MAYBE * Step 1: InnermostRuleRemoval MAYBE + Considered Problem: - Strict TRS: if(false(),x,y) -> s(minus(p(x),y)) if(true(),x,y) -> 0() le(0(),y) -> true() le(p(s(x)),x) -> le(x,x) le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,y) -> if(le(x,y),x,y) p(0()) -> s(s(0())) p(p(s(x))) -> p(x) p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. le(p(s(x)),x) -> le(x,x) p(p(s(x))) -> p(x) All above mentioned rules can be savely removed. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: if(false(),x,y) -> s(minus(p(x),y)) if(true(),x,y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,y) -> if(le(x,y),x,y) p(0()) -> s(s(0())) p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1}, uargs(minus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if) = [1] x1 + [1] x2 + [0] p(le) = [5] p(minus) = [1] x1 + [0] p(p) = [1] x1 + [9] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: le(0(),y) = [5] > [0] = true() le(s(x),0()) = [5] > [0] = false() p(0()) = [9] > [0] = s(s(0())) p(s(x)) = [1] x + [9] > [1] x + [0] = x Following rules are (at-least) weakly oriented: if(false(),x,y) = [1] x + [0] >= [1] x + [9] = s(minus(p(x),y)) if(true(),x,y) = [1] x + [0] >= [0] = 0() le(s(x),s(y)) = [5] >= [5] = le(x,y) minus(x,y) = [1] x + [0] >= [1] x + [5] = if(le(x,y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: if(false(),x,y) -> s(minus(p(x),y)) if(true(),x,y) -> 0() le(s(x),s(y)) -> le(x,y) minus(x,y) -> if(le(x,y),x,y) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() p(0()) -> s(s(0())) p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1}, uargs(minus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(if) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [12] p(p) = [1] x1 + [13] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: minus(x,y) = [1] x + [12] > [1] x + [0] = if(le(x,y),x,y) Following rules are (at-least) weakly oriented: if(false(),x,y) = [1] x + [0] >= [1] x + [25] = s(minus(p(x),y)) if(true(),x,y) = [1] x + [0] >= [1] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) p(0()) = [14] >= [1] = s(s(0())) p(s(x)) = [1] x + [13] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: if(false(),x,y) -> s(minus(p(x),y)) if(true(),x,y) -> 0() le(s(x),s(y)) -> le(x,y) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() minus(x,y) -> if(le(x,y),x,y) p(0()) -> s(s(0())) p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1}, uargs(minus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [1] p(if) = [1] x1 + [1] x2 + [11] p(le) = [1] p(minus) = [1] x1 + [12] p(p) = [1] x1 + [1] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: if(true(),x,y) = [1] x + [11] > [0] = 0() Following rules are (at-least) weakly oriented: if(false(),x,y) = [1] x + [12] >= [1] x + [13] = s(minus(p(x),y)) le(0(),y) = [1] >= [0] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,y) = [1] x + [12] >= [1] x + [12] = if(le(x,y),x,y) p(0()) = [1] >= [0] = s(s(0())) p(s(x)) = [1] x + [1] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: Failure MAYBE + Considered Problem: - Strict TRS: if(false(),x,y) -> s(minus(p(x),y)) le(s(x),s(y)) -> le(x,y) - Weak TRS: if(true(),x,y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() minus(x,y) -> if(le(x,y),x,y) p(0()) -> s(s(0())) p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE