YES(O(1),O(n^2)) 171.39/60.06 YES(O(1),O(n^2)) 171.39/60.06 171.39/60.06 We are left with following problem, upon which TcT provides the 171.39/60.06 certificate YES(O(1),O(n^2)). 171.39/60.06 171.39/60.06 Strict Trs: 171.39/60.06 { g(c(x, s(y))) -> g(c(s(x), y)) 171.39/60.06 , f(x) -> x 171.39/60.06 , f(c(s(x), y)) -> f(c(x, s(y))) 171.39/60.06 , f(f(x)) -> f(d(f(x))) } 171.39/60.06 Obligation: 171.39/60.06 derivational complexity 171.39/60.06 Answer: 171.39/60.06 YES(O(1),O(n^2)) 171.39/60.06 171.39/60.06 We use the processor 'matrix interpretation of dimension 1' to 171.39/60.06 orient following rules strictly. 171.39/60.06 171.39/60.06 Trs: { f(x) -> x } 171.39/60.06 171.39/60.06 The induced complexity on above rules (modulo remaining rules) is 171.39/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 171.39/60.06 component(s). 171.39/60.06 171.39/60.06 Sub-proof: 171.39/60.06 ---------- 171.39/60.06 TcT has computed the following triangular matrix interpretation. 171.39/60.06 171.39/60.06 [g](x1) = [1] x1 + [0] 171.39/60.06 171.39/60.06 [c](x1, x2) = [1] x1 + [1] x2 + [0] 171.39/60.06 171.39/60.06 [s](x1) = [1] x1 + [0] 171.39/60.06 171.39/60.06 [f](x1) = [1] x1 + [1] 171.39/60.06 171.39/60.06 [d](x1) = [1] x1 + [0] 171.39/60.06 171.39/60.06 The order satisfies the following ordering constraints: 171.39/60.06 171.39/60.06 [g(c(x, s(y)))] = [1] x + [1] y + [0] 171.39/60.06 >= [1] x + [1] y + [0] 171.39/60.06 = [g(c(s(x), y))] 171.39/60.06 171.39/60.06 [f(x)] = [1] x + [1] 171.39/60.06 > [1] x + [0] 171.39/60.06 = [x] 171.39/60.06 171.39/60.06 [f(c(s(x), y))] = [1] x + [1] y + [1] 171.39/60.06 >= [1] x + [1] y + [1] 171.39/60.06 = [f(c(x, s(y)))] 171.39/60.06 171.39/60.06 [f(f(x))] = [1] x + [2] 171.39/60.06 >= [1] x + [2] 171.39/60.06 = [f(d(f(x)))] 171.39/60.06 171.39/60.06 171.39/60.06 We return to the main proof. 171.39/60.06 171.39/60.06 We are left with following problem, upon which TcT provides the 171.39/60.06 certificate YES(O(1),O(n^2)). 171.39/60.06 171.39/60.06 Strict Trs: 171.39/60.06 { g(c(x, s(y))) -> g(c(s(x), y)) 171.39/60.06 , f(c(s(x), y)) -> f(c(x, s(y))) 171.39/60.06 , f(f(x)) -> f(d(f(x))) } 171.39/60.06 Weak Trs: { f(x) -> x } 171.39/60.06 Obligation: 171.39/60.06 derivational complexity 171.39/60.06 Answer: 171.39/60.06 YES(O(1),O(n^2)) 171.39/60.06 171.39/60.06 We use the processor 'matrix interpretation of dimension 2' to 171.39/60.06 orient following rules strictly. 171.39/60.06 171.39/60.06 Trs: { f(c(s(x), y)) -> f(c(x, s(y))) } 171.39/60.06 171.39/60.06 The induced complexity on above rules (modulo remaining rules) is 171.39/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 171.39/60.06 component(s). 171.39/60.06 171.39/60.06 Sub-proof: 171.39/60.06 ---------- 171.39/60.06 TcT has computed the following triangular matrix interpretation. 