YES(O(1),O(n^1)) 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict Trs: 0.00/0.97 { minus(X, 0()) -> X 0.00/0.97 , minus(s(X), s(Y)) -> p(minus(X, Y)) 0.00/0.97 , p(s(X)) -> X 0.00/0.97 , div(0(), s(Y)) -> 0() 0.00/0.97 , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 Obligation: 0.00/0.97 runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 The weightgap principle applies (using the following nonconstant 0.00/0.97 growth matrix-interpretation) 0.00/0.97 0.00/0.97 The following argument positions are usable: 0.00/0.97 Uargs(minus) = {1}, Uargs(s) = {1}, Uargs(p) = {1}, 0.00/0.97 Uargs(div) = {1} 0.00/0.97 0.00/0.97 TcT has computed the following matrix interpretation satisfying 0.00/0.97 not(EDA) and not(IDA(1)). 0.00/0.97 0.00/0.97 [minus](x1, x2) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [0] = [0] 0.00/0.97 0.00/0.97 [s](x1) = [1] x1 + [1] 0.00/0.97 0.00/0.97 [p](x1) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [div](x1, x2) = [1] x1 + [0] 0.00/0.97 0.00/0.97 The order satisfies the following ordering constraints: 0.00/0.97 0.00/0.97 [minus(X, 0())] = [1] X + [0] 0.00/0.97 >= [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [minus(s(X), s(Y))] = [1] X + [1] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [p(minus(X, Y))] 0.00/0.97 0.00/0.97 [p(s(X))] = [1] X + [1] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [div(0(), s(Y))] = [0] 0.00/0.97 >= [0] 0.00/0.97 = [0()] 0.00/0.97 0.00/0.97 [div(s(X), s(Y))] = [1] X + [1] 0.00/0.97 >= [1] X + [1] 0.00/0.97 = [s(div(minus(X, Y), s(Y)))] 0.00/0.97 0.00/0.97 0.00/0.97 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict Trs: 0.00/0.97 { minus(X, 0()) -> X 0.00/0.97 , div(0(), s(Y)) -> 0() 0.00/0.97 , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 Weak Trs: 0.00/0.97 { minus(s(X), s(Y)) -> p(minus(X, Y)) 0.00/0.97 , p(s(X)) -> X } 0.00/0.97 Obligation: 0.00/0.97 runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 The weightgap principle applies (using the following nonconstant 0.00/0.97 growth matrix-interpretation) 0.00/0.97 0.00/0.97 The following argument positions are usable: 0.00/0.97 Uargs(minus) = {1}, Uargs(s) = {1}, Uargs(p) = {1}, 0.00/0.97 Uargs(div) = {1} 0.00/0.97 0.00/0.97 TcT has computed the following matrix interpretation satisfying 0.00/0.97 not(EDA) and not(IDA(1)). 0.00/0.97 0.00/0.97 [minus](x1, x2) = [1] x1 + [1] 0.00/0.97 0.00/0.97 [0] = [4] 0.00/0.97 0.00/0.97 [s](x1) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [p](x1) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [div](x1, x2) = [1] x1 + [0] 0.00/0.97 0.00/0.97 The order satisfies the following ordering constraints: 0.00/0.97 0.00/0.97 [minus(X, 0())] = [1] X + [1] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [minus(s(X), s(Y))] = [1] X + [1] 0.00/0.97 >= [1] X + [1] 0.00/0.97 = [p(minus(X, Y))] 0.00/0.97 0.00/0.97 [p(s(X))] = [1] X + [0] 0.00/0.97 >= [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [div(0(), s(Y))] = [4] 0.00/0.97 >= [4] 0.00/0.97 = [0()] 0.00/0.97 0.00/0.97 [div(s(X), s(Y))] = [1] X + [0] 0.00/0.97 ? [1] X + [1] 0.00/0.97 = [s(div(minus(X, Y), s(Y)))] 0.00/0.97 0.00/0.97 0.00/0.97 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict Trs: 0.00/0.97 { div(0(), s(Y)) -> 0() 0.00/0.97 , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 Weak Trs: 0.00/0.97 { minus(X, 0()) -> X 0.00/0.97 , minus(s(X), s(Y)) -> p(minus(X, Y)) 0.00/0.97 , p(s(X)) -> X } 0.00/0.97 Obligation: 0.00/0.97 runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 The weightgap principle applies (using the following nonconstant 0.00/0.97 growth matrix-interpretation) 0.00/0.97 0.00/0.97 The following argument positions are usable: 0.00/0.97 Uargs(minus) = {1}, Uargs(s) = {1}, Uargs(p) = {1}, 0.00/0.97 Uargs(div) = {1} 0.00/0.97 0.00/0.97 TcT has computed the following matrix interpretation satisfying 0.00/0.97 not(EDA) and not(IDA(1)). 