YES(O(1),O(n^2)) 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(n^2)). 167.92/60.07 167.92/60.07 Strict Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , f(x, g(y, z)) -> g(f(x, y), z) 167.92/60.07 , rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We use the processor 'matrix interpretation of dimension 1' to 167.92/60.07 orient following rules strictly. 167.92/60.07 167.92/60.07 Trs: 167.92/60.07 { rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) } 167.92/60.07 167.92/60.07 The induced complexity on above rules (modulo remaining rules) is 167.92/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.92/60.07 component(s). 167.92/60.07 167.92/60.07 Sub-proof: 167.92/60.07 ---------- 167.92/60.07 TcT has computed the following triangular matrix interpretation. 167.92/60.07 167.92/60.07 [norm](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [nil] = [0] 167.92/60.07 167.92/60.07 [0] = [0] 167.92/60.07 167.92/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 [s](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 [rem](x1, x2) = [1] x1 + [1] x2 + [1] 167.92/60.07 167.92/60.07 The order satisfies the following ordering constraints: 167.92/60.07 167.92/60.07 [norm(nil())] = [0] 167.92/60.07 >= [0] 167.92/60.07 = [0()] 167.92/60.07 167.92/60.07 [norm(g(x, y))] = [1] x + [1] y + [0] 167.92/60.07 >= [1] x + [0] 167.92/60.07 = [s(norm(x))] 167.92/60.07 167.92/60.07 [f(x, nil())] = [1] x + [0] 167.92/60.07 >= [1] x + [0] 167.92/60.07 = [g(nil(), x)] 167.92/60.07 167.92/60.07 [f(x, g(y, z))] = [1] x + [1] y + [1] z + [0] 167.92/60.07 >= [1] x + [1] y + [1] z + [0] 167.92/60.07 = [g(f(x, y), z)] 167.92/60.07 167.92/60.07 [rem(nil(), y)] = [1] y + [1] 167.92/60.07 > [0] 167.92/60.07 = [nil()] 167.92/60.07 167.92/60.07 [rem(g(x, y), 0())] = [1] x + [1] y + [1] 167.92/60.07 > [1] x + [1] y + [0] 167.92/60.07 = [g(x, y)] 167.92/60.07 167.92/60.07 [rem(g(x, y), s(z))] = [1] x + [1] y + [1] z + [1] 167.92/60.07 >= [1] x + [1] z + [1] 167.92/60.07 = [rem(x, z)] 167.92/60.07 167.92/60.07 167.92/60.07 We return to the main proof. 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(n^2)). 167.92/60.07 167.92/60.07 Strict Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , f(x, g(y, z)) -> g(f(x, y), z) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Weak Trs: 167.92/60.07 { rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We use the processor 'matrix interpretation of dimension 1' to 167.92/60.07 orient following rules strictly. 167.92/60.07 167.92/60.07 Trs: { norm(nil()) -> 0() } 167.92/60.07 167.92/60.07 The induced complexity on above rules (modulo remaining rules) is 167.92/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.92/60.07 component(s). 167.92/60.07 167.92/60.07 Sub-proof: 167.92/60.07 ---------- 167.92/60.07 TcT has computed the following triangular matrix interpretation. 167.92/60.07 167.92/60.07 [norm](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [nil] = [1] 167.92/60.07 167.92/60.07 [0] = [0] 167.92/60.07 167.92/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 [s](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 [rem](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 The order satisfies the following ordering constraints: 167.92/60.07 167.92/60.07 [norm(nil())] = [1] 167.92/60.07 > [0] 167.92/60.07 = [0()] 167.92/60.07 167.92/60.07 [norm(g(x, y))] = [1] x + [1] y + [0] 167.92/60.07 >= [1] x + [0] 167.92/60.07 = [s(norm(x))] 167.92/60.07 167.92/60.07 [f(x, nil())] = [1] x + [1] 167.92/60.07 >= [1] x + [1] 167.92/60.07 = [g(nil(), x)] 167.92/60.07 167.92/60.07 [f(x, g(y, z))] = [1] x + [1] y + [1] z + [0] 167.92/60.07 >= [1] x + [1] y + [1] z + [0] 167.92/60.07 = [g(f(x, y), z)] 167.92/60.07 167.92/60.07 [rem(nil(), y)] = [1] y + [1] 167.92/60.07 >= [1] 167.92/60.07 = [nil()] 167.92/60.07 167.92/60.07 [rem(g(x, y), 0())] = [1] x + [1] y + [0] 167.92/60.07 >= [1] x + [1] y + [0] 167.92/60.07 = [g(x, y)] 167.92/60.07 167.92/60.07 [rem(g(x, y), s(z))] = [1] x + [1] y + [1] z + [0] 167.92/60.07 >= [1] x + [1] z + [0] 167.92/60.07 = [rem(x, z)] 167.92/60.07 167.92/60.07 167.92/60.07 We return to the main proof. 