(0) Obligation:
Clauses:
append([], Ys, Ys).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
sublist(X, Y) :- ','(append(P, X1, Y), append(X2, X, P)).
Query: sublist(g,g)
(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(2) Obligation:
Complex Complexity Dependency Tuples Problem
MAX
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
Complex Complexity Dependency Tuples Problem
MULTIPLY
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
(3) MaxProof (BOTH BOUNDS(ID, ID) transformation)
Took the maximum complexity of the problems.
(4) Complex Obligation (MAX)
(5) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
(6) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(7) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(U10'(f12_in(z3, z0), z2, z3, z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
S tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F1_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, U2', F11_IN, U5', F12_IN, F1_IN, U9'
Compound Symbols:
c2, c3, c1, c4, c2, c5, c7
(8) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
(9) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F1_IN(z0, z1) → c7
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(10) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F1_IN(z0, z1) → c7
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U9'(f11_out1(z0, z1), z2, z3) → c7
U5'(f11_out1(z0, z1), z2, z3) → c4
Now S is empty
(11) BOUNDS(O(1), O(1))
(12) Obligation:
Complex Complexity Dependency Tuples Problem
MULTIPLY
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
(13) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)
Multiplied the complexity of the problems.
(14) Complex Obligation (MULT)
(15) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
(16) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(17) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(U10'(f12_in(z3, z0), z2, z3, z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
S tuples:
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, U2', F11_IN, U5', F12_IN, F1_IN, U9'
Compound Symbols:
c2, c3, c1, c4, c2, c5, c7
(18) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
(19) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(20) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U5'(f11_out1(z0, z1), z2, z3) → c4
We considered the (Usable) Rules:
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
And the Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = 0
POL(F11_IN(x1)) = [2]
POL(F12_IN(x1, x2)) = [1] + x1
POL(F1_IN(x1, x2)) = [2] + [2]x1 + [3]x2
POL(F7_IN(x1, x2)) = [2] + [3]x1 + [2]x2
POL(U2'(x1, x2, x3)) = [2] + x2
POL(U4(x1, x2)) = 0
POL(U5'(x1, x2, x3)) = [2] + [3]x2 + [2]x3
POL(U9'(x1, x2, x3)) = [2] + [2]x2 + x3
POL([]) = 0
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3) = 0
POL(c4) = 0
POL(c5(x1)) = x1
POL(c7) = 0
POL(c7(x1)) = x1
POL(f11_in(x1)) = 0
POL(f11_out1(x1, x2)) = 0
(21) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
K tuples:
U5'(f11_out1(z0, z1), z2, z3) → c4
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(22) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
We considered the (Usable) Rules:
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
And the Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [3] + x2
POL(F11_IN(x1)) = 0
POL(F12_IN(x1, x2)) = 0
POL(F1_IN(x1, x2)) = [3] + [2]x1 + [3]x2
POL(F7_IN(x1, x2)) = [2] + [3]x1 + [2]x2
POL(U2'(x1, x2, x3)) = [1] + [3]x2 + [2]x3
POL(U4(x1, x2)) = [2]
POL(U5'(x1, x2, x3)) = x3
POL(U9'(x1, x2, x3)) = 0
POL([]) = 0
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3) = 0
POL(c4) = 0
POL(c5(x1)) = x1
POL(c7) = 0
POL(c7(x1)) = x1
POL(f11_in(x1)) = [3] + [2]x1
POL(f11_out1(x1, x2)) = [2]
(23) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
K tuples:
U5'(f11_out1(z0, z1), z2, z3) → c4
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(24) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
We considered the (Usable) Rules:
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
And the Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [1] + x2
POL(F11_IN(x1)) = [1] + [2]x1
POL(F12_IN(x1, x2)) = x1
POL(F1_IN(x1, x2)) = [1] + [2]x1 + [3]x2
POL(F7_IN(x1, x2)) = [1] + [2]x1 + [2]x2
POL(U2'(x1, x2, x3)) = [2]x3
POL(U4(x1, x2)) = 0
POL(U5'(x1, x2, x3)) = x2 + [2]x3
POL(U9'(x1, x2, x3)) = [1] + x3
POL([]) = 0
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3) = 0
POL(c4) = 0
POL(c5(x1)) = x1
POL(c7) = 0
POL(c7(x1)) = x1
POL(f11_in(x1)) = 0
POL(f11_out1(x1, x2)) = 0
(25) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:none
K tuples:
U5'(f11_out1(z0, z1), z2, z3) → c4
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(26) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(27) BOUNDS(O(1), O(1))
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F1_IN(z0, z1) → c(U1'(f7_in(z1, z0), z0, z1), F7_IN(z1, z0))
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
F7_IN(z0, z1) → c3(U5'(f11_in(z0), z0, z1), F11_IN(z0))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
S tuples:
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c6(U10'(f12_in(z3, z0), z2, z3, z0, z1), F12_IN(z3, z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F1_IN, F7_IN, U2', F11_IN, U5', F12_IN, U9'
Compound Symbols:
c, c2, c3, c1, c3, c4, c2, c5, c6
(29) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
U2'(f11_out1(z0, z1), z2, z3) → c3(U3'(f12_in(z3, z0), z2, z3, z0, z1))
F11_IN(.(z0, z1)) → c1(U4'(f11_in(z1), .(z0, z1)), F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4(U6'(f12_in(z3, z0), z2, z3, z0, z1))
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(U1'(f7_in(z1, z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(U10'(f12_in(z3, z0), z2, z3, z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
S tuples:
F12_IN(.(z0, z1), .(z0, z1)) → c1(U7'(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(U8'(f12_in(z0, z2), z0, .(z1, z2)), F12_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(U10'(f12_in(z3, z0), z2, z3, z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, U2', F11_IN, U5', F12_IN, F1_IN, U9'
Compound Symbols:
c2, c3, c1, c4, c2, c5, c7
(31) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
U9'(f11_out1(z0, z1), z2, z3) → c7
K tuples:none
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
We considered the (Usable) Rules:
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
And the Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = 0
POL(F11_IN(x1)) = [2]
POL(F12_IN(x1, x2)) = x1
POL(F1_IN(x1, x2)) = [3] + [3]x1 + [3]x2
POL(F7_IN(x1, x2)) = [2] + [2]x2
POL(U2'(x1, x2, x3)) = [2] + [2]x3
POL(U4(x1, x2)) = 0
POL(U5'(x1, x2, x3)) = [1] + x3
POL(U9'(x1, x2, x3)) = x3
POL([]) = 0
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3) = 0
POL(c4) = 0
POL(c5(x1)) = x1
POL(c7) = 0
POL(c7(x1)) = x1
POL(f11_in(x1)) = 0
POL(f11_out1(x1, x2)) = 0
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
U9'(f11_out1(z0, z1), z2, z3) → c7
K tuples:
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(35) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U9'(f11_out1(z0, z1), z2, z3) → c7
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
K tuples:
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U9'(f11_out1(z0, z1), z2, z3) → c7
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
We considered the (Usable) Rules:
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
And the Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [1] + x2
POL(F11_IN(x1)) = [1]
POL(F12_IN(x1, x2)) = x1 + [2]x2
POL(F1_IN(x1, x2)) = [3] + [2]x1 + [2]x2
POL(F7_IN(x1, x2)) = [3] + [2]x1 + [2]x2
POL(U2'(x1, x2, x3)) = [3] + [2]x3
POL(U4(x1, x2)) = [1] + x1
POL(U5'(x1, x2, x3)) = [2] + x2 + x3
POL(U9'(x1, x2, x3)) = [2]x1 + x3
POL([]) = [1]
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3) = 0
POL(c4) = 0
POL(c5(x1)) = x1
POL(c7) = 0
POL(c7(x1)) = x1