0.00/0.97 0.00/0.97 [minus](x1, x2) = [1] x1 + [4] 0.00/0.97 0.00/0.97 [0] = [4] 0.00/0.97 0.00/0.97 [s](x1) = [1] x1 + [4] 0.00/0.97 0.00/0.97 [p](x1) = [1] x1 + [4] 0.00/0.97 0.00/0.97 [div](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.97 0.00/0.97 The order satisfies the following ordering constraints: 0.00/0.97 0.00/0.97 [minus(X, 0())] = [1] X + [4] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [minus(s(X), s(Y))] = [1] X + [8] 0.00/0.97 >= [1] X + [8] 0.00/0.97 = [p(minus(X, Y))] 0.00/0.97 0.00/0.97 [p(s(X))] = [1] X + [8] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [div(0(), s(Y))] = [1] Y + [8] 0.00/0.97 > [4] 0.00/0.97 = [0()] 0.00/0.97 0.00/0.97 [div(s(X), s(Y))] = [1] X + [1] Y + [8] 0.00/0.97 ? [1] X + [1] Y + [12] 0.00/0.97 = [s(div(minus(X, Y), s(Y)))] 0.00/0.97 0.00/0.97 0.00/0.97 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict Trs: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 Weak Trs: 0.00/0.97 { minus(X, 0()) -> X 0.00/0.97 , minus(s(X), s(Y)) -> p(minus(X, Y)) 0.00/0.97 , p(s(X)) -> X 0.00/0.97 , div(0(), s(Y)) -> 0() } 0.00/0.97 Obligation: 0.00/0.97 runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.97 orient following rules strictly. 0.00/0.97 0.00/0.97 Trs: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 0.00/0.97 The induced complexity on above rules (modulo remaining rules) is 0.00/0.97 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.97 component(s). 0.00/0.97 0.00/0.97 Sub-proof: 0.00/0.97 ---------- 0.00/0.97 The following argument positions are usable: 0.00/0.97 Uargs(minus) = {1}, Uargs(s) = {1}, Uargs(p) = {1}, 0.00/0.97 Uargs(div) = {1} 0.00/0.97 0.00/0.97 TcT has computed the following constructor-based matrix 0.00/0.97 interpretation satisfying not(EDA). 0.00/0.97 0.00/0.97 [minus](x1, x2) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [0] = [0] 0.00/0.97 0.00/0.97 [s](x1) = [1] x1 + [4] 0.00/0.97 0.00/0.97 [p](x1) = [1] x1 + [0] 0.00/0.97 0.00/0.97 [div](x1, x2) = [2] x1 + [0] 0.00/0.97 0.00/0.97 The order satisfies the following ordering constraints: 0.00/0.97 0.00/0.97 [minus(X, 0())] = [1] X + [0] 0.00/0.97 >= [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [minus(s(X), s(Y))] = [1] X + [4] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [p(minus(X, Y))] 0.00/0.97 0.00/0.97 [p(s(X))] = [1] X + [4] 0.00/0.97 > [1] X + [0] 0.00/0.97 = [X] 0.00/0.97 0.00/0.97 [div(0(), s(Y))] = [0] 0.00/0.97 >= [0] 0.00/0.97 = [0()] 0.00/0.97 0.00/0.97 [div(s(X), s(Y))] = [2] X + [8] 0.00/0.97 > [2] X + [4] 0.00/0.97 = [s(div(minus(X, Y), s(Y)))] 0.00/0.97 0.00/0.97 0.00/0.97 We return to the main proof. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(1)). 0.00/0.97 0.00/0.97 Weak Trs: 0.00/0.97 { minus(X, 0()) -> X 0.00/0.97 , minus(s(X), s(Y)) -> p(minus(X, Y)) 0.00/0.97 , p(s(X)) -> X 0.00/0.97 , div(0(), s(Y)) -> 0() 0.00/0.97 , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 0.00/0.97 Obligation: 0.00/0.97 runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(1)) 0.00/0.97 0.00/0.97 Empty rules are trivially bounded 0.00/0.97 0.00/0.97 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.97 EOF