WORST_CASE(?, O(n^1))
<?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>g</name><arg><funapp><name>S</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>g</name><arg><var>x</var></arg><arg><funapp><name>S</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>f</name><arg><var>y</var></arg><arg><funapp><name>S</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>f</name><arg><funapp><name>S</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>g</name><arg><funapp><name>0</name></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><var>x2</var></rhs></rule><rule><lhs><funapp><name>f</name><arg><var>x1</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>g</name><arg><var>x1</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></trs><strategy><innermost/></strategy></trsInput><runtimeComplexity><signature><symbol><name>S</name><arity>1</arity></symbol><symbol><name>0</name><arity>0</arity></symbol></signature><signature><symbol><name>g</name><arity>2</arity></symbol><symbol><name>f</name><arity>2</arity></symbol></signature></runtimeComplexity><polynomial>1</polynomial></complexityInput></input><cpfVersion>2.1</cpfVersion><proof><complexityProof><bounds><type><match/></type><bound>2</bound><finalStates><state>0</state></finalStates><treeAutomaton><finalStates><state>0</state><state>1</state><state>2</state></finalStates><transitions><transition><lhs><name>S</name><height>0</height><state>0</state></lhs><rhs><state>0</state></rhs></transition><transition><lhs><name>0</name><height>0</height></lhs><rhs><state>0</state></rhs></transition><transition><lhs><name>g</name><height>0</height><state>0</state><state>0</state></lhs><rhs><state>1</state></rhs></transition><transition><lhs><name>f</name><height>0</height><state>0</state><state>0</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>0</state></lhs><rhs><state>3</state></rhs></transition><transition><lhs><name>g</name><height>1</height><state>0</state><state>3</state></lhs><rhs><state>1</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>0</state></lhs><rhs><state>4</state></rhs></transition><transition><lhs><name>f</name><height>1</height><state>4</state><state>0</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>0</name><height>1</height></lhs><rhs><state>5</state></rhs></transition><transition><lhs><name>g</name><height>1</height><state>0</state><state>5</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>3</state></lhs><rhs><state>3</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>5</state></lhs><rhs><state>3</state></rhs></transition><transition><lhs><name>g</name><height>1</height><state>0</state><state>3</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>4</state></lhs><rhs><state>4</state></rhs></transition><transition><lhs><name>g</name><height>1</height><state>4</state><state>5</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>S</name><height>2</height><state>5</state></lhs><rhs><state>6</state></rhs></transition><transition><lhs><name>g</name><height>2</height><state>0</state><state>6</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>g</name><height>2</height><state>4</state><state>6</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><name>S</name><height>1</height><state>6</state></lhs><rhs><state>3</state></rhs></transition><transition><lhs><name>S</name><height>2</height><state>6</state></lhs><rhs><state>6</state></rhs></transition><transition><lhs><state>0</state></lhs><rhs><state>1</state></rhs></transition><transition><lhs><state>3</state></lhs><rhs><state>1</state></rhs></transition><transition><lhs><state>3</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><state>5</state></lhs><rhs><state>2</state></rhs></transition><transition><lhs><state>6</state></lhs><rhs><state>2</state></rhs></transition></transitions></treeAutomaton></bounds></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 % (1 real / 0 unknown / 0 assumptions / 1 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>