WORST_CASE(?,O(n^1))
proof of input_zfBK40Kdjf.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(10) CdtProblem
(11) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 103 ms]
(16) CdtProblem
(17) CdtKnowledgeProof [FINISHED, 0 ms]
(18) BOUNDS(1, 1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   if(true, t, e) -> t
   if(false, t, e) -> e
   member(x, nil) -> false
   member(x, cons(y, ys)) -> if(eq(x, y), true, member(x, ys))
   eq(nil, nil) -> true
   eq(O(x), 0(y)) -> eq(x, y)
   eq(0(x), 1(y)) -> false
   eq(1(x), 0(y)) -> false
   eq(1(x), 1(y)) -> eq(x, y)
   negate(0(x)) -> 1(x)
   negate(1(x)) -> 0(x)
   choice(cons(x, xs)) -> x
   choice(cons(x, xs)) -> choice(xs)
   guess(nil) -> nil
   guess(cons(clause, cnf)) -> cons(choice(clause), guess(cnf))
   verify(nil) -> true
   verify(cons(l, ls)) -> if(member(negate(l), ls), false, verify(ls))
   sat(cnf) -> satck(cnf, guess(cnf))
   satck(cnf, assign) -> if(verify(assign), assign, unsat)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   IF(true, z0, z1) -> c
   IF(false, z0, z1) -> c1
   MEMBER(z0, nil) -> c2
   MEMBER(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), true, member(z0, z2)), EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), true, member(z0, z2)), MEMBER(z0, z2))
   EQ(nil, nil) -> c5
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(0(z0), 1(z1)) -> c7
   EQ(1(z0), 0(z1)) -> c8
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   NEGATE(0(z0)) -> c10
   NEGATE(1(z0)) -> c11
   CHOICE(cons(z0, z1)) -> c12
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(nil) -> c14
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   VERIFY(nil) -> c17
   VERIFY(cons(z0, z1)) -> c18(IF(member(negate(z0), z1), false, verify(z1)), MEMBER(negate(z0), z1), NEGATE(z0))
   VERIFY(cons(z0, z1)) -> c19(IF(member(negate(z0), z1), false, verify(z1)), VERIFY(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   SATCK(z0, z1) -> c21(IF(verify(z1), z1, unsat), VERIFY(z1))
S tuples:
   IF(true, z0, z1) -> c
   IF(false, z0, z1) -> c1
   MEMBER(z0, nil) -> c2
   MEMBER(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), true, member(z0, z2)), EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), true, member(z0, z2)), MEMBER(z0, z2))
   EQ(nil, nil) -> c5
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(0(z0), 1(z1)) -> c7
   EQ(1(z0), 0(z1)) -> c8
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   NEGATE(0(z0)) -> c10
   NEGATE(1(z0)) -> c11
   CHOICE(cons(z0, z1)) -> c12
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(nil) -> c14
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   VERIFY(nil) -> c17
   VERIFY(cons(z0, z1)) -> c18(IF(member(negate(z0), z1), false, verify(z1)), MEMBER(negate(z0), z1), NEGATE(z0))
   VERIFY(cons(z0, z1)) -> c19(IF(member(negate(z0), z1), false, verify(z1)), VERIFY(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   SATCK(z0, z1) -> c21(IF(verify(z1), z1, unsat), VERIFY(z1))
K tuples:none
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   IF_3, MEMBER_2, EQ_2, NEGATE_1, CHOICE_1, GUESS_1, VERIFY_1, SAT_1, SATCK_2

Compound Symbols:   c, c1, c2, c3_2, c4_2, c5, c6_1, c7, c8, c9_1, c10, c11, c12, c13_1, c14, c15_1, c16_1, c17, c18_3, c19_2, c20_2, c21_2


----------------------------------------

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 11 trailing nodes:
   NEGATE(0(z0)) -> c10
   EQ(1(z0), 0(z1)) -> c8
   IF(true, z0, z1) -> c
   EQ(nil, nil) -> c5
   IF(false, z0, z1) -> c1
   NEGATE(1(z0)) -> c11
   EQ(0(z0), 1(z1)) -> c7
   VERIFY(nil) -> c17
   MEMBER(z0, nil) -> c2
   CHOICE(cons(z0, z1)) -> c12
   GUESS(nil) -> c14

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   MEMBER(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), true, member(z0, z2)), EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), true, member(z0, z2)), MEMBER(z0, z2))
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   VERIFY(cons(z0, z1)) -> c18(IF(member(negate(z0), z1), false, verify(z1)), MEMBER(negate(z0), z1), NEGATE(z0))
   VERIFY(cons(z0, z1)) -> c19(IF(member(negate(z0), z1), false, verify(z1)), VERIFY(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   SATCK(z0, z1) -> c21(IF(verify(z1), z1, unsat), VERIFY(z1))
S tuples:
   MEMBER(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), true, member(z0, z2)), EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), true, member(z0, z2)), MEMBER(z0, z2))
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   VERIFY(cons(z0, z1)) -> c18(IF(member(negate(z0), z1), false, verify(z1)), MEMBER(negate(z0), z1), NEGATE(z0))
   VERIFY(cons(z0, z1)) -> c19(IF(member(negate(z0), z1), false, verify(z1)), VERIFY(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   SATCK(z0, z1) -> c21(IF(verify(z1), z1, unsat), VERIFY(z1))
K tuples:none
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   MEMBER_2, EQ_2, CHOICE_1, GUESS_1, VERIFY_1, SAT_1, SATCK_2

Compound Symbols:   c3_2, c4_2, c6_1, c9_1, c13_1, c15_1, c16_1, c18_3, c19_2, c20_2, c21_2


----------------------------------------

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
S tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   SAT(z0) -> c20(SATCK(z0, guess(z0)), GUESS(z0))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
K tuples:none
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, SAT_1, MEMBER_2, VERIFY_1, SATCK_2

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c20_2, c3_1, c4_1, c18_1, c19_1, c21_1


----------------------------------------

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SAT(z0) -> c(GUESS(z0))
S tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SAT(z0) -> c(GUESS(z0))
K tuples:none
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, MEMBER_2, VERIFY_1, SATCK_2, SAT_1

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c3_1, c4_1, c18_1, c19_1, c21_1, c_1


----------------------------------------

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SAT(z0) -> c(GUESS(z0))

----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
S tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
K tuples:none
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, MEMBER_2, VERIFY_1, SATCK_2, SAT_1

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c3_1, c4_1, c18_1, c19_1, c21_1, c_1


----------------------------------------

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SATCK(z0, z1) -> c21(VERIFY(z1))

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
S tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
K tuples:
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SATCK(z0, z1) -> c21(VERIFY(z1))
Defined Rule Symbols:   if_3, member_2, eq_2, negate_1, choice_1, guess_1, verify_1, sat_1, satck_2

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, MEMBER_2, VERIFY_1, SATCK_2, SAT_1

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c3_1, c4_1, c18_1, c19_1, c21_1, c_1


----------------------------------------

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   member(z0, nil) -> false
   member(z0, cons(z1, z2)) -> if(eq(z0, z1), true, member(z0, z2))
   eq(nil, nil) -> true
   eq(O(z0), 0(z1)) -> eq(z0, z1)
   eq(0(z0), 1(z1)) -> false
   eq(1(z0), 0(z1)) -> false
   eq(1(z0), 1(z1)) -> eq(z0, z1)
   verify(nil) -> true
   verify(cons(z0, z1)) -> if(member(negate(z0), z1), false, verify(z1))
   sat(z0) -> satck(z0, guess(z0))
   satck(z0, z1) -> if(verify(z1), z1, unsat)

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
S tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
K tuples:
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SATCK(z0, z1) -> c21(VERIFY(z1))
Defined Rule Symbols:   negate_1, guess_1, choice_1

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, MEMBER_2, VERIFY_1, SATCK_2, SAT_1

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c3_1, c4_1, c18_1, c19_1, c21_1, c_1


----------------------------------------

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
We considered the (Usable) Rules:
   negate(1(z0)) -> 0(z0)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   choice(cons(z0, z1)) -> choice(z1)
   choice(cons(z0, z1)) -> z0
   negate(0(z0)) -> 1(z0)
And the Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0(x_1)) = [1] + x_1
   POL(1(x_1)) = [1] + x_1
   POL(CHOICE(x_1)) = [1] + x_1
   POL(EQ(x_1, x_2)) = [1] + x_1 + x_2
   POL(GUESS(x_1)) = x_1
   POL(MEMBER(x_1, x_2)) = [1] + x_1 + x_2
   POL(O(x_1)) = [1] + x_1
   POL(SAT(x_1)) = [1] + x_1
   POL(SATCK(x_1, x_2)) = [1] + x_2
   POL(VERIFY(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(choice(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(guess(x_1)) = x_1
   POL(negate(x_1)) = x_1
   POL(nil) = [1]

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   negate(0(z0)) -> 1(z0)
   negate(1(z0)) -> 0(z0)
   guess(nil) -> nil
   guess(cons(z0, z1)) -> cons(choice(z0), guess(z1))
   choice(cons(z0, z1)) -> z0
   choice(cons(z0, z1)) -> choice(z1)
Tuples:
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   SAT(z0) -> c(SATCK(z0, guess(z0)))
S tuples:
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
K tuples:
   SAT(z0) -> c(SATCK(z0, guess(z0)))
   SATCK(z0, z1) -> c21(VERIFY(z1))
   EQ(O(z0), 0(z1)) -> c6(EQ(z0, z1))
   EQ(1(z0), 1(z1)) -> c9(EQ(z0, z1))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
   GUESS(cons(z0, z1)) -> c16(GUESS(z1))
   MEMBER(z0, cons(z1, z2)) -> c3(EQ(z0, z1))
   MEMBER(z0, cons(z1, z2)) -> c4(MEMBER(z0, z2))
   VERIFY(cons(z0, z1)) -> c18(MEMBER(negate(z0), z1))
   VERIFY(cons(z0, z1)) -> c19(VERIFY(z1))
Defined Rule Symbols:   negate_1, guess_1, choice_1

Defined Pair Symbols:   EQ_2, CHOICE_1, GUESS_1, MEMBER_2, VERIFY_1, SATCK_2, SAT_1

Compound Symbols:   c6_1, c9_1, c13_1, c15_1, c16_1, c3_1, c4_1, c18_1, c19_1, c21_1, c_1


----------------------------------------

(17) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   GUESS(cons(z0, z1)) -> c15(CHOICE(z0))
   CHOICE(cons(z0, z1)) -> c13(CHOICE(z1))
Now S is empty
----------------------------------------

(18)
BOUNDS(1, 1)