* Step 1: Sum WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w))) h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w)))))) h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))) t(x) -> x t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x))))) t(t(x)) -> t(c(t(x),x)) - Signature: {h/3,t/1} / {0/0,c/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {h,t} and constructors {0,c,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: InnermostRuleRemoval WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w))) h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w)))))) h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))) t(x) -> x t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x))))) t(t(x)) -> t(c(t(x),x)) - Signature: {h/3,t/1} / {0/0,c/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {h,t} and constructors {0,c,s} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w))) h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w)))))) h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))) t(t(x)) -> t(c(t(x),x)) All above mentioned rules can be savely removed. * Step 3: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: t(x) -> x t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x))))) - Signature: {h/3,t/1} / {0/0,c/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {h,t} and constructors {0,c,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs t#(x) -> c_1() t#(x) -> c_2() Weak DPs and mark the set of starting terms. * Step 4: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: t#(x) -> c_1() t#(x) -> c_2() - Weak TRS: t(x) -> x t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x))))) - Signature: {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/0,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: t#(x) -> c_1() t#(x) -> c_2() * Step 5: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: t#(x) -> c_1() t#(x) -> c_2() - Signature: {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/0,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:t#(x) -> c_1() 2:S:t#(x) -> c_2() The dependency graph contains no loops, we remove all dependency pairs. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/0,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))