POL(f11_in(x1)) = [1] + x1
POL(f11_out1(x1, x2)) = x1
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(z0, z1) → U1(f7_in(z1, z0), z0, z1)
U1(f7_out1(z0, z1, z2), z3, z4) → f1_out1
f7_in(z0, z1) → U2(f11_in(z0), z0, z1)
f7_in(z0, z1) → U5(f11_in(z0), z0, z1)
f7_in(z0, z1) → U9(f11_in(z0), z0, z1)
U2(f11_out1(z0, z1), z2, z3) → U3(f12_in(z3, z0), z2, z3, z0, z1)
U3(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f11_in(z0) → f11_out1([], z0)
f11_in(.(z0, z1)) → U4(f11_in(z1), .(z0, z1))
U4(f11_out1(z0, z1), .(z2, z3)) → f11_out1(.(z2, z0), z1)
U5(f11_out1(z0, z1), z2, z3) → U6(f12_in(z3, z0), z2, z3, z0, z1)
U6(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
f12_in(z0, z0) → f12_out1([])
f12_in(.(z0, z1), .(z0, z1)) → U7(f12_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f12_in(z0, .(z1, z2)) → U8(f12_in(z0, z2), z0, .(z1, z2))
U7(f12_out1(z0), .(z1, z2), .(z1, z2)) → f12_out1(.(z1, z0))
U8(f12_out1(z0), z1, .(z2, z3)) → f12_out1(.(z2, z0))
U9(f11_out1(z0, z1), z2, z3) → U10(f12_in(z3, z0), z2, z3, z0, z1)
U10(f12_out1(z0), z1, z2, z3, z4) → f7_out1(z3, z4, z0)
Tuples:
F7_IN(z0, z1) → c2(U2'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
F1_IN(z0, z1) → c7(F7_IN(z1, z0))
F7_IN(z0, z1) → c7(U5'(f11_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F11_IN(z0))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U2'(f11_out1(z0, z1), z2, z3) → c3
F11_IN(.(z0, z1)) → c1(F11_IN(z1))
U5'(f11_out1(z0, z1), z2, z3) → c4
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
F1_IN(z0, z1) → c7
U9'(f11_out1(z0, z1), z2, z3) → c7
S tuples:none
K tuples:
F7_IN(z0, z1) → c5(U9'(f11_in(z0), z0, z1))
U9'(f11_out1(z0, z1), z2, z3) → c7(F12_IN(z3, z0))
U9'(f11_out1(z0, z1), z2, z3) → c7
F12_IN(.(z0, z1), .(z0, z1)) → c1(F12_IN(.(z0, z1), z1))
F12_IN(z0, .(z1, z2)) → c2(F12_IN(z0, z2))
Defined Rule Symbols:
f1_in, U1, f7_in, U2, U3, f11_in, U4, U5, U6, f12_in, U7, U8, U9, U10
Defined Pair Symbols:
F7_IN, F1_IN, U9', U2', F11_IN, U5', F12_IN
Compound Symbols:
c2, c5, c7, c3, c1, c4, c7
(39) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(40) BOUNDS(O(1), O(1))
(41) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(42) Obligation:
Complex Complexity Dependency Tuples Problem
MAX
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
Complex Complexity Dependency Tuples Problem
MULTIPLY
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
(43) MaxProof (BOTH BOUNDS(ID, ID) transformation)
Took the maximum complexity of the problems.
(44) Complex Obligation (MAX)
(45) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
(46) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 4 of 18 dangling nodes:
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
(47) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
S tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12
(48) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(49) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F2_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
S tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F2_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, U3', F30_IN, U7', F45_IN, F180_IN, F140_IN, F2_IN, F10_IN, F8_IN
Compound Symbols:
c6, c7, c1, c4, c, c6, c11, c12, c2
(50) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 13 trailing tuple parts
(51) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
S tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, F2_IN, F10_IN, F8_IN, F180_IN, U3', F30_IN, U7', F45_IN, F140_IN
Compound Symbols:
c6, c2, c7, c1, c4, c, c11, c12, c2
(52) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F10_IN(.(z0, z1), z2) → c2
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
Now S is empty
(53) BOUNDS(O(1), O(1))
(54) Obligation:
Complex Complexity Dependency Tuples Problem
MULTIPLY
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
(55) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)
Multiplied the complexity of the problems.
