WORST_CASE(?,O(n^2))
proof of input_ecIqu2taF1.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 145 ms]
(10) CdtProblem
(11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms]
(12) CdtProblem
(13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 173 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 181 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 156 ms]
(18) CdtProblem
(19) CdtKnowledgeProof [FINISHED, 0 ms]
(20) BOUNDS(1, 1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   reach(x, y, empty, h) -> false
   reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
   if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
   if_reach_2(true, x, y, edge(u, v, i), h) -> true
   if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
   if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

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:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
   reach(z0, z1, empty, z2) -> false
   reach(z0, z1, edge(z2, z3, z4), z5) -> if_reach_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(true, z0, z1, edge(z2, z3, z4), z5) -> if_reach_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(false, z0, z1, edge(z2, z3, z4), z5) -> reach(z0, z1, z4, edge(z2, z3, z5))
   if_reach_2(true, z0, z1, edge(z2, z3, z4), z5) -> true
   if_reach_2(false, z0, z1, edge(z2, z3, z4), z5) -> or(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty))
Tuples:
   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   OR(true, z0) -> c4
   OR(false, z0) -> c5
   UNION(empty, z0) -> c6
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, empty, z2) -> c8
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(true, z0, z1, edge(z2, z3, z4), z5) -> c12
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   OR(true, z0) -> c4
   OR(false, z0) -> c5
   UNION(empty, z0) -> c6
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, empty, z2) -> c8
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(true, z0, z1, edge(z2, z3, z4), z5) -> c12
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, union_2, reach_4, if_reach_1_5, if_reach_2_5

Defined Pair Symbols:   EQ_2, OR_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c, c1, c2, c3_1, c4, c5, c6, c7_1, c8, c9_2, c10_2, c11_1, c12, c13_2, c14_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   IF_REACH_2(true, z0, z1, edge(z2, z3, z4), z5) -> c12
   REACH(z0, z1, empty, z2) -> c8
   UNION(empty, z0) -> c6
   EQ(s(z0), 0) -> c2
   EQ(0, s(z0)) -> c1
   EQ(0, 0) -> c
   OR(true, z0) -> c4
   OR(false, z0) -> c5

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
   reach(z0, z1, empty, z2) -> false
   reach(z0, z1, edge(z2, z3, z4), z5) -> if_reach_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(true, z0, z1, edge(z2, z3, z4), z5) -> if_reach_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(false, z0, z1, edge(z2, z3, z4), z5) -> reach(z0, z1, z4, edge(z2, z3, z5))
   if_reach_2(true, z0, z1, edge(z2, z3, z4), z5) -> true
   if_reach_2(false, z0, z1, edge(z2, z3, z4), z5) -> or(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(OR(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty)), REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, union_2, reach_4, if_reach_1_5, if_reach_2_5

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_2, c14_3


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
   reach(z0, z1, empty, z2) -> false
   reach(z0, z1, edge(z2, z3, z4), z5) -> if_reach_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(true, z0, z1, edge(z2, z3, z4), z5) -> if_reach_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(false, z0, z1, edge(z2, z3, z4), z5) -> reach(z0, z1, z4, edge(z2, z3, z5))
   if_reach_2(true, z0, z1, edge(z2, z3, z4), z5) -> true
   if_reach_2(false, z0, z1, edge(z2, z3, z4), z5) -> or(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, union_2, reach_4, if_reach_1_5, if_reach_2_5

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   or(true, z0) -> true
   or(false, z0) -> z0
   reach(z0, z1, empty, z2) -> false
   reach(z0, z1, edge(z2, z3, z4), z5) -> if_reach_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(true, z0, z1, edge(z2, z3, z4), z5) -> if_reach_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5)
   if_reach_1(false, z0, z1, edge(z2, z3, z4), z5) -> reach(z0, z1, z4, edge(z2, z3, z5))
   if_reach_2(true, z0, z1, edge(z2, z3, z4), z5) -> true
   if_reach_2(false, z0, z1, edge(z2, z3, z4), z5) -> or(reach(z0, z1, z4, z5), reach(z3, z1, union(z4, z5), empty))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
K tuples:none
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
We considered the (Usable) Rules:
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(0, 0) -> true
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_4 + x_5
   POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_4 + x_5
   POL(REACH(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(UNION(x_1, x_2)) = 0
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(edge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0
   POL(union(x_1, x_2)) = x_1 + x_2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
K tuples:
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
We considered the (Usable) Rules:
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4 + x_5
   POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_4 + x_5
   POL(REACH(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(UNION(x_1, x_2)) = 0
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(edge(x_1, x_2, x_3)) = [1] + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]
   POL(union(x_1, x_2)) = x_1 + x_2

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
K tuples:
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
We considered the (Usable) Rules:
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = x_5^2 + [2]x_4*x_5 + x_4^2 + x_2*x_4
   POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_5^2 + [2]x_4*x_5 + x_4^2 + x_2*x_4
   POL(REACH(x_1, x_2, x_3, x_4)) = x_4^2 + [2]x_3*x_4 + x_1*x_3 + x_3^2
   POL(UNION(x_1, x_2)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(edge(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(union(x_1, x_2)) = x_1 + x_2

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
K tuples:
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
We considered the (Usable) Rules:
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = [2]x_1
   POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = [2]x_3 + x_5^2 + [2]x_4*x_5 + [2]x_3*x_5 + x_4^2 + [2]x_3*x_4 + [2]x_2*x_4
   POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_5^2 + [2]x_4*x_5 + [2]x_3*x_5 + x_4^2 + [2]x_3*x_4 + [2]x_2*x_4
   POL(REACH(x_1, x_2, x_3, x_4)) = [2]x_1 + [2]x_2 + x_4^2 + [2]x_3*x_4 + [2]x_2*x_4 + [2]x_1*x_3 + x_3^2 + [2]x_2*x_3
   POL(UNION(x_1, x_2)) = 0
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(edge(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0
   POL(union(x_1, x_2)) = x_1 + x_2

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
K tuples:
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
We considered the (Usable) Rules:
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = [2]x_4 + x_5^2 + [2]x_4*x_5 + x_4^2 + x_2*x_4
   POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_5^2 + [2]x_4*x_5 + x_4^2 + x_2*x_4
   POL(REACH(x_1, x_2, x_3, x_4)) = [2]x_3 + x_4^2 + [2]x_3*x_4 + x_1*x_3 + x_3^2
   POL(UNION(x_1, x_2)) = 0
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(edge(x_1, x_2, x_3)) = [2] + x_2 + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(union(x_1, x_2)) = x_1 + x_2

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   union(empty, z0) -> z0
   union(edge(z0, z1, z2), z3) -> edge(z0, z1, union(z2, z3))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
S tuples:
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
K tuples:
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c13(REACH(z0, z1, z4, z5))
   IF_REACH_2(false, z0, z1, edge(z2, z3, z4), z5) -> c14(REACH(z3, z1, union(z4, z5), empty), UNION(z4, z5))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   UNION(edge(z0, z1, z2), z3) -> c7(UNION(z2, z3))
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
Defined Rule Symbols:   eq_2, union_2

Defined Pair Symbols:   EQ_2, UNION_2, REACH_4, IF_REACH_1_5, IF_REACH_2_5

Compound Symbols:   c3_1, c7_1, c9_2, c10_2, c11_1, c13_1, c14_2


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(19) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   REACH(z0, z1, edge(z2, z3, z4), z5) -> c9(IF_REACH_1(eq(z0, z2), z0, z1, edge(z2, z3, z4), z5), EQ(z0, z2))
   IF_REACH_1(true, z0, z1, edge(z2, z3, z4), z5) -> c10(IF_REACH_2(eq(z1, z3), z0, z1, edge(z2, z3, z4), z5), EQ(z1, z3))
   IF_REACH_1(false, z0, z1, edge(z2, z3, z4), z5) -> c11(REACH(z0, z1, z4, edge(z2, z3, z5)))
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
Now S is empty
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(20)
BOUNDS(1, 1)