/export/starexec/sandbox/solver/bin/starexec_run_tct_rci_cert /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(n^3))
<?xml version="1.0" encoding="UTF-8"?>
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<certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="cpf.xsd"><input><complexityInput><trsInput><trs><rules><rule><lhs><funapp><name>add0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>add0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>goal</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>Nil</name></funapp></rhs></rule></rules></trs><strategy><innermost/></strategy><relativeRules><rules/></relativeRules></trsInput><runtimeComplexity><signature><symbol><name>Cons</name><arity>2</arity></symbol><symbol><name>Nil</name><arity>0</arity></symbol><symbol><name>S</name><arity>0</arity></symbol></signature><signature><symbol><name>add0</name><arity>2</arity></symbol><symbol><name>goal</name><arity>2</arity></symbol><symbol><name>mul0</name><arity>2</arity></symbol></signature></runtimeComplexity><polynomial>3</polynomial></complexityInput></input><cpfVersion>2.2</cpfVersion><proof><complexityProof><dtTransformation><strictDTs><ruleWithDT><rule><lhs><funapp><name>add0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>add0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>goal</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>Nil</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></ruleWithDT></strictDTs><weakDTs/><innermostLhss><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></innermostLhss><complexityProof><usableRules><nonUsableRules><rules><rule><lhs><funapp><name>goal</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule></rules></nonUsableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><naturals/></domain><dimension>1</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>add0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>4</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>goal</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mul0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>5</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>add0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>14</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>4</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>mul0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>13</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>4</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>3</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>9</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><naturals/></domain><dimension>1</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>3</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>add0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>12</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>goal</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mul0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>add0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>mul0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>4</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><naturals/></domain><dimension>1</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>8</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>8</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>add0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>goal</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mul0</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>4</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>add0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>mul0</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>2</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>2</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>8</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>0</arity><polynomial><sum><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>3</degree></polynomial></type><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>add0</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>goal</name><arity>2</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>mul0</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>add0</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>mul0</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>add0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>add0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>add0</name><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>mul0</name><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>Nil</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c1</name><arg><funapp><sharp><name>add0</name></sharp><arg><var>xs</var></arg><arg><funapp><name>Cons</name><arg><funapp><name>S</name></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c4</name><arg><funapp><sharp><name>add0</name></sharp><arg><funapp><name>mul0</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><sharp><name>mul0</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mul0</name></sharp><arg><funapp><name>Nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>c5</name></funapp></rhs></rule></rules></usableRules><complexityProof><rIsEmpty/></complexityProof></ruleShifting></complexityProof></ruleShifting></complexityProof></ruleShifting></complexityProof></ruleShifting></complexityProof></usableRules></complexityProof></dtTransformation></complexityProof></proof><origin><proofOrigin><tool><name>TcT</name><version/></tool></proofOrigin></origin></certificationProblem>