WORST_CASE(Omega(n^1),O(n^3))
proof of input_2CzBQ3REYT.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^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) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxWeightedTrs
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedTrs
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
    (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) SimplificationProof [BOTH BOUNDS(ID, ID), 3 ms]
    (24) CpxRNTS
    (25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 268 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 3749 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 812 ms]
    (50) CpxRNTS
    (51) FinalProof [FINISHED, 0 ms]
    (52) BOUNDS(1, n^3)
(53) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(54) CdtProblem
    (55) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CpxRelTRS
    (57) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxRelTRS
    (59) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) typed CpxTrs
    (61) OrderProof [LOWER BOUND(ID), 13 ms]
    (62) typed CpxTrs
    (63) RewriteLemmaProof [LOWER BOUND(ID), 323 ms]
    (64) BEST
        (65) proven lower bound
            (66) LowerBoundPropagationProof [FINISHED, 0 ms]
            (67) BOUNDS(n^1, INF)
        (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
            (72) typed CpxTrs
            (73) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
            (74) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^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:
   F(0, z0, s(z1), z2) -> c10
   IF(true, z0, z1) -> c5
   <='(s(z0), 0) -> c3
   IF(false, z0, z1) -> c6
   -'(z0, 0) -> c
   F(0, z0, 0, z1) -> c9
   <='(0, z0) -> c2
   PERFECTP(0) -> c7

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

(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) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   -'(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 (relative) TRS S consists of the following rules:

   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(11) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   <='(s(z0), s(z1)) -> c4(<='(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, -(z1, s(z0)), z2), -'(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), -(z1, z0), z2, z3), -'(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3)) [1]
   -(s(z0), s(z1)) -> -(z0, z1) [0]
   -(z0, 0) -> z0 [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   <='(s(z0), s(z1)) -> c4(<='(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, minus(z1, s(z0)), z2), -'(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), minus(z1, z0), z2, z3), -'(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3)) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   <='(s(z0), s(z1)) -> c4(<='(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, minus(z1, s(z0)), z2), -'(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), minus(z1, z0), z2, z3), -'(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3)) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
<=' :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
F :: s:0 -> s:0 -> s:0 -> s:0 -> c11:c12:c13:c14
0 :: s:0
c11 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
minus :: s:0 -> s:0 -> s:0
c12 :: c4 -> c11:c12:c13:c14
c13 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
c14 :: c11:c12:c13:c14 -> c11:c12:c13:c14

Rewrite Strategy: INNERMOST
----------------------------------------

(17) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   -'_2
   <='_2
   F_4

(c) The following functions are completely defined:

   minus_2

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]

And the following fresh constants:    const, const1, const2

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

(18)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   <='(s(z0), s(z1)) -> c4(<='(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, minus(z1, s(z0)), z2), -'(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), minus(z1, z0), z2, z3), -'(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3)) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
<=' :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
F :: s:0 -> s:0 -> s:0 -> s:0 -> c11:c12:c13:c14
0 :: s:0
c11 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
minus :: s:0 -> s:0 -> s:0
c12 :: c4 -> c11:c12:c13:c14
c13 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
c14 :: c11:c12:c13:c14 -> c11:c12:c13:c14
const :: c1
const1 :: c4
const2 :: c11:c12:c13:c14

Rewrite Strategy: INNERMOST
----------------------------------------

(19) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(20)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   <='(s(z0), s(z1)) -> c4(<='(z0, z1)) [1]
   F(s(z0), 0, s(z0'), z2) -> c11(F(z0, z2, minus(z0', z0), z2), -'(s(z0'), s(z0))) [1]
   F(s(z0), 0, z1, z2) -> c11(F(z0, z2, 0, z2), -'(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c12(<='(z0, z1)) [1]
   F(s(s(z1')), s(s(z0'')), z2, z3) -> c13(F(s(s(z1')), minus(z0'', z1'), z2, z3), -'(s(z0''), s(z1'))) [1]
   F(s(0), s(z1), z2, z3) -> c13(F(s(0), z1, z2, z3), -'(z1, 0)) [1]
   F(s(z0), s(z1), z2, z3) -> c13(F(s(z0), 0, z2, z3), -'(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(z0, z3, z2, z3)) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
<=' :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
F :: s:0 -> s:0 -> s:0 -> s:0 -> c11:c12:c13:c14
0 :: s:0
c11 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
minus :: s:0 -> s:0 -> s:0
c12 :: c4 -> c11:c12:c13:c14
c13 :: c11:c12:c13:c14 -> c1 -> c11:c12:c13:c14
c14 :: c11:c12:c13:c14 -> c11:c12:c13:c14
const :: c1
const1 :: c4
const2 :: c11:c12:c13:c14

Rewrite Strategy: INNERMOST
----------------------------------------

(21) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   const => 0
   const1 => 0
   const2 => 0

