WORST_CASE(?, O(n^3))
<?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><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>S</name><arg><var>x</var></arg></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><funapp><name>+</name><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>x2</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>add0</name><arg><funapp><name>0</name></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><var>x2</var></rhs></rule></rules></trs><strategy><innermost/></strategy><relativeRules><rules><rule><lhs><funapp><name>+</name><arg><var>x</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>S</name><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>S</name><arg><var>y</var></arg></funapp></rhs></rule></rules></relativeRules></trsInput><runtimeComplexity><signature><symbol><name>S</name><arity>1</arity></symbol><symbol><name>0</name><arity>0</arity></symbol></signature><signature><symbol><name>add0</name><arity>2</arity></symbol><symbol><name>+</name><arity>2</arity></symbol></signature></runtimeComplexity><polynomial>3</polynomial></complexityInput></input><cpfVersion>2.1</cpfVersion><proof><complexityProof><dtTransformation><strictDTs><ruleWithDT><rule><lhs><funapp><name>add0</name><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>+</name><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name>add0</name></sharp><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>add0</name><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><var>z0</var></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c3</name></funapp></rhs></rule></ruleWithDT></strictDTs><weakDTs><ruleWithDT><rule><lhs><funapp><name>+</name><arg><var>z0</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>S</name><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>+</name></sharp><arg><var>z0</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>+</name><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>S</name><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule></ruleWithDT></weakDTs><innermostLhss><funapp><sharp><name>+</name></sharp><arg><var>z0</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z0</var></arg></funapp><funapp><sharp><name>add0</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp><funapp><sharp><name>add0</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></innermostLhss><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>3</degree></polynomial></type><interpret><name>c</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</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>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>add0</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><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><polynomial><variable>2</variable></polynomial></product></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></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>+</name></sharp><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>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</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>1</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></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c3</name></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>+</name></sharp><arg><var>z0</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name>add0</name></sharp><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c3</name></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>c</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</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>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></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>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</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>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>+</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></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><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name>add0</name></sharp><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>+</name></sharp><arg><var>z0</var></arg><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>+</name></sharp><arg><funapp><name>S</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>add0</name><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name>add0</name></sharp><arg><var>z1</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>add0</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c3</name></funapp></rhs></rule></rules></usableRules><complexityProof><rIsEmpty/></complexityProof></ruleShifting></complexityProof></ruleShifting></complexityProof></dtTransformation></complexityProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
</version><strategy>Statistics for single proof: 100.00 % (4 real / 0 unknown / 0 assumptions / 4 total proof steps)</strategy><url>http://aprove.informatik.rwth-aachen.de</url></tool><toolUser><firstName>John</firstName><lastName>Doe</lastName></toolUser></proofOrigin><inputOrigin/></origin></certificationProblem>