(56) Complex Obligation (MULT)
(57) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
(58) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(59) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
S tuples:
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, U3', F30_IN, U7', F8_IN, F45_IN, F180_IN, F140_IN, U16', F2_IN, F10_IN
Compound Symbols:
c6, c7, c1, c4, c, c6, c11, c12, c15, c16, c18, c2
(60) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 16 trailing tuple parts
(61) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
S tuples:
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, F2_IN, F10_IN, F8_IN, F180_IN, U3', F30_IN, U7', F45_IN, F140_IN, U16'
Compound Symbols:
c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2
(62) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
We considered the (Usable) Rules:
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
And the Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [1]
POL(F10_IN(x1, x2)) = [2] + [2]x1 + x2
POL(F140_IN(x1, x2, x3)) = [3]
POL(F180_IN(x1, x2)) = [2] + [2]x1
POL(F24_IN(x1, x2, x3)) = [3] + x2
POL(F2_IN(x1, x2)) = [2] + [3]x1 + [3]x2
POL(F30_IN(x1)) = 0
POL(F45_IN(x1, x2)) = [3]
POL(F8_IN(x1, x2)) = [2] + [2]x1 + [2]x2
POL(U16'(x1, x2, x3, x4)) = x3
POL(U3'(x1, x2, x3, x4)) = 0
POL(U6(x1, x2)) = [2]x2
POL(U7'(x1, x2, x3, x4)) = [2]
POL([]) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c15(x1)) = x1
POL(c16) = 0
POL(c18) = 0
POL(c2) = 0
POL(c2(x1)) = x1
POL(c4) = 0
POL(c6) = 0
POL(c6(x1)) = x1
POL(c7) = 0
POL(f30_in(x1)) = [2]
POL(f30_out1(x1, x2)) = 0
(63) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
S tuples:
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
K tuples:
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, F2_IN, F10_IN, F8_IN, F180_IN, U3', F30_IN, U7', F45_IN, F140_IN, U16'
Compound Symbols:
c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2
(64) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
We considered the (Usable) Rules:
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
And the Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [2] + x2
POL(F10_IN(x1, x2)) = [1] + [2]x1 + [3]x2
POL(F140_IN(x1, x2, x3)) = [1]
POL(F180_IN(x1, x2)) = x2
POL(F24_IN(x1, x2, x3)) = [1] + x1 + [3]x2
POL(F2_IN(x1, x2)) = [3] + [3]x1 + [3]x2
POL(F30_IN(x1)) = x1
POL(F45_IN(x1, x2)) = [1]
POL(F8_IN(x1, x2)) = [2] + [3]x1 + [3]x2
POL(U16'(x1, x2, x3, x4)) = [3]x3
POL(U3'(x1, x2, x3, x4)) = x2 + [3]x3
POL(U6(x1, x2)) = 0
POL(U7'(x1, x2, x3, x4)) = x2
POL([]) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c15(x1)) = x1
POL(c16) = 0
POL(c18) = 0
POL(c2) = 0
POL(c2(x1)) = x1
POL(c4) = 0
POL(c6) = 0
POL(c6(x1)) = x1
POL(c7) = 0
POL(f30_in(x1)) = 0
POL(f30_out1(x1, x2)) = 0
(65) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
S tuples:none
K tuples:
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, F2_IN, F10_IN, F8_IN, F180_IN, U3', F30_IN, U7', F45_IN, F140_IN, U16'
Compound Symbols:
c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2
(66) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(67) BOUNDS(O(1), O(1))
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F2_IN(z0, z1) → c(U1'(f8_in(z0, z1), z0, z1), F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c4(U2'(f24_in(z1, z2, z0), .(z0, z1), z2), F24_IN(z1, z2, z0))
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c9(U5'(f9_in(z0), f10_in(z1, z0), z0, z1), F9_IN(z0), F10_IN(z1, z0))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
F24_IN(z0, z1, z2) → c3(U7'(f30_in(z0), z0, z1, z2), F30_IN(z0))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
S tuples:
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(.(z0, z1), .(z0, z1)) → c3(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)), F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c5(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))), F45_IN(z0, z1))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F2_IN, F10_IN, F24_IN, U3', F8_IN, F30_IN, U7', F45_IN, F180_IN, F140_IN, U16'
Compound Symbols:
c, c4, c6, c7, c9, c1, c3, c4, c5, c6, c11, c12, c15, c16, c18
(69) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7(U4'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F30_IN(.(z0, z1)) → c1(U6'(f30_in(z1), .(z0, z1)), F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4(U8'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c6(U9'(f9_in(z0), f10_in(z1, z0), z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F2_IN(z0, z1) → c2(U1'(f8_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(U2'(f24_in(z1, z2, z0), .(z0, z1), z2))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(U5'(f9_in(z0), f10_in(z1, z0), z0, z1))
F8_IN(z0, z1) → c2(F9_IN(z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
S tuples:
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(U10'(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))), F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(U13'(f180_in(z0, z3), z0, .(z1, .(z2, z3))), F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(U14'(f45_in(z0, z1), z0, z1, .(z0, z1)), F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(U15'(f140_in(z0, z1, z3), z0, z1, .(z2, z3)), F140_IN(z0, z1, z3))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16(U17'(f31_in(z3, z4, z0), z2, z3, z4, z0, z1))
F8_IN(z0, z1) → c18(U18'(f9_in(z0), f10_in(z1, z0), z0, z1))
F180_IN(.(z0, z1), .(z0, z1)) → c2(U11'(f45_in(z0, z1), .(z0, z1), .(z0, z1)))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(U12'(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1))))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, U3', F30_IN, U7', F8_IN, F45_IN, F180_IN, F140_IN, U16', F2_IN, F10_IN
Compound Symbols:
c6, c7, c1, c4, c, c6, c11, c12, c15, c16, c18, c2
(71) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 16 trailing tuple parts
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(z0, z1) → U1(f8_in(z0, z1), z0, z1)
U1(f8_out1(z0), z1, z2) → f2_out1
U1(f8_out2(z0, z1, z2), z3, z4) → f2_out1
f9_in([]) → f9_out1([])
f10_in(.(z0, z1), z2) → U2(f24_in(z1, z2, z0), .(z0, z1), z2)
U2(f24_out1(z0, z1, z2), .(z3, z4), z5) → f10_out1(.(z3, z0), z1, z2)
f24_in(z0, z1, z2) → U3(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U7(f30_in(z0), z0, z1, z2)
f24_in(z0, z1, z2) → U16(f30_in(z0), z0, z1, z2)
U3(f30_out1(z0, z1), z2, z3, z4) → U4(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U4(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
f8_in(z0, z1) → U5(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U9(f9_in(z0), f10_in(z1, z0), z0, z1)
f8_in(z0, z1) → U18(f9_in(z0), f10_in(z1, z0), z0, z1)
U5(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U5(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f30_in(z0) → f30_out1([], z0)
f30_in(.(z0, z1)) → U6(f30_in(z1), .(z0, z1))
U6(f30_out1(z0, z1), .(z2, z3)) → f30_out1(.(z2, z0), z1)
U7(f30_out1(z0, z1), z2, z3, z4) → U8(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U8(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U9(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U9(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
f45_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → U10(f140_in(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8), z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))))
U10(f140_out1(z0), z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z9)))))))) → f45_out1(.(z2, .(z3, .(z4, .(z5, .(z6, .(z7, .(z8, z0))))))))
f180_in(z0, z0) → f180_out1([])
f180_in(.(z0, z1), .(z0, z1)) → U11(f45_in(z0, z1), .(z0, z1), .(z0, z1))
f180_in(z0, .(z1, z0)) → f180_out1(.(z1, []))
f180_in(.(z0, z1), .(z2, .(z0, z1))) → U12(f45_in(z0, z1), .(z0, z1), .(z2, .(z0, z1)))
f180_in(z0, .(z1, .(z2, z3))) → U13(f180_in(z0, z3), z0, .(z1, .(z2, z3)))
U11(f45_out1(z0), .(z1, z2), .(z1, z2)) → f180_out1(.(z1, z0))
U12(f45_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f180_out1(.(z3, .(z1, z0)))
U13(f180_out1(z0), z1, .(z2, .(z3, z4))) → f180_out1(.(z2, .(z3, z0)))
f140_in(z0, z1, .(z0, z1)) → f140_out1([])
f140_in(z0, z1, .(z0, z1)) → U14(f45_in(z0, z1), z0, z1, .(z0, z1))
f140_in(z0, z1, .(z2, z3)) → U15(f140_in(z0, z1, z3), z0, z1, .(z2, z3))
U14(f45_out1(z0), z1, z2, .(z1, z2)) → f140_out1(.(z1, z0))
U15(f140_out1(z0), z1, z2, .(z3, z4)) → f140_out1(.(z3, z0))
U16(f30_out1(z0, z1), z2, z3, z4) → U17(f31_in(z3, z4, z0), z2, z3, z4, z0, z1)
U17(f31_out1(z0), z1, z2, z3, z4, z5) → f24_out1(z4, z5, z0)
U18(f9_out1(z0), z1, z2, z3) → f8_out1(z0)
U18(z0, f10_out1(z1, z2, z3), z4, z5) → f8_out2(z1, z2, z3)
Tuples:
F24_IN(z0, z1, z2) → c6(U3'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F2_IN(z0, z1) → c2(F8_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c2(F24_IN(z1, z2, z0))
F8_IN(z0, z1) → c2(F10_IN(z1, z0))
F24_IN(z0, z1, z2) → c2(U7'(f30_in(z0), z0, z1, z2))
F24_IN(z0, z1, z2) → c2(F30_IN(z0))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
U3'(f30_out1(z0, z1), z2, z3, z4) → c7
F30_IN(.(z0, z1)) → c1(F30_IN(z1))
U7'(f30_out1(z0, z1), z2, z3, z4) → c4
F8_IN(z0, z1) → c6
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F2_IN(z0, z1) → c2
F10_IN(.(z0, z1), z2) → c2
F8_IN(z0, z1) → c2
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
S tuples:
F24_IN(z0, z1, z2) → c15(U16'(f30_in(z0), z0, z1, z2))
F180_IN(.(z0, z1), .(z0, z1)) → c2(F45_IN(z0, z1))
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2(F45_IN(z0, z1))
F45_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8)))))))) → c(F140_IN(z0, .(z1, .(z2, .(z3, .(z4, .(z5, .(z6, .(z7, z8))))))), z8))
F180_IN(z0, .(z1, .(z2, z3))) → c6(F180_IN(z0, z3))
F140_IN(z0, z1, .(z0, z1)) → c11(F45_IN(z0, z1))
F140_IN(z0, z1, .(z2, z3)) → c12(F140_IN(z0, z1, z3))
U16'(f30_out1(z0, z1), z2, z3, z4) → c16
F8_IN(z0, z1) → c18
F180_IN(.(z0, z1), .(z0, z1)) → c2
F180_IN(.(z0, z1), .(z2, .(z0, z1))) → c2
K tuples:none
Defined Rule Symbols:
f2_in, U1, f9_in, f10_in, U2, f24_in, U3, U4, f8_in, U5, f30_in, U6, U7, U8, U9, f45_in, U10, f180_in, U11, U12, U13, f140_in, U14, U15, U16, U17, U18
Defined Pair Symbols:
F24_IN, F2_IN, F10_IN, F8_IN, F180_IN, U3', F30_IN, U7', F45_IN, F140_IN, U16'
Compound Symbols:
c6, c15, c2, c7, c1, c4, c6, c, c11, c12, c16, c18, c2