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(n^2)). 167.92/60.07 167.92/60.07 Strict Trs: 167.92/60.07 { norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , f(x, g(y, z)) -> g(f(x, y), z) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Weak Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We use the processor 'matrix interpretation of dimension 1' to 167.92/60.07 orient following rules strictly. 167.92/60.07 167.92/60.07 Trs: { f(x, nil()) -> g(nil(), x) } 167.92/60.07 167.92/60.07 The induced complexity on above rules (modulo remaining rules) is 167.92/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.92/60.07 component(s). 167.92/60.07 167.92/60.07 Sub-proof: 167.92/60.07 ---------- 167.92/60.07 TcT has computed the following triangular matrix interpretation. 167.92/60.07 167.92/60.07 [norm](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [nil] = [0] 167.92/60.07 167.92/60.07 [0] = [0] 167.92/60.07 167.92/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 [s](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [2] 167.92/60.07 167.92/60.07 [rem](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 The order satisfies the following ordering constraints: 167.92/60.07 167.92/60.07 [norm(nil())] = [0] 167.92/60.07 >= [0] 167.92/60.07 = [0()] 167.92/60.07 167.92/60.07 [norm(g(x, y))] = [1] x + [1] y + [0] 167.92/60.07 >= [1] x + [0] 167.92/60.07 = [s(norm(x))] 167.92/60.07 167.92/60.07 [f(x, nil())] = [1] x + [2] 167.92/60.07 > [1] x + [0] 167.92/60.07 = [g(nil(), x)] 167.92/60.07 167.92/60.07 [f(x, g(y, z))] = [1] x + [1] y + [1] z + [2] 167.92/60.07 >= [1] x + [1] y + [1] z + [2] 167.92/60.07 = [g(f(x, y), z)] 167.92/60.07 167.92/60.07 [rem(nil(), y)] = [1] y + [0] 167.92/60.07 >= [0] 167.92/60.07 = [nil()] 167.92/60.07 167.92/60.07 [rem(g(x, y), 0())] = [1] x + [1] y + [0] 167.92/60.07 >= [1] x + [1] y + [0] 167.92/60.07 = [g(x, y)] 167.92/60.07 167.92/60.07 [rem(g(x, y), s(z))] = [1] x + [1] y + [1] z + [0] 167.92/60.07 >= [1] x + [1] z + [0] 167.92/60.07 = [rem(x, z)] 167.92/60.07 167.92/60.07 167.92/60.07 We return to the main proof. 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(n^2)). 167.92/60.07 167.92/60.07 Strict Trs: 167.92/60.07 { norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, g(y, z)) -> g(f(x, y), z) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Weak Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We use the processor 'matrix interpretation of dimension 1' to 167.92/60.07 orient following rules strictly. 167.92/60.07 167.92/60.07 Trs: 167.92/60.07 { norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 167.92/60.07 The induced complexity on above rules (modulo remaining rules) is 167.92/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.92/60.07 component(s). 167.92/60.07 167.92/60.07 Sub-proof: 167.92/60.07 ---------- 167.92/60.07 TcT has computed the following triangular matrix interpretation. 167.92/60.07 167.92/60.07 [norm](x1) = [1] x1 + [2] 167.92/60.07 167.92/60.07 [nil] = [2] 167.92/60.07 167.92/60.07 [0] = [2] 167.92/60.07 167.92/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [2] 167.92/60.07 167.92/60.07 [s](x1) = [1] x1 + [0] 167.92/60.07 167.92/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [2] 167.92/60.07 167.92/60.07 [rem](x1, x2) = [1] x1 + [1] x2 + [0] 167.92/60.07 167.92/60.07 The order satisfies the following ordering constraints: 167.92/60.07 167.92/60.07 [norm(nil())] = [4] 167.92/60.07 > [2] 167.92/60.07 = [0()] 167.92/60.07 167.92/60.07 [norm(g(x, y))] = [1] x + [1] y + [4] 167.92/60.07 > [1] x + [2] 167.92/60.07 = [s(norm(x))] 167.92/60.07 167.92/60.07 [f(x, nil())] = [1] x + [4] 167.92/60.07 >= [1] x + [4] 167.92/60.07 = [g(nil(), x)] 167.92/60.07 167.92/60.07 [f(x, g(y, z))] = [1] x + [1] y + [1] z + [4] 167.92/60.07 >= [1] x + [1] y + [1] z + [4] 167.92/60.07 = [g(f(x, y), z)] 167.92/60.07 167.92/60.07 [rem(nil(), y)] = [1] y + [2] 167.92/60.07 >= [2] 167.92/60.07 = [nil()] 167.92/60.07 167.92/60.07 [rem(g(x, y), 0())] = [1] x + [1] y + [4] 167.92/60.07 > [1] x + [1] y + [2] 167.92/60.07 = [g(x, y)] 167.92/60.07 167.92/60.07 [rem(g(x, y), s(z))] = [1] x + [1] y + [1] z + [2] 167.92/60.07 > [1] x + [1] z + [0] 167.92/60.07 = [rem(x, z)] 167.92/60.07 167.92/60.07 167.92/60.07 We return to the main proof. 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(n^2)). 167.92/60.07 167.92/60.07 Strict Trs: { f(x, g(y, z)) -> g(f(x, y), z) } 167.92/60.07 Weak Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(n^2)) 167.92/60.07 167.92/60.07 We use the processor 'matrix interpretation of dimension 2' to 167.92/60.07 orient following rules strictly. 167.92/60.07 167.92/60.07 Trs: { f(x, g(y, z)) -> g(f(x, y), z) } 167.92/60.07 167.92/60.07 The induced complexity on above rules (modulo remaining rules) is 167.92/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 167.92/60.07 component(s). 167.92/60.07 167.92/60.07 Sub-proof: 167.92/60.07 ---------- 167.92/60.07 TcT has computed the following triangular matrix interpretation. 167.92/60.07 167.92/60.07 [norm](x1) = [1 0] x1 + [0] 167.92/60.07 [0 0] [0] 167.92/60.07 167.92/60.07 [nil] = [0] 167.92/60.07 [0] 167.92/60.07 167.92/60.07 [0] = [0] 167.92/60.07 [0] 167.92/60.07 167.92/60.07 [g](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 167.92/60.07 [0 1] [0 1] [1] 167.92/60.07 167.92/60.07 [s](x1) = [1 0] x1 + [0] 167.92/60.07 [0 0] [0] 167.92/60.07 167.92/60.07 [f](x1, x2) = [1 2] x1 + [1 1] x2 + [0] 167.92/60.07 [0 1] [0 1] [1] 167.92/60.07 167.92/60.07 [rem](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 167.92/60.07 [0 1] [0 0] [2] 167.92/60.07 167.92/60.07 The order satisfies the following ordering constraints: 167.92/60.07 167.92/60.07 [norm(nil())] = [0] 167.92/60.07 [0] 167.92/60.07 >= [0] 167.92/60.07 [0] 167.92/60.07 = [0()] 167.92/60.07 167.92/60.07 [norm(g(x, y))] = [1 0] x + [1 1] y + [0] 167.92/60.07 [0 0] [0 0] [0] 167.92/60.07 >= [1 0] x + [0] 167.92/60.07 [0 0] [0] 167.92/60.07 = [s(norm(x))] 167.92/60.07 167.92/60.07 [f(x, nil())] = [1 2] x + [0] 167.92/60.07 [0 1] [1] 167.92/60.07 >= [1 1] x + [0] 167.92/60.07 [0 1] [1] 167.92/60.07 = [g(nil(), x)] 167.92/60.07 167.92/60.07 [f(x, g(y, z))] = [1 2] x + [1 1] y + [1 2] z + [1] 167.92/60.07 [0 1] [0 1] [0 1] [2] 167.92/60.07 > [1 2] x + [1 1] y + [1 1] z + [0] 167.92/60.07 [0 1] [0 1] [0 1] [2] 167.92/60.07 = [g(f(x, y), z)] 167.92/60.07 167.92/60.07 [rem(nil(), y)] = [1 0] y + [0] 167.92/60.07 [0 0] [2] 167.92/60.07 >= [0] 167.92/60.07 [0] 167.92/60.07 = [nil()] 167.92/60.07 167.92/60.07 [rem(g(x, y), 0())] = [1 0] x + [1 1] y + [0] 167.92/60.07 [0 1] [0 1] [3] 167.92/60.07 >= [1 0] x + [1 1] y + [0] 167.92/60.07 [0 1] [0 1] [1] 167.92/60.07 = [g(x, y)] 167.92/60.07 167.92/60.07 [rem(g(x, y), s(z))] = [1 0] x + [1 1] y + [1 0] z + [0] 167.92/60.07 [0 1] [0 1] [0 0] [3] 167.92/60.07 >= [1 0] x + [1 0] z + [0] 167.92/60.07 [0 1] [0 0] [2] 167.92/60.07 = [rem(x, z)] 167.92/60.07 167.92/60.07 167.92/60.07 We return to the main proof. 167.92/60.07 167.92/60.07 We are left with following problem, upon which TcT provides the 167.92/60.07 certificate YES(O(1),O(1)). 167.92/60.07 167.92/60.07 Weak Trs: 167.92/60.07 { norm(nil()) -> 0() 167.92/60.07 , norm(g(x, y)) -> s(norm(x)) 167.92/60.07 , f(x, nil()) -> g(nil(), x) 167.92/60.07 , f(x, g(y, z)) -> g(f(x, y), z) 167.92/60.07 , rem(nil(), y) -> nil() 167.92/60.07 , rem(g(x, y), 0()) -> g(x, y) 167.92/60.07 , rem(g(x, y), s(z)) -> rem(x, z) } 167.92/60.07 Obligation: 167.92/60.07 derivational complexity 167.92/60.07 Answer: 167.92/60.07 YES(O(1),O(1)) 167.92/60.07 167.92/60.07 Empty rules are trivially bounded 167.92/60.07 167.92/60.07 Hurray, we answered YES(O(1),O(n^2)) 167.92/60.08 EOF