WORST_CASE(?,O(n^3))
proof of input_ZGnQ1Ype8W.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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)), 100 ms]
(10) CdtProblem
(11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 61 ms]
(12) CdtProblem
(13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 603 ms]
(14) CdtProblem
(15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) BOUNDS(1, 1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   <=(0, y) -> true
   <=(s(x), 0) -> false
   <=(s(x), s(y)) -> <=(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, -(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

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:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   <=(0, z0) -> true
   <=(s(z0), 0) -> false
   <=(s(z0), s(z1)) -> <=(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, -(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(0, z0) -> c2
   <='(s(z0), 0) -> c3
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   PERFECTP(0) -> c7
   PERFECTP(s(z0)) -> c8(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c9
   F(0, z0, s(z1), z2) -> c10
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), <='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(0, z0) -> c2
   <='(s(z0), 0) -> c3
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   PERFECTP(0) -> c7
   PERFECTP(s(z0)) -> c8(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c9
   F(0, z0, s(z1), z2) -> c10
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), <='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   -_2, <=_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   -'_2, <='_2, IF_3, PERFECTP_1, F_4

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6, c7, c8_1, c9, c10, c11_2, c12_2, c13_3, c14_2


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PERFECTP(s(z0)) -> c8(F(z0, s(0), s(z0), s(z0)))
Removed 8 trailing nodes:
   IF(false, z0, z1) -> c6
   PERFECTP(0) -> c7
   <='(0, z0) -> c2
   <='(s(z0), 0) -> c3
   -'(z0, 0) -> c
   F(0, z0, s(z1), z2) -> c10
   F(0, z0, 0, z1) -> c9
   IF(true, z0, z1) -> c5

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   <=(0, z0) -> true
   <=(s(z0), 0) -> false
   <=(s(z0), s(z1)) -> <=(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, -(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), <='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), <='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(IF(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   -_2, <=_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_2, c13_3, c14_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   <=(0, z0) -> true
   <=(s(z0), 0) -> false
   <=(s(z0), s(z1)) -> <=(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, -(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   -_2, <=_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_1, c13_2, c14_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   <=(0, z0) -> true
   <=(s(z0), 0) -> false
   <=(s(z0), s(z1)) -> <=(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, -(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(<=(z0, z1), f(s(z0), -(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   -_2

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_1, c13_2, c14_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(<='(x_1, x_2)) = [1] + x_1
   POL(F(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_3
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
K tuples:
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_1, c13_2, c14_1


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = [2]
   POL(<='(x_1, x_2)) = [1] + x_2
   POL(F(x_1, x_2, x_3, x_4)) = [2] + x_2 + x_1*x_4
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(s(x_1)) = [2] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
K tuples:
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_1, c13_2, c14_1


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = [1] + x_2
   POL(0) = [1]
   POL(<='(x_1, x_2)) = x_1 + x_2 + x_1*x_2 + x_1^2 + x_1^3
   POL(F(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_2*x_4 + x_1*x_4 + x_1*x_2 + x_2*x_3 + x_1^3 + x_1^2*x_3 + x_1^2*x_4 + x_1*x_4^2 + x_1*x_3*x_4
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
S tuples:none
K tuples:
   <='(s(z0), s(z1)) -> c4(<='(z0, z1))
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3))
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0))
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   -'_2, <='_2, F_4

Compound Symbols:   c1_1, c4_1, c11_2, c12_1, c13_2, c14_1


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(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(16)
BOUNDS(1, 1)