MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: div(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> 0() if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> 0() if2(true(),x,y) -> s(div(minus(x,y),y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [15] p(div) = [1] x1 + [3] p(false) = [0] p(ge) = [1] x1 + [4] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [1] x1 + [0] p(if2) = [1] x1 + [0] p(minus) = [2] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: ge(x,0()) = [1] x + [4] > [0] = true() ge(0(),s(x)) = [19] > [0] = false() minus(s(x),y) = [2] > [0] = if(gt(s(x),y),x,y) Following rules are (at-least) weakly oriented: div(x,y) = [1] x + [3] >= [1] x + [4] = if1(ge(x,y),x,y) ge(s(x),s(y)) = [1] x + [4] >= [1] x + [4] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [15] = 0() if(true(),x,y) = [0] >= [2] = s(minus(x,y)) if1(false(),x,y) = [0] >= [15] = 0() if1(true(),x,y) = [0] >= [0] = if2(gt(y,0()),x,y) if2(false(),x,y) = [0] >= [15] = 0() if2(true(),x,y) = [0] >= [5] = s(div(minus(x,y),y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: div(x,y) -> if1(ge(x,y),x,y) ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> 0() if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> 0() if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [5] p(false) = [0] p(ge) = [7] p(gt) = [3] p(if) = [1] x1 + [1] p(if1) = [1] x1 + [0] p(if2) = [1] x1 + [0] p(minus) = [4] p(s) = [1] x1 + [4] p(true) = [0] Following rules are strictly oriented: gt(0(),y) = [3] > [0] = false() gt(s(x),0()) = [3] > [0] = true() if(false(),x,y) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: div(x,y) = [1] x + [5] >= [7] = if1(ge(x,y),x,y) ge(x,0()) = [7] >= [0] = true() ge(0(),s(x)) = [7] >= [0] = false() ge(s(x),s(y)) = [7] >= [7] = ge(x,y) gt(s(x),s(y)) = [3] >= [3] = gt(x,y) if(true(),x,y) = [1] >= [8] = s(minus(x,y)) if1(false(),x,y) = [0] >= [0] = 0() if1(true(),x,y) = [0] >= [3] = if2(gt(y,0()),x,y) if2(false(),x,y) = [0] >= [0] = 0() if2(true(),x,y) = [0] >= [13] = s(div(minus(x,y),y)) minus(s(x),y) = [4] >= [4] = if(gt(s(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: div(x,y) -> if1(ge(x,y),x,y) ge(s(x),s(y)) -> ge(x,y) gt(s(x),s(y)) -> gt(x,y) if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> 0() if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> 0() if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() gt(0(),y) -> false() gt(s(x),0()) -> true() if(false(),x,y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] x2 + [5] p(false) = [5] p(ge) = [5] p(gt) = [5] p(if) = [1] x1 + [1] p(if1) = [1] x1 + [1] x3 + [0] p(if2) = [1] x1 + [1] x3 + [2] p(minus) = [6] p(s) = [1] x1 + [4] p(true) = [1] Following rules are strictly oriented: if1(false(),x,y) = [1] y + [5] > [0] = 0() if2(false(),x,y) = [1] y + [7] > [0] = 0() Following rules are (at-least) weakly oriented: div(x,y) = [1] x + [1] y + [5] >= [1] y + [5] = if1(ge(x,y),x,y) ge(x,0()) = [5] >= [1] = true() ge(0(),s(x)) = [5] >= [5] = false() ge(s(x),s(y)) = [5] >= [5] = ge(x,y) gt(0(),y) = [5] >= [5] = false() gt(s(x),0()) = [5] >= [1] = true() gt(s(x),s(y)) = [5] >= [5] = gt(x,y) if(false(),x,y) = [6] >= [0] = 0() if(true(),x,y) = [2] >= [10] = s(minus(x,y)) if1(true(),x,y) = [1] y + [1] >= [1] y + [7] = if2(gt(y,0()),x,y) if2(true(),x,y) = [1] y + [3] >= [1] y + [15] = s(div(minus(x,y),y)) minus(s(x),y) = [6] >= [6] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: div(x,y) -> if1(ge(x,y),x,y) ge(s(x),s(y)) -> ge(x,y) gt(s(x),s(y)) -> gt(x,y) if(true(),x,y) -> s(minus(x,y)) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() gt(0(),y) -> false() gt(s(x),0()) -> true() if(false(),x,y) -> 0() if1(false(),x,y) -> 0() if2(false(),x,y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(div) = [1] x1 + [0] p(false) = [0] p(ge) = [1] x1 + [0] p(gt) = [2] p(if) = [1] x1 + [2] p(if1) = [1] x1 + [1] p(if2) = [1] x1 + [2] p(minus) = [4] p(s) = [1] x1 + [3] p(true) = [0] Following rules are strictly oriented: ge(s(x),s(y)) = [1] x + [3] > [1] x + [0] = ge(x,y) Following rules are (at-least) weakly oriented: div(x,y) = [1] x + [0] >= [1] x + [1] = if1(ge(x,y),x,y) ge(x,0()) = [1] x + [0] >= [0] = true() ge(0(),s(x)) = [1] >= [0] = false() gt(0(),y) = [2] >= [0] = false() gt(s(x),0()) = [2] >= [0] = true() gt(s(x),s(y)) = [2] >= [2] = gt(x,y) if(false(),x,y) = [2] >= [1] = 0() if(true(),x,y) = [2] >= [7] = s(minus(x,y)) if1(false(),x,y) = [1] >= [1] = 0() if1(true(),x,y) = [1] >= [4] = if2(gt(y,0()),x,y) if2(false(),x,y) = [2] >= [1] = 0() if2(true(),x,y) = [2] >= [7] = s(div(minus(x,y),y)) minus(s(x),y) = [4] >= [4] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap MAYBE + Considered Problem: - Strict TRS: div(x,y) -> if1(ge(x,y),x,y) gt(s(x),s(y)) -> gt(x,y) if(true(),x,y) -> s(minus(x,y)) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() if(false(),x,y) -> 0() if1(false(),x,y) -> 0() if2(false(),x,y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [2] x2 + [3] p(false) = [0] p(ge) = [1] x1 + [0] p(gt) = [0] p(if) = [1] x1 + [1] p(if1) = [1] x1 + [2] x3 + [0] p(if2) = [1] x1 + [2] x3 + [1] p(minus) = [1] p(s) = [1] x1 + [2] p(true) = [0] Following rules are strictly oriented: div(x,y) = [1] x + [2] y + [3] > [1] x + [2] y + [0] = if1(ge(x,y),x,y) Following rules are (at-least) weakly oriented: ge(x,0()) = [1] x + [0] >= [0] = true() ge(0(),s(x)) = [0] >= [0] = false() ge(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [1] >= [0] = 0() if(true(),x,y) = [1] >= [3] = s(minus(x,y)) if1(false(),x,y) = [2] y + [0] >= [0] = 0() if1(true(),x,y) = [2] y + [0] >= [2] y + [1] = if2(gt(y,0()),x,y) if2(false(),x,y) = [2] y + [1] >= [0] = 0() if2(true(),x,y) = [2] y + [1] >= [2] y + [6] = s(div(minus(x,y),y)) minus(s(x),y) = [1] >= [1] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap MAYBE + Considered Problem: - Strict TRS: gt(s(x),s(y)) -> gt(x,y) if(true(),x,y) -> s(minus(x,y)) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: div(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() if(false(),x,y) -> 0() if1(false(),x,y) -> 0() if2(false(),x,y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,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(div) = {1}, uargs(if) = {1}, uargs(if1) = {1}, uargs(if2) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] x2 + [7] p(false) = [0] p(ge) = [1] x1 + [1] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [1] x1 + [1] x3 + [6] p(if2) = [1] x1 + [1] x3 + [1] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: if1(true(),x,y) = [1] y + [6] > [1] y + [1] = if2(gt(y,0()),x,y) Following rules are (at-least) weakly oriented: div(x,y) = [1] x + [1] y + [7] >= [1] x + [1] y + [7] = if1(ge(x,y),x,y) ge(x,0()) = [1] x + [1] >= [0] = true() ge(0(),s(x)) = [1] >= [0] = false() ge(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [0] = 0() if(true(),x,y) = [0] >= [0] = s(minus(x,y)) if1(false(),x,y) = [1] y + [6] >= [0] = 0() if2(false(),x,y) = [1] y + [1] >= [0] = 0() if2(true(),x,y) = [1] y + [1] >= [1] y + [7] = s(div(minus(x,y),y)) minus(s(x),y) = [0] >= [0] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: Failure MAYBE + Considered Problem: - Strict TRS: gt(s(x),s(y)) -> gt(x,y) if(true(),x,y) -> s(minus(x,y)) if2(true(),x,y) -> s(div(minus(x,y),y)) - Weak TRS: div(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() if(false(),x,y) -> 0() if1(false(),x,y) -> 0() if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,minus} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE