YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84
0.00/0.84 0.00/0.840 CpxTRS0.00/0.84
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.84
↳2 CdtProblem0.00/0.84
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.84
↳4 CdtProblem0.00/0.85
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))0.00/0.85
↳6 CdtProblem0.00/0.85
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.85
↳8 CdtProblem0.00/0.85
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))0.00/0.85
↳10 BOUNDS(O(1), O(1))0.00/0.85
f(x, nil) → g(nil, x) 0.00/0.85
f(x, g(y, z)) → g(f(x, y), z) 0.00/0.85
++(x, nil) → x 0.00/0.85
++(x, g(y, z)) → g(++(x, y), z) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(x, y)) → false 0.00/0.85
mem(nil, y) → false 0.00/0.85
mem(g(x, y), z) → or(=(y, z), mem(x, z)) 0.00/0.85
mem(x, max(x)) → not(null(x)) 0.00/0.85
max(g(g(nil, x), y)) → max'(x, y) 0.00/0.85
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)
Tuples:
f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
S tuples:
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MEM(z0, max(z0)) → c8(NULL(z0)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2)))
K tuples:none
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MEM(z0, max(z0)) → c8(NULL(z0)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2)))
f, ++, null, mem, max
F, ++', MEM, MAX
c1, c3, c7, c8, c10
Tuples:
f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
S tuples:
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
K tuples:none
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
f, ++, null, mem, max
F, ++', MEM, MAX
c1, c3, c7, c10, c8
MEM(z0, max(z0)) → c8
Tuples:
f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
S tuples:
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
K tuples:none
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
f, ++, null, mem, max
F, ++', MEM, MAX
c1, c3, c7, c10, c8
We considered the (Usable) Rules:none
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
The order we found is given by the following interpretation:
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
POL(++'(x1, x2)) = [3]x2 0.00/0.85
POL(F(x1, x2)) = [3]x2 0.00/0.85
POL(MAX(x1)) = [5]x1 0.00/0.85
POL(MEM(x1, x2)) = [3] + [3]x1 + [3]x2 0.00/0.85
POL(c1(x1)) = x1 0.00/0.85
POL(c10(x1)) = x1 0.00/0.85
POL(c3(x1)) = x1 0.00/0.85
POL(c7(x1)) = x1 0.00/0.85
POL(c8) = 0 0.00/0.85
POL(g(x1, x2)) = [3] + x1 + x2 0.00/0.85
POL(max(x1)) = [3] 0.00/0.85
POL(u) = [3]
Tuples:
f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
S tuples:none
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
Defined Rule Symbols:
F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
f, ++, null, mem, max
F, ++', MEM, MAX
c1, c3, c7, c10, c8