MAYBE * Step 1: InnermostRuleRemoval MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(s(X)) -> X prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() prod(s(X),Y) -> add(Y,prod(X,Y)) s(X) -> n__s(X) zero(0()) -> true() zero(s(X)) -> false() - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__s,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(s(X),Y) -> s(add(X,Y)) p(s(X)) -> X prod(s(X),Y) -> add(Y,prod(X,Y)) zero(s(X)) -> false() All above mentioned rules can be savely removed. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) add(0(),X) -> X fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() s(X) -> n__s(X) zero(0()) -> true() - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__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) = {1,3}, uargs(n__prod) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [9] p(add) = [2] x2 + [0] p(fact) = [0] p(false) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [5] p(n__prod) = [1] x2 + [0] p(n__s) = [0] p(p) = [0] p(prod) = [1] x2 + [0] p(s) = [0] p(true) = [0] p(zero) = [0] Following rules are strictly oriented: activate(X) = [1] X + [9] > [1] X + [0] = X activate(n__prod(X1,X2)) = [1] X2 + [9] > [1] X2 + [0] = prod(X1,X2) activate(n__s(X)) = [9] > [0] = s(X) Following rules are (at-least) weakly oriented: add(0(),X) = [2] X + [0] >= [1] X + [0] = X fact(X) = [0] >= [5] = if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) = [1] X + [1] Y + [5] >= [1] Y + [9] = activate(Y) if(true(),X,Y) = [1] X + [1] Y + [5] >= [1] X + [9] = activate(X) prod(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__prod(X1,X2) prod(0(),X) = [1] X + [0] >= [0] = 0() s(X) = [0] >= [0] = n__s(X) zero(0()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: add(0(),X) -> X fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() s(X) -> n__s(X) zero(0()) -> true() - Weak TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__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) = {1,3}, uargs(n__prod) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [1] x1 + [2] p(add) = [12] x1 + [1] x2 + [3] p(fact) = [8] x1 + [1] p(false) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(n__prod) = [1] x2 + [4] p(n__s) = [1] x1 + [6] p(p) = [0] p(prod) = [1] x2 + [6] p(s) = [1] x1 + [0] p(true) = [0] p(zero) = [2] x1 + [2] Following rules are strictly oriented: add(0(),X) = [1] X + [27] > [1] X + [0] = X prod(X1,X2) = [1] X2 + [6] > [1] X2 + [4] = n__prod(X1,X2) prod(0(),X) = [1] X + [6] > [2] = 0() zero(0()) = [6] > [0] = true() Following rules are (at-least) weakly oriented: activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__prod(X1,X2)) = [1] X2 + [6] >= [1] X2 + [6] = prod(X1,X2) activate(n__s(X)) = [1] X + [8] >= [1] X + [0] = s(X) fact(X) = [8] X + [1] >= [2] X + [15] = if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [2] = activate(Y) if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [2] = activate(X) s(X) = [1] X + [0] >= [1] X + [6] = n__s(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: fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) s(X) -> n__s(X) - Weak TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) add(0(),X) -> X prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() zero(0()) -> true() - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__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) = {1,3}, uargs(n__prod) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(activate) = [1] x1 + [10] p(add) = [2] x2 + [0] p(fact) = [1] x1 + [0] p(false) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [4] p(n__prod) = [1] x1 + [1] x2 + [1] p(n__s) = [1] x1 + [0] p(p) = [12] p(prod) = [1] x1 + [1] x2 + [6] p(s) = [1] x1 + [10] p(true) = [2] p(zero) = [2] Following rules are strictly oriented: s(X) = [1] X + [10] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [10] >= [1] X + [0] = X activate(n__prod(X1,X2)) = [1] X1 + [1] X2 + [11] >= [1] X1 + [1] X2 + [6] = prod(X1,X2) activate(n__s(X)) = [1] X + [10] >= [1] X + [10] = s(X) add(0(),X) = [2] X + [0] >= [1] X + [0] = X fact(X) = [1] X + [0] >= [1] X + [22] = if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) = [1] X + [1] Y + [4] >= [1] Y + [10] = activate(Y) if(true(),X,Y) = [1] X + [1] Y + [6] >= [1] X + [10] = activate(X) prod(X1,X2) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [1] = n__prod(X1,X2) prod(0(),X) = [1] X + [9] >= [3] = 0() zero(0()) = [2] >= [2] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap MAYBE + Considered Problem: - Strict TRS: fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) - Weak TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) add(0(),X) -> X prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() s(X) -> n__s(X) zero(0()) -> true() - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__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) = {1,3}, uargs(n__prod) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(add) = [2] x2 + [0] p(fact) = [1] x1 + [0] p(false) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [13] p(n__prod) = [1] x1 + [1] x2 + [0] p(n__s) = [0] p(p) = [0] p(prod) = [1] x1 + [1] x2 + [0] p(s) = [0] p(true) = [8] p(zero) = [13] Following rules are strictly oriented: if(false(),X,Y) = [1] X + [1] Y + [13] > [1] Y + [0] = activate(Y) if(true(),X,Y) = [1] X + [1] Y + [21] > [1] X + [0] = activate(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__prod(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = prod(X1,X2) activate(n__s(X)) = [0] >= [0] = s(X) add(0(),X) = [2] X + [0] >= [1] X + [0] = X fact(X) = [1] X + [0] >= [1] X + [26] = if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) prod(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__prod(X1,X2) prod(0(),X) = [1] X + [0] >= [0] = 0() s(X) = [0] >= [0] = n__s(X) zero(0()) = [13] >= [8] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: Failure MAYBE + Considered Problem: - Strict TRS: fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X)))) - Weak TRS: activate(X) -> X activate(n__prod(X1,X2)) -> prod(X1,X2) activate(n__s(X)) -> s(X) add(0(),X) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() s(X) -> n__s(X) zero(0()) -> true() - Signature: {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0 ,false,n__prod,n__s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE