YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict 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() , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) , divp(x, y) -> =(rem(x, y), 0()) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { prime^#(0()) -> c_1() , prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , prime1^#(x, s(0())) -> c_5() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(0()) -> c_1() , prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , prime1^#(x, s(0())) -> c_5() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Strict 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() , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) , divp(x, y) -> =(rem(x, y), 0()) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(0()) -> c_1() , prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , prime1^#(x, s(0())) -> c_5() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [0] = [0] [0] [s](x1) = [0] [0] [prime^#](x1) = [0] [0] [c_1] = [1] [0] [c_2] = [1] [0] [c_3](x1) = [1 0] x1 + [1] [0 1] [0] [prime1^#](x1, x2) = [0] [0] [c_4] = [1] [0] [c_5] = [1] [0] [c_6](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] [divp^#](x1, x2) = [0 0] x2 + [1] [2 2] [0] [c_7](x1, x2) = [0 0] x2 + [0] [1 1] [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(0())] = [0] [0] ? [1] [0] = [c_1()] [prime^#(s(0()))] = [0] [0] ? [1] [0] = [c_2()] [prime^#(s(s(x)))] = [0] [0] ? [1] [0] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, 0())] = [0] [0] ? [1] [0] = [c_4()] [prime1^#(x, s(0()))] = [0] [0] ? [1] [0] = [c_5()] [prime1^#(x, s(s(y)))] = [0] [0] ? [0 0] x + [1] [2 2] [0] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0 0] y + [1] [2 2] [0] > [0 0] y + [0] [1 1] [0] = [c_7(x, y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { prime^#(0()) -> c_1() , prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , prime1^#(x, s(0())) -> c_5() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(?,O(n^1)) We employ 'linear path analysis' using the following approximated dependency graph: ->{3,7,6} [ ? ] | |->{1} [ YES(O(1),O(n^1)) ] | |->{2} [ YES(O(1),O(n^1)) ] | |->{4} [ YES(O(1),O(n^1)) ] | `->{5} [ YES(O(1),O(n^1)) ] Here dependency-pairs are as follows: Strict DPs: { 1: prime^#(0()) -> c_1() , 2: prime^#(s(0())) -> c_2() , 3: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , 4: prime1^#(x, 0()) -> c_4() , 5: prime1^#(x, s(0())) -> c_5() , 6: prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { 7: divp^#(x, y) -> c_7(x, y) } * Path {3,7,6}->{1}: YES(O(1),O(n^1)) ----------------------------------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(0()) -> c_1() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime^#(0()) -> c_1() , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [2] [false] = [0] [s](x1) = [2] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [4] x1 + [0] [c_1] = [6] [c_2] = [0] [c_3](x1) = [4] x1 + [7] [prime1^#](x1, x2) = [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [4] x1 + [4] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(0())] = [8] > [6] = [c_1()] [prime^#(s(s(x)))] = [8] > [7] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [0] >= [0] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(0()) -> c_1() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(0()) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [0] [false] = [0] [s](x1) = [1] x1 + [2] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [2] x1 + [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [1] [prime1^#](x1, x2) = [1] x2 + [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [1] x1 + [1] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [2] x + [15] > [1] x + [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [1] y + [4] > [1] y + [2] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded * Path {3,7,6}->{2}: YES(O(1),O(n^1)) ----------------------------------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime^#(s(0())) -> c_2() , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [7] [false] = [0] [s](x1) = [0] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [4] [c_1] = [0] [c_2] = [2] [c_3](x1) = [4] x1 + [3] [prime1^#](x1, x2) = [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [4] x1 + [4] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(0()))] = [4] > [2] = [c_2()] [prime^#(s(s(x)))] = [4] > [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [0] >= [0] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(0())) -> c_2() , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(s(0())) -> c_2() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [0] [false] = [0] [s](x1) = [1] x1 + [2] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [2] x1 + [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [1] [prime1^#](x1, x2) = [1] x2 + [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [1] x1 + [1] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [2] x + [15] > [1] x + [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [1] y + [4] > [1] y + [2] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded * Path {3,7,6}->{4}: YES(O(1),O(n^1)) ----------------------------------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , 2: prime1^#(x, 0()) -> c_4() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [1] [false] = [0] [s](x1) = [0] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [4] x1 + [3] [prime1^#](x1, x2) = [2] x2 + [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [2] x1 + [4] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [7] > [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, 0())] = [2] > [0] = [c_4()] [prime1^#(x, s(s(y)))] = [0] >= [0] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, 0()) -> c_4() , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime1^#(x, 0()) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [0] [false] = [0] [s](x1) = [1] x1 + [2] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [2] x1 + [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [1] [prime1^#](x1, x2) = [1] x2 + [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [1] x1 + [1] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [2] x + [15] > [1] x + [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [1] y + [4] > [1] y + [2] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded * Path {3,7,6}->{5}: YES(O(1),O(n^1)) ----------------------------------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(0())) -> c_5() , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , 2: prime1^#(x, s(0())) -> c_5() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [7] [false] = [0] [s](x1) = [0] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [2] x1 + [1] [prime1^#](x1, x2) = [1] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [4] x1 + [1] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [7] > [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(0()))] = [1] > [0] = [c_5()] [prime1^#(x, s(s(y)))] = [1] >= [1] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(0())) -> c_5() , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime1^#(x, s(0())) -> c_5() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [prime](x1) = [7] x1 + [0] [0] = [0] [false] = [0] [s](x1) = [1] x1 + [2] [prime1](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [not](x1) = [1] x1 + [0] [divp](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] x2 + [0] [rem](x1, x2) = [1] x1 + [1] x2 + [0] [prime^#](x1) = [2] x1 + [7] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [1] [prime1^#](x1, x2) = [1] x2 + [0] [c_4] = [0] [c_5] = [0] [c_6](x1, x2) = [1] x1 + [1] x2 + [0] [divp^#](x1, x2) = [0] [c_7](x1, x2) = [0] The following symbols are considered usable {prime^#, prime1^#, divp^#} The order satisfies the following ordering constraints: [prime^#(s(s(x)))] = [2] x + [15] > [1] x + [3] = [c_3(prime1^#(s(s(x)), s(x)))] [prime1^#(x, s(s(y)))] = [1] y + [4] > [1] y + [2] = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] [divp^#(x, y)] = [0] >= [0] = [c_7(x, y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) , prime1^#(x, s(s(y))) -> c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) , divp^#(x, y) -> c_7(x, y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))