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 13} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 34}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 7}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x3]]]]], {"CriticalPairLemma", 35} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 13}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 36} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 35}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 12}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 37} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 36}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, 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-> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic]}, "SummaryPanel"], DynamicModuleValues:>{}], "]"}], ProofObject["WolframModelLogic", CircleTimes[1, 1, 1] == CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]], And[ ForAll[{$CellContext`x, $CellContext`y, $CellContext`z}, CircleTimes[$CellContext`x, $CellContext`x, $CellContext`y] == CircleDot[ CircleTimes[$CellContext`y, $CellContext`y, $CellContext`x], CircleDot[ CircleTimes[$CellContext`y, $CellContext`z, $CellContext`y], CircleTimes[$CellContext`x, $CellContext`y, $CellContext`z]]]], ForAll[{$CellContext`x, $CellContext`y, $CellContext`z, $CellContext`w}, CircleDot[ CircleTimes[$CellContext`x, $CellContext`y, $CellContext`x], CircleTimes[$CellContext`z, $CellContext`w, $CellContext`y]] == CircleTimes[$CellContext`w, $CellContext`z, $CellContext`y]], ForAll[{\[FormalA], \[FormalB], \[FormalC]}, CircleDot[\[FormalA], CircleDot[\[FormalB], \[FormalC]]] == CircleDot[ CircleDot[\[FormalA], \[FormalB]], \[FormalC]]], ForAll[{\[FormalA], \[FormalB]}, CircleDot[\[FormalA], \[FormalB]] == CircleDot[\[FormalB], \[FormalA]]]], {{"Axiom", 1} -> Association[ "Statement" -> CircleDot[$CellContext`x1, $CellContext`x2] == CircleDot[$CellContext`x2, $CellContext`x1], "Proof" -> Association[]], {"Axiom", 2} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x1, $CellContext`x2], $CellContext`x3], "Proof" -> Association[]], {"Axiom", 3} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x2]] == CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2], "Proof" -> Association[]], {"Axiom", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[]], {"Hypothesis", 1} -> Association["Statement" -> CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[]], {"CriticalPairLemma", 1} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x1, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 2} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x2, $CellContext`x1], $CellContext`x3], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$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[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"SubstitutionLemma", 1} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 2}, "Position" -> 2, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]]], { "CriticalPairLemma", 3} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x4]], "Proof" -> Association[ "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x2], \ $CellContext`x5]] == CircleDot[ CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2], \ $CellContext`x5], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$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[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Input" -> {"Axiom", 4}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], { "CriticalPairLemma", 5} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 6} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 7} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"SubstitutionLemma", 2}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 8} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 7}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"SubstitutionLemma", 3} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 8}, "Position" -> 2, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], { "CriticalPairLemma", 9} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 3}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 3}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 10} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 9}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 7}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 11} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x4], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 10}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "Side" -> 2, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 10}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 12} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x4, $CellContext`x5]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 11}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 13} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 12}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 4} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 5}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 12}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]], { "SubstitutionLemma", 5} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x5, Blank[]]]] -> CircleDot[ CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2], \ $CellContext`x5]), "OutputExpression" -> CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], 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CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x5, Blank[]]]] -> CircleDot[ CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2], \ $CellContext`x5]), "OutputExpression" -> CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> {-1, 1}, "Rule" -> 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{"CriticalPairLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x5, Blank[]]]] -> CircleDot[ CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2], \ $CellContext`x5]), "OutputExpression" -> CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 17} -> Association["Statement" -> CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 16}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, 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1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 18}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 20} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 19}, "Position" -> {1, 2}, "Construct" -> {"CriticalPairLemma", 12}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], 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Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 5} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 6} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"SubstitutionLemma", 2}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 7} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 6}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"SubstitutionLemma", 4} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 7}, "Position" -> 2, "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], { "CriticalPairLemma", 8} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 4}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 9} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 8}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 6}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 10} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x4], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 9}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "Side" -> 2, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 9}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 11} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x4, $CellContext`x4, $CellContext`x5]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 10}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 12} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x2, Blank[]]]] -> CircleTimes[$CellContext`x4, $CellContext`x3, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 5} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 4}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]], { "SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> 1, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> {1, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> {1, 2, 2}, "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 11} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {1, 2, 2, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 13} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 12}, "Position" -> {1, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]] -> CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 2, 4], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 14} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 13}, "Position" -> {1, 2, 2, 2, 2, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 2], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 15} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 14}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[2, 2, 1]]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 16} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 15}, "Position" -> {1, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[4, 2, 2]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 17} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 16}, "Position" -> {1, 2, 2, 2, 2, 2}, "Construct" -> {"CriticalPairLemma", 12}, "Orientation" -> 1, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]] -> CircleTimes[1, 1, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 4]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 18} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 17}, "Position" -> {1, 1}, "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> {-1, 1}, "Rule" -> (CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]]] == CircleTimes[1, 1, 1]]], { "SubstitutionLemma", 19} -> Association["Statement" -> CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 1, 1], CircleTimes[1, 1, 1]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 18}, "Position" -> {1, 1}, "Construct" -> {"CriticalPairLemma", 11}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x5, Blank[]]]] -> CircleTimes[$CellContext`x2, $CellContext`x1, 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Association["Statement" -> CircleDot[ CircleTimes[1, 2, 1], CircleDot[ CircleTimes[1, 1, 1], CircleDot[ CircleTimes[1, 3, 1], CircleDot[ CircleTimes[1, 1, 3], CircleDot[ CircleTimes[2, 2, 1], CircleDot[ CircleTimes[2, 4, 2], CircleTimes[1, 2, 4]]]]]]] == CircleTimes[1, 1, 1], "Proof" -> Association[]], {"CriticalPairLemma", 1} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x1, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 2} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[ CircleDot[$CellContext`x2, $CellContext`x1], $CellContext`x3], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$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[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"SubstitutionLemma", 1} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 2}, "Position" -> 2, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x2, CircleDot[$CellContext`x1, $CellContext`x3]]]], { "CriticalPairLemma", 3} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"Axiom", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3]], \ $CellContext`x4]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4], "Proof" -> Association[ "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$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[$CellContext`x2, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 2} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 4}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x3], $CellContext`x4]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]]], {"CriticalPairLemma", 5} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 3} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x3]]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 5}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x4, CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x3]]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x1]]]], {"CriticalPairLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 7} -> Association["Statement" -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]] == CircleDot[$CellContext`x3, CircleDot[$CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "Side" -> 2, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]]] == CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[$CellContext`x3, CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x4, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, \ $CellContext`x4]]]] == CircleDot[$CellContext`x3, CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 1}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 7}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 11} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2]], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 7}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], "MatchingConstruct" -> {"Axiom", 3}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 4} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], \ $CellContext`x4]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 11}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, \ $CellContext`x3], $CellContext`x4]]] == CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x1]]]], {"CriticalPairLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 10}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"CriticalPairLemma", 13} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 8}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 14} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 13}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 15} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 14}, "Orientation" -> {-1, 1}, "Rule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 16} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 3}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x4, Blank[]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x4, Blank[]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 3}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"CriticalPairLemma", 17} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 16}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 18} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 17}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 7}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 19} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 17}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 20} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 18}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"CriticalPairLemma", 21} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 19}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 15}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 22} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 21}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 23} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 9}, "MatchingOrientation" -> 1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x4, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x4, Blank[]]]]]] -> CircleDot[$CellContext`x3, CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]), "MatchingSide" -> 1]], {"SubstitutionLemma", 5} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 23}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 3}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]], { "CriticalPairLemma", 24} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 5}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"Axiom", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 25} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 24}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 20}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 26} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 25}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 12}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 27} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x2, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 26}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 20}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 28} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 25}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 6} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 28}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x2]]]]], {"SubstitutionLemma", 7} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], { "CriticalPairLemma", 29} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 27}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "MatchingSide" -> 1]], {"SubstitutionLemma", 8} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x2]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 29}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]]] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x4, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x4], CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x2]]]]], {"SubstitutionLemma", 9} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> 2, "Construct" -> {"CriticalPairLemma", 6}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x3]]] == CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]]], { "CriticalPairLemma", 30} -> Association["Statement" -> CircleDot[ CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2]], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 4}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[$CellContext`x4, CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 22}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 10} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 30}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> CircleDot[$CellContext`x1, CircleDot[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]]]], {"SubstitutionLemma", 11} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 10}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, \ $CellContext`x1]]]]], {"CriticalPairLemma", 31} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 7}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingConstruct" -> {"CriticalPairLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x3, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 32} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"SubstitutionLemma", 11}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "MatchingSide" -> 1]], {"SubstitutionLemma", 12} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 32}, "Position" -> 2, "Construct" -> {"SubstitutionLemma", 9}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]), "OutputExpression" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x1, $CellContext`x2], CircleTimes[$CellContext`x3, $CellContext`x3, \ $CellContext`x1]]] == CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], { "CriticalPairLemma", 33} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x1, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 12}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 34} -> Association["Statement" -> CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]]] == CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x3, $CellContext`x2], CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"SubstitutionLemma", 2}, "Orientation" -> 1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]]]] -> CircleDot[ CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1], \ $CellContext`x4]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], Pattern[$CellContext`x4, Blank[]]], "MatchingConstruct" -> {"CriticalPairLemma", 33}, "MatchingOrientation" -> -1, "MatchingRule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x1]), "MatchingSide" -> 1]], {"SubstitutionLemma", 13} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]]], "Proof" -> Association[ "Input" -> {"CriticalPairLemma", 34}, "Position" -> 1, "Construct" -> {"CriticalPairLemma", 31}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x2, $CellContext`x2, $CellContext`x3]), "OutputExpression" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ 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Pattern[$CellContext`x2, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]]], "MatchingConstruct" -> {"SubstitutionLemma", 1}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ Pattern[$CellContext`x1, Blank[]], CircleDot[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]]], CircleDot[ Pattern[$CellContext`x2, Blank[]], CircleDot[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 36} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1], CircleTimes[$CellContext`x3, $CellContext`x3, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 35}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x3, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 12}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 1]], {"CriticalPairLemma", 37} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 36}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]]] -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2]), "Side" -> 1, "Subpattern" -> CircleDot[ CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x3, Blank[]], Pattern[$CellContext`x1, Blank[]]]], "MatchingConstruct" -> {"CriticalPairLemma", 20}, "MatchingOrientation" -> {-1, 1}, "MatchingRule" -> TwoWayRule[ CircleDot[ CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]]], CircleTimes[ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]], CircleDot[ CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]], CircleTimes[ Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x1, Blank[]]]]], "MatchingSide" -> 2]], {"CriticalPairLemma", 38} -> Association[ "Statement" -> CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x2] == CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x3, $CellContext`x1], CircleDot[ CircleTimes[$CellContext`x1, $CellContext`x1, $CellContext`x3], CircleTimes[$CellContext`x1, $CellContext`x2, $CellContext`x1]]], "Proof" -> Association[ "Construct" -> {"CriticalPairLemma", 37}, "Orientation" -> -1, "Rule" -> (CircleDot[ CircleTimes[ 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