WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(divp) = [0] p(false) = [0] p(not) = [1] x1 + [0] p(prime) = [0] p(prime1) = [8] x2 + [3] p(rem) = [4] p(s) = [3] p(true) = [0] Following rules are strictly oriented: prime1(x,0()) = [3] > [0] = false() prime1(x,s(0())) = [27] > [0] = true() Following rules are (at-least) weakly oriented: divp(x,y) = [0] >= [4] = =(rem(x,y),0()) prime(0()) = [0] >= [0] = false() prime(s(0())) = [0] >= [0] = false() prime(s(s(x))) = [0] >= [27] = prime1(s(s(x)),s(x)) prime1(x,s(s(y))) = [27] >= [27] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Weak TRS: prime1(x,0()) -> false() prime1(x,s(0())) -> true() - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [8] p(=) = [4] p(and) = [1] x1 + [1] x2 + [2] p(divp) = [3] p(false) = [0] p(not) = [1] x1 + [3] p(prime) = [1] x1 + [2] p(prime1) = [1] x1 + [1] x2 + [7] p(rem) = [1] x1 + [1] x2 + [1] p(s) = [9] p(true) = [0] Following rules are strictly oriented: prime(0()) = [10] > [0] = false() prime(s(0())) = [11] > [0] = false() Following rules are (at-least) weakly oriented: divp(x,y) = [3] >= [4] = =(rem(x,y),0()) prime(s(s(x))) = [11] >= [25] = prime1(s(s(x)),s(x)) prime1(x,0()) = [1] x + [15] >= [0] = false() prime1(x,s(0())) = [1] x + [16] >= [0] = true() prime1(x,s(s(y))) = [1] x + [16] >= [1] x + [24] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Weak TRS: prime(0()) -> false() prime(s(0())) -> false() prime1(x,0()) -> false() prime1(x,s(0())) -> true() - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [5] p(divp) = [0] p(false) = [2] p(not) = [1] x1 + [0] p(prime) = [2] x1 + [15] p(prime1) = [2] x1 + [8] p(rem) = [0] p(s) = [8] p(true) = [1] Following rules are strictly oriented: prime(s(s(x))) = [31] > [24] = prime1(s(s(x)),s(x)) Following rules are (at-least) weakly oriented: divp(x,y) = [0] >= [0] = =(rem(x,y),0()) prime(0()) = [15] >= [2] = false() prime(s(0())) = [31] >= [2] = false() prime1(x,0()) = [2] x + [8] >= [2] = false() prime1(x,s(0())) = [2] x + [8] >= [1] = true() prime1(x,s(s(y))) = [2] x + [8] >= [2] x + [13] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Weak TRS: prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(divp) = [1] p(false) = [0] p(not) = [1] x1 + [0] p(prime) = [9] x1 + [3] p(prime1) = [8] x2 + [0] p(rem) = [1] x1 + [0] p(s) = [2] p(true) = [1] Following rules are strictly oriented: divp(x,y) = [1] > [0] = =(rem(x,y),0()) Following rules are (at-least) weakly oriented: prime(0()) = [3] >= [0] = false() prime(s(0())) = [21] >= [0] = false() prime(s(s(x))) = [21] >= [16] = prime1(s(s(x)),s(x)) prime1(x,0()) = [0] >= [0] = false() prime1(x,s(0())) = [16] >= [1] = true() prime1(x,s(s(y))) = [16] >= [17] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Weak TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(=) = [4] p(and) = [1] x1 + [1] x2 + [1] p(divp) = [5] p(false) = [0] p(not) = [1] x1 + [1] p(prime) = [4] x1 + [1] p(prime1) = [4] x2 + [9] p(rem) = [1] x2 + [0] p(s) = [1] x1 + [2] p(true) = [1] Following rules are strictly oriented: prime1(x,s(s(y))) = [4] y + [25] > [4] y + [24] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Following rules are (at-least) weakly oriented: divp(x,y) = [5] >= [4] = =(rem(x,y),0()) prime(0()) = [9] >= [0] = false() prime(s(0())) = [17] >= [0] = false() prime(s(s(x))) = [4] x + [17] >= [4] x + [17] = prime1(s(s(x)),s(x)) prime1(x,0()) = [17] >= [0] = false() prime1(x,s(0())) = [25] >= [1] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))