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   <='(z, z') -{ 1 }-> 1 + <='(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z3, z2, z3) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, minus(z0', z0), z2) + -'(1 + z0', 1 + z0) :|: z0' >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z0', z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 0, z2) + -'(z1, 1 + z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + z0, 0, z2, z3) + -'(z1, z0) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z1, z2, z3) + -'(z1, 0) :|: z'' = z2, z1 >= 0, z = 1 + 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + z1'), minus(z0'', z1'), z2, z3) + -'(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + (1 + z0''), z1' >= 0, z = 1 + (1 + z1'), z0'' >= 0, z4 = z3, z2 >= 0, z3 >= 0
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(23) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

(25) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { <=' }
   { -' }
   { minus }
   { F }

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {<='}, {-'}, {minus}, {F}

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {<='}, {-'}, {minus}, {F}

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: <='
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {<='}, {-'}, {minus}, {F}
Previous analysis results are:
  <=': runtime: ?, size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: <='
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {-'}, {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {-'}, {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: -'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {-'}, {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: ?, size: O(1) [0]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: -'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + -'(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + -'(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + -'(z' - 1, 0) :|: z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (z - 1), 0, z'', z4) + -'(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + -'(1 + (z' - 2), 1 + (z - 2)) :|: z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + s1 :|: s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + s3 :|: s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + s1 :|: s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + s3 :|: s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: ?, size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + s1 :|: s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), minus(z' - 2, z - 2), z'', z4) + s3 :|: s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, s6, z4) + s1 :|: s6 >= 0, s6 <= z'' - 1, s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s7, z'', z4) + s3 :|: s7 >= 0, s7 <= z' - 2, s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 3*z + 9*z*z4 + 3*z'

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, s6, z4) + s1 :|: s6 >= 0, s6 <= z'' - 1, s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s7, z'', z4) + s3 :|: s7 >= 0, s7 <= z' - 2, s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  F: runtime: ?, size: O(n^2) [1 + 3*z + 9*z*z4 + 3*z']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 5*z + z*z' + z*z4 + 3*z^2 + z^2*z4 + 2*z' + 2*z4

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   <='(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z', z'', z4) -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, s6, z4) + s1 :|: s6 >= 0, s6 <= z'' - 1, s1 >= 0, s1 <= 0, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, z' - 1, z'', z4) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s7, z'', z4) + s3 :|: s7 >= 0, s7 <= z' - 2, s3 >= 0, s3 <= 0, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   minus(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  <=': runtime: O(n^1) [z'], size: O(1) [0]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  F: runtime: O(n^3) [4 + 5*z + z*z' + z*z4 + 3*z^2 + z^2*z4 + 2*z' + 2*z4], size: O(n^2) [1 + 3*z + 9*z*z4 + 3*z']

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

(51) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(52)
BOUNDS(1, n^3)

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

(53) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(54)
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


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

(55) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(56)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   -'(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))

The (relative) TRS S consists of the following 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))

Rewrite Strategy: INNERMOST
----------------------------------------

(57) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(58)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   -'(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))

The (relative) TRS S consists of the following 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))

Rewrite Strategy: INNERMOST
----------------------------------------

(59) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(60)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14

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

(61) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-', <=', F, -, <=, f

They will be analysed ascendingly in the following order:
-' < F
<=' < F
- < F
<= < F
f < F
- < f
<= < f

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

(62)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
-', <=', F, -, <=, f

They will be analysed ascendingly in the following order:
-' < F
<=' < F
- < F
<= < F
f < F
- < f
<= < f

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15)) -> gen_c:c18_15(n13_15), rt in Omega(1 + n13_15)

Induction Base:
-'(gen_0':s9_15(0), gen_0':s9_15(0)) ->_R^Omega(1)
c

Induction Step:
-'(gen_0':s9_15(+(n13_15, 1)), gen_0':s9_15(+(n13_15, 1))) ->_R^Omega(1)
c1(-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15))) ->_IH
c1(gen_c:c18_15(c14_15))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(64)
Complex Obligation (BEST)

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

(65)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
-', <=', F, -, <=, f

They will be analysed ascendingly in the following order:
-' < F
<=' < F
- < F
<= < F
f < F
- < f
<= < f

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

(66) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(67)
BOUNDS(n^1, INF)

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

(68)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Lemmas:
-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15)) -> gen_c:c18_15(n13_15), rt in Omega(1 + n13_15)


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
<=', F, -, <=, f

They will be analysed ascendingly in the following order:
<=' < F
- < F
<= < F
f < F
- < f
<= < f

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<='(gen_0':s9_15(n442_15), gen_0':s9_15(n442_15)) -> gen_c2:c3:c410_15(n442_15), rt in Omega(1 + n442_15)

Induction Base:
<='(gen_0':s9_15(0), gen_0':s9_15(0)) ->_R^Omega(1)
c2

Induction Step:
<='(gen_0':s9_15(+(n442_15, 1)), gen_0':s9_15(+(n442_15, 1))) ->_R^Omega(1)
c4(<='(gen_0':s9_15(n442_15), gen_0':s9_15(n442_15))) ->_IH
c4(gen_c2:c3:c410_15(c443_15))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(70)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Lemmas:
-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15)) -> gen_c:c18_15(n13_15), rt in Omega(1 + n13_15)
<='(gen_0':s9_15(n442_15), gen_0':s9_15(n442_15)) -> gen_c2:c3:c410_15(n442_15), rt in Omega(1 + n442_15)


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
-, F, <=, f

They will be analysed ascendingly in the following order:
- < F
<= < F
f < F
- < f
<= < f

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s9_15(n1131_15), gen_0':s9_15(n1131_15)) -> gen_0':s9_15(0), rt in Omega(0)

Induction Base:
-(gen_0':s9_15(0), gen_0':s9_15(0)) ->_R^Omega(0)
gen_0':s9_15(0)

Induction Step:
-(gen_0':s9_15(+(n1131_15, 1)), gen_0':s9_15(+(n1131_15, 1))) ->_R^Omega(0)
-(gen_0':s9_15(n1131_15), gen_0':s9_15(n1131_15)) ->_IH
gen_0':s9_15(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(72)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Lemmas:
-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15)) -> gen_c:c18_15(n13_15), rt in Omega(1 + n13_15)
<='(gen_0':s9_15(n442_15), gen_0':s9_15(n442_15)) -> gen_c2:c3:c410_15(n442_15), rt in Omega(1 + n442_15)
-(gen_0':s9_15(n1131_15), gen_0':s9_15(n1131_15)) -> gen_0':s9_15(0), rt in Omega(0)


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
<=, F, f

They will be analysed ascendingly in the following order:
<= < F
f < F
<= < f

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<=(gen_0':s9_15(n1779_15), gen_0':s9_15(n1779_15)) -> true, rt in Omega(0)

Induction Base:
<=(gen_0':s9_15(0), gen_0':s9_15(0)) ->_R^Omega(0)
true

Induction Step:
<=(gen_0':s9_15(+(n1779_15, 1)), gen_0':s9_15(+(n1779_15, 1))) ->_R^Omega(0)
<=(gen_0':s9_15(n1779_15), gen_0':s9_15(n1779_15)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(74)
Obligation:
Innermost TRS:
Rules:
-'(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))
-(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))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
<=' :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> true:false -> true:false -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
PERFECTP :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c9:c10:c11:c12:c13:c14 -> c7:c8
F :: 0':s -> 0':s -> 0':s -> 0':s -> c9:c10:c11:c12:c13:c14
c9 :: c9:c10:c11:c12:c13:c14
c10 :: c9:c10:c11:c12:c13:c14
c11 :: c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
- :: 0':s -> 0':s -> 0':s
c12 :: c5:c6 -> c2:c3:c4 -> c9:c10:c11:c12:c13:c14
<= :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c13 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c:c1 -> c9:c10:c11:c12:c13:c14
c14 :: c5:c6 -> c9:c10:c11:c12:c13:c14 -> c9:c10:c11:c12:c13:c14
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c5:c64_15 :: c5:c6
hole_true:false5_15 :: true:false
hole_c7:c86_15 :: c7:c8
hole_c9:c10:c11:c12:c13:c147_15 :: c9:c10:c11:c12:c13:c14
gen_c:c18_15 :: Nat -> c:c1
gen_0':s9_15 :: Nat -> 0':s
gen_c2:c3:c410_15 :: Nat -> c2:c3:c4
gen_c9:c10:c11:c12:c13:c1411_15 :: Nat -> c9:c10:c11:c12:c13:c14


Lemmas:
-'(gen_0':s9_15(n13_15), gen_0':s9_15(n13_15)) -> gen_c:c18_15(n13_15), rt in Omega(1 + n13_15)
<='(gen_0':s9_15(n442_15), gen_0':s9_15(n442_15)) -> gen_c2:c3:c410_15(n442_15), rt in Omega(1 + n442_15)
-(gen_0':s9_15(n1131_15), gen_0':s9_15(n1131_15)) -> gen_0':s9_15(0), rt in Omega(0)
<=(gen_0':s9_15(n1779_15), gen_0':s9_15(n1779_15)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c18_15(0) <=> c
gen_c:c18_15(+(x, 1)) <=> c1(gen_c:c18_15(x))
gen_0':s9_15(0) <=> 0'
gen_0':s9_15(+(x, 1)) <=> s(gen_0':s9_15(x))
gen_c2:c3:c410_15(0) <=> c2
gen_c2:c3:c410_15(+(x, 1)) <=> c4(gen_c2:c3:c410_15(x))
gen_c9:c10:c11:c12:c13:c1411_15(0) <=> c9
gen_c9:c10:c11:c12:c13:c1411_15(+(x, 1)) <=> c11(gen_c9:c10:c11:c12:c13:c1411_15(x), c)


The following defined symbols remain to be analysed:
f, F

They will be analysed ascendingly in the following order:
f < F