171.39/60.06 171.39/60.06 [g](x1) = [1 0] x1 + [0] 171.39/60.06 [0 0] [0] 171.39/60.06 171.39/60.06 [c](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 171.39/60.06 [0 1] [0 0] [2] 171.39/60.06 171.39/60.06 [s](x1) = [1 0] x1 + [0] 171.39/60.06 [0 1] [2] 171.39/60.06 171.39/60.06 [f](x1) = [1 1] x1 + [0] 171.39/60.06 [0 1] [0] 171.39/60.06 171.39/60.06 [d](x1) = [1 0] x1 + [0] 171.39/60.06 [0 0] [0] 171.39/60.06 171.39/60.06 The order satisfies the following ordering constraints: 171.39/60.06 171.39/60.06 [g(c(x, s(y)))] = [1 0] x + [1 0] y + [0] 171.39/60.06 [0 0] [0 0] [0] 171.39/60.06 >= [1 0] x + [1 0] y + [0] 171.39/60.06 [0 0] [0 0] [0] 171.39/60.06 = [g(c(s(x), y))] 171.39/60.06 171.39/60.06 [f(x)] = [1 1] x + [0] 171.39/60.06 [0 1] [0] 171.39/60.06 >= [1 0] x + [0] 171.39/60.06 [0 1] [0] 171.39/60.06 = [x] 171.39/60.06 171.39/60.06 [f(c(s(x), y))] = [1 1] x + [1 0] y + [4] 171.39/60.06 [0 1] [0 0] [4] 171.39/60.06 > [1 1] x + [1 0] y + [2] 171.39/60.06 [0 1] [0 0] [2] 171.39/60.06 = [f(c(x, s(y)))] 171.39/60.06 171.39/60.06 [f(f(x))] = [1 2] x + [0] 171.39/60.06 [0 1] [0] 171.39/60.06 >= [1 1] x + [0] 171.39/60.07 [0 0] [0] 171.39/60.07 = [f(d(f(x)))] 171.39/60.07 171.39/60.07 171.39/60.07 We return to the main proof. 171.39/60.07 171.39/60.07 We are left with following problem, upon which TcT provides the 171.39/60.07 certificate YES(O(1),O(n^2)). 171.39/60.07 171.39/60.07 Strict Trs: 171.39/60.07 { g(c(x, s(y))) -> g(c(s(x), y)) 171.39/60.07 , f(f(x)) -> f(d(f(x))) } 171.39/60.07 Weak Trs: 171.39/60.07 { f(x) -> x 171.39/60.07 , f(c(s(x), y)) -> f(c(x, s(y))) } 171.39/60.07 Obligation: 171.39/60.07 derivational complexity 171.39/60.07 Answer: 171.39/60.07 YES(O(1),O(n^2)) 171.39/60.07 171.39/60.07 We use the processor 'matrix interpretation of dimension 2' to 171.39/60.07 orient following rules strictly. 171.39/60.07 171.39/60.07 Trs: { f(f(x)) -> f(d(f(x))) } 171.39/60.07 171.39/60.07 The induced complexity on above rules (modulo remaining rules) is 171.39/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 171.39/60.07 component(s). 171.39/60.07 171.39/60.07 Sub-proof: 171.39/60.07 ---------- 171.39/60.07 TcT has computed the following triangular matrix interpretation. 171.39/60.07 171.39/60.07 [g](x1) = [1 0] x1 + [0] 171.39/60.07 [0 0] [0] 171.39/60.07 171.39/60.07 [c](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 171.39/60.07 [0 0] [0 0] [0] 171.39/60.07 171.39/60.07 [s](x1) = [1 0] x1 + [0] 171.39/60.07 [0 0] [0] 171.39/60.07 171.39/60.07 [f](x1) = [1 2] x1 + [0] 171.39/60.07 [0 1] [2] 171.39/60.07 171.39/60.07 [d](x1) = [1 1] x1 + [1] 171.39/60.07 [0 0] [0] 171.39/60.07 171.39/60.07 The order satisfies the following ordering constraints: 171.39/60.07 171.39/60.07 [g(c(x, s(y)))] = [1 0] x + [1 0] y + [0] 171.39/60.07 [0 0] [0 0] [0] 171.39/60.07 >= [1 0] x + [1 0] y + [0] 171.39/60.07 [0 0] [0 0] [0] 171.39/60.07 = [g(c(s(x), y))] 171.39/60.07 171.39/60.07 [f(x)] = [1 2] x + [0] 171.39/60.07 [0 1] [2] 171.39/60.07 >= [1 0] x + [0] 171.39/60.07 [0 1] [0] 171.39/60.07 = [x] 171.39/60.07 171.39/60.07 [f(c(s(x), y))] = [1 0] x + [1 0] y + [0] 171.39/60.07 [0 0] [0 0] [2] 171.39/60.07 >= [1 0] x + [1 0] y + [0] 171.39/60.07 [0 0] [0 0] [2] 171.39/60.07 = [f(c(x, s(y)))] 171.39/60.07 171.39/60.07 [f(f(x))] = [1 4] x + [4] 171.39/60.07 [0 1] [4] 171.39/60.07 > [1 3] x + [3] 171.39/60.07 [0 0] [2] 171.39/60.07 = [f(d(f(x)))] 171.39/60.07 171.39/60.07 171.39/60.07 We return to the main proof. 171.39/60.07 171.39/60.07 We are left with following problem, upon which TcT provides the 171.39/60.07 certificate YES(O(1),O(n^2)). 171.39/60.07 171.39/60.07 Strict Trs: { g(c(x, s(y))) -> g(c(s(x), y)) } 171.39/60.07 Weak Trs: 171.39/60.07 { f(x) -> x 171.39/60.07 , f(c(s(x), y)) -> f(c(x, s(y))) 171.39/60.07 , f(f(x)) -> f(d(f(x))) } 171.39/60.07 Obligation: 171.39/60.07 derivational complexity 171.39/60.07 Answer: 171.39/60.07 YES(O(1),O(n^2)) 171.39/60.07 171.39/60.07 We use the processor 'matrix interpretation of dimension 2' to 171.39/60.07 orient following rules strictly. 171.39/60.07 171.39/60.07 Trs: { g(c(x, s(y))) -> g(c(s(x), y)) } 171.39/60.07 171.39/60.07 The induced complexity on above rules (modulo remaining rules) is 171.39/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 171.39/60.07 component(s). 171.39/60.07 171.39/60.07 Sub-proof: 171.39/60.07 ---------- 171.39/60.07 TcT has computed the following triangular matrix interpretation. 171.39/60.07 171.39/60.07 [g](x1) = [1 0] x1 + [0] 171.39/60.07 [0 0] [0] 171.39/60.07 171.39/60.07 [c](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 171.39/60.07 [0 1] [0 0] [2] 171.39/60.07 171.39/60.07 [s](x1) = [1 0] x1 + [0] 171.39/60.07 [0 1] [1] 171.39/60.07 171.39/60.07 [f](x1) = [1 2] x1 + [0] 171.39/60.07 [0 1] [1] 171.39/60.07 171.39/60.07 [d](x1) = [1 0] x1 + [0] 171.39/60.07 [0 0] [0] 171.39/60.07 171.39/60.07 The order satisfies the following ordering constraints: 171.39/60.07 171.39/60.07 [g(c(x, s(y)))] = [1 0] x + [1 1] y + [1] 171.39/60.07 [0 0] [0 0] [0] 171.39/60.07 > [1 0] x + [1 1] y + [0] 171.39/60.07 [0 0] [0 0] [0] 171.39/60.07 = [g(c(s(x), y))] 171.39/60.07 171.39/60.07 [f(x)] = [1 2] x + [0] 171.39/60.07 [0 1] [1] 171.39/60.07 >= [1 0] x + [0] 171.39/60.07 [0 1] [0] 171.39/60.07 = [x] 171.39/60.07 171.39/60.07 [f(c(s(x), y))] = [1 2] x + [1 1] y + [6] 171.39/60.07 [0 1] [0 0] [4] 171.39/60.07 > [1 2] x + [1 1] y + [5] 171.39/60.07 [0 1] [0 0] [3] 171.39/60.07 = [f(c(x, s(y)))] 171.39/60.07 171.39/60.07 [f(f(x))] = [1 4] x + [2] 171.39/60.07 [0 1] [2] 171.39/60.07 > [1 2] x + [0] 171.39/60.07 [0 0] [1] 171.39/60.07 = [f(d(f(x)))] 171.39/60.07 171.39/60.07 171.39/60.07 We return to the main proof. 171.39/60.07 171.39/60.07 We are left with following problem, upon which TcT provides the 171.39/60.07 certificate YES(O(1),O(1)). 171.39/60.07 171.39/60.07 Weak Trs: 171.39/60.07 { g(c(x, s(y))) -> g(c(s(x), y)) 171.39/60.07 , f(x) -> x 171.39/60.07 , f(c(s(x), y)) -> f(c(x, s(y))) 171.39/60.07 , f(f(x)) -> f(d(f(x))) } 171.39/60.07 Obligation: 171.39/60.07 derivational complexity 171.39/60.07 Answer: 171.39/60.07 YES(O(1),O(1)) 171.39/60.07 171.39/60.07 Empty rules are trivially bounded 171.39/60.07 171.39/60.07 Hurray, we answered YES(O(1),O(n^2)) 171.39/60.07 EOF