MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: cond(false(),x,y) -> 0() cond(true(),x,y) -> s(minus(x,s(y))) ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) minus(x,y) -> cond(ge(x,s(y)),x,y) - Signature: {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cond) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(cond) = [1] x1 + [3] x2 + [3] p(false) = [0] p(ge) = [1] p(minus) = [3] x1 + [5] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: cond(false(),x,y) = [3] x + [3] > [0] = 0() ge(u,0()) = [1] > [0] = true() ge(0(),s(v)) = [1] > [0] = false() minus(x,y) = [3] x + [5] > [3] x + [4] = cond(ge(x,s(y)),x,y) Following rules are (at-least) weakly oriented: cond(true(),x,y) = [3] x + [3] >= [3] x + [6] = s(minus(x,s(y))) ge(s(u),s(v)) = [1] >= [1] = ge(u,v) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: Failure MAYBE + Considered Problem: - Strict TRS: cond(true(),x,y) -> s(minus(x,s(y))) ge(s(u),s(v)) -> ge(u,v) - Weak TRS: cond(false(),x,y) -> 0() ge(u,0()) -> true() ge(0(),s(v)) -> false() minus(x,y) -> cond(ge(x,s(y)),x,y) - Signature: {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE