WORST_CASE(Omega(n^1),O(n^3))
proof of input_eK8TQN0rKu.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 4606 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 994 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^3)
(51) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CdtProblem
    (53) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxRelTRS
    (55) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CpxRelTRS
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) typed CpxTrs
    (59) OrderProof [LOWER BOUND(ID), 6 ms]
    (60) typed CpxTrs
    (61) RewriteLemmaProof [LOWER BOUND(ID), 335 ms]
    (62) BEST
        (63) proven lower bound
            (64) LowerBoundPropagationProof [FINISHED, 0 ms]
            (65) BOUNDS(n^1, INF)
        (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
            (72) 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:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(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, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(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:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, IF_3, PERFECTP_1, F_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11, c12_2, c13_2, c14_3, c15_2


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
Removed 9 trailing nodes:
   IF(true, z0, z1) -> c6
   F(0, z0, s(z1), z2) -> c11
   LE(s(z0), 0) -> c4
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   MINUS(0, z0) -> c
   F(0, z0, 0, z1) -> c10
   LE(0, z0) -> c3
   MINUS(s(z0), 0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_2, c14_3, c15_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


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

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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(s(z0), 0) -> s(z0)
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_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:

   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))

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

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

   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(s(z0), 0) -> s(z0) [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(s(z0), 0) -> s(z0) [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
LE :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
F :: s:0 -> s:0 -> s:0 -> s:0 -> c12:c13:c14:c15
0 :: s:0
c12 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
minus :: s:0 -> s:0 -> s:0
c13 :: c5 -> c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
c15 :: c12:c13:c14:c15 -> c12:c13:c14:c15

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

(15) 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:

   MINUS_2
   LE_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

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

(16)
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:

   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
LE :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
F :: s:0 -> s:0 -> s:0 -> s:0 -> c12:c13:c14:c15
0 :: s:0
c12 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
minus :: s:0 -> s:0 -> s:0
c13 :: c5 -> c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
c15 :: c12:c13:c14:c15 -> c12:c13:c14:c15
const :: c2
const1 :: c5
const2 :: c12:c13:c14:c15

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

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

(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:

   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   F(s(z0), 0, 0, z2) -> c12(F(z0, z2, 0, z2), MINUS(0, s(z0))) [1]
   F(s(z0), 0, s(z0'), z2) -> c12(F(z0, z2, minus(z0', z0), z2), MINUS(s(z0'), s(z0))) [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, 0, z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) [1]
   F(s(z0), s(0), z2, z3) -> c14(F(s(z0), 0, z2, z3), MINUS(0, z0)) [1]
   F(s(s(z1')), s(s(z0'')), z2, z3) -> c14(F(s(s(z1')), minus(z0'', z1'), z2, z3), MINUS(s(z0''), s(z1'))) [1]
   F(s(0), s(s(z01)), z2, z3) -> c14(F(s(0), s(z01), z2, z3), MINUS(s(z01), 0)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), 0, z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
LE :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
F :: s:0 -> s:0 -> s:0 -> s:0 -> c12:c13:c14:c15
0 :: s:0
c12 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
minus :: s:0 -> s:0 -> s:0
c13 :: c5 -> c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c2 -> c12:c13:c14:c15
c15 :: c12:c13:c14:c15 -> c12:c13:c14:c15
const :: c2
const1 :: c5
const2 :: c12:c13:c14:c15

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

(19) 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

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 + LE(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, 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 + F(z0, z2, minus(z0', z0), z2) + MINUS(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) + MINUS(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(z0, z2, 0, z2) + MINUS(0, 1 + z0) :|: z'' = 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + z0, 0, z2, z3) + MINUS(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 + z0, 0, z2, z3) + MINUS(0, z0) :|: z'' = z2, z = 1 + z0, z' = 1 + 0, z0 >= 0, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, 1 + z01, z2, z3) + MINUS(1 + z01, 0) :|: z'' = z2, z01 >= 0, z = 1 + 0, z' = 1 + (1 + z01), z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + (1 + z1'), minus(z0'', z1'), z2, z3) + MINUS(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + (1 + z0''), z1' >= 0, z = 1 + (1 + z1'), z0'' >= 0, z4 = z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0, z' = 0


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

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

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

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


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

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

   { minus }
   { LE }
   { MINUS }
   { F }

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

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

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

Function symbols to be analyzed: {minus}, {LE}, {MINUS}, {F}

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

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

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

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

Function symbols to be analyzed: {minus}, {LE}, {MINUS}, {F}

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

(27) 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

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

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

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

Function symbols to be analyzed: {minus}, {LE}, {MINUS}, {F}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(29) 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

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

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

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

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

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

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

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

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

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo 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:

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

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

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

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

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

   F(z, z', z'', z4) -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 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, s, z4) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s6 :|: s6 >= 0, s6 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, 1 + (z' - 2), z'', z4) + s9 :|: s9 >= 0, s9 <= 0, z' - 2 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s10 :|: s10 >= 0, s10 <= 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) + s7 :|: s7 >= 0, s7 <= 0, z' = 1 + 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s', z'', z4) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= z' - 2, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   LE(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0

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

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

(45) 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 + 4*z + 16*z*z4 + 4*z'

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

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

   F(z, z', z'', z4) -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 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, s, z4) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s6 :|: s6 >= 0, s6 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, 1 + (z' - 2), z'', z4) + s9 :|: s9 >= 0, s9 <= 0, z' - 2 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s10 :|: s10 >= 0, s10 <= 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) + s7 :|: s7 >= 0, s7 <= 0, z' = 1 + 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s', z'', z4) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= z' - 2, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   LE(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]
  F: runtime: ?, size: O(n^2) [1 + 4*z + 16*z*z4 + 4*z']

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

(47) 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

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

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

   F(z, z', z'', z4) -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 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, s, z4) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 + z }-> 1 + F(z - 1, z4, 0, z4) + s6 :|: s6 >= 0, s6 <= 0, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(1 + 0, 1 + (z' - 2), z'', z4) + s9 :|: s9 >= 0, s9 <= 0, z' - 2 >= 0, z = 1 + 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (z - 1), 0, z'', z4) + s10 :|: s10 >= 0, s10 <= 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) + s7 :|: s7 >= 0, s7 <= 0, z' = 1 + 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ z }-> 1 + F(1 + (1 + (z - 2)), s', z'', z4) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= z' - 2, z - 2 >= 0, z' - 2 >= 0, z'' >= 0, z4 >= 0
   LE(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]
  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 + 4*z + 16*z*z4 + 4*z']

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

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

(50)
BOUNDS(1, n^3)

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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, IF_3, PERFECTP_1, F_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11, c12_2, c13_2, c14_3, c15_2


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

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

(54)
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:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

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

(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:

   MINUS(0', z0) -> c
   MINUS(s(z0), 0') -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0', z0) -> c3
   LE(s(z0), 0') -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0') -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
   F(0', z0, 0', z1) -> c10
   F(0', z0, s(z1), z2) -> c11
   F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0', z0) -> 0'
   minus(s(z0), 0') -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

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

(58)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15

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

(59) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, LE, F, minus, le, f

They will be analysed ascendingly in the following order:
MINUS < F
LE < F
minus < F
le < F
f < F
minus < f
le < f

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

(60)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


The following defined symbols remain to be analysed:
MINUS, LE, F, minus, le, f

They will be analysed ascendingly in the following order:
MINUS < F
LE < F
minus < F
le < F
f < F
minus < f
le < f

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16)) -> gen_c:c1:c28_16(n13_16), rt in Omega(1 + n13_16)

Induction Base:
MINUS(gen_0':s9_16(0), gen_0':s9_16(0)) ->_R^Omega(1)
c

Induction Step:
MINUS(gen_0':s9_16(+(n13_16, 1)), gen_0':s9_16(+(n13_16, 1))) ->_R^Omega(1)
c2(MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16))) ->_IH
c2(gen_c:c1:c28_16(c14_16))

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

(62)
Complex Obligation (BEST)

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

(63)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


The following defined symbols remain to be analysed:
MINUS, LE, F, minus, le, f

They will be analysed ascendingly in the following order:
MINUS < F
LE < F
minus < F
le < F
f < F
minus < f
le < f

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

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

(65)
BOUNDS(n^1, INF)

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

(66)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16)) -> gen_c:c1:c28_16(n13_16), rt in Omega(1 + n13_16)


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


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

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

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s9_16(n647_16), gen_0':s9_16(n647_16)) -> gen_c3:c4:c510_16(n647_16), rt in Omega(1 + n647_16)

Induction Base:
LE(gen_0':s9_16(0), gen_0':s9_16(0)) ->_R^Omega(1)
c3

Induction Step:
LE(gen_0':s9_16(+(n647_16, 1)), gen_0':s9_16(+(n647_16, 1))) ->_R^Omega(1)
c5(LE(gen_0':s9_16(n647_16), gen_0':s9_16(n647_16))) ->_IH
c5(gen_c3:c4:c510_16(c648_16))

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

(68)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16)) -> gen_c:c1:c28_16(n13_16), rt in Omega(1 + n13_16)
LE(gen_0':s9_16(n647_16), gen_0':s9_16(n647_16)) -> gen_c3:c4:c510_16(n647_16), rt in Omega(1 + n647_16)


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


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

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

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s9_16(n1356_16), gen_0':s9_16(n1356_16)) -> gen_0':s9_16(0), rt in Omega(0)

Induction Base:
minus(gen_0':s9_16(0), gen_0':s9_16(0)) ->_R^Omega(0)
0'

Induction Step:
minus(gen_0':s9_16(+(n1356_16, 1)), gen_0':s9_16(+(n1356_16, 1))) ->_R^Omega(0)
minus(gen_0':s9_16(n1356_16), gen_0':s9_16(n1356_16)) ->_IH
gen_0':s9_16(0)

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

(70)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16)) -> gen_c:c1:c28_16(n13_16), rt in Omega(1 + n13_16)
LE(gen_0':s9_16(n647_16), gen_0':s9_16(n647_16)) -> gen_c3:c4:c510_16(n647_16), rt in Omega(1 + n647_16)
minus(gen_0':s9_16(n1356_16), gen_0':s9_16(n1356_16)) -> gen_0':s9_16(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


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

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

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s9_16(n2185_16), gen_0':s9_16(n2185_16)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s9_16(0), gen_0':s9_16(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s9_16(+(n2185_16, 1)), gen_0':s9_16(+(n2185_16, 1))) ->_R^Omega(0)
le(gen_0':s9_16(n2185_16), gen_0':s9_16(n2185_16)) ->_IH
true

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:
MINUS(0', z0) -> c
MINUS(s(z0), 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7
PERFECTP(0') -> c8
PERFECTP(s(z0)) -> c9(F(z0, s(0'), s(z0), s(z0)))
F(0', z0, 0', z1) -> c10
F(0', z0, s(z1), z2) -> c11
F(s(z0), 0', z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
minus(0', z0) -> 0'
minus(s(z0), 0') -> s(z0)
minus(s(z0), s(z1)) -> minus(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(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, minus(z1, s(z0)), z2)
f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0':s -> 0':s -> 0':s -> 0':s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0':s -> 0':s -> 0':s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0':s -> 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
hole_c:c1:c21_16 :: c:c1:c2
hole_0':s2_16 :: 0':s
hole_c3:c4:c53_16 :: c3:c4:c5
hole_c6:c74_16 :: c6:c7
hole_true:false5_16 :: true:false
hole_c8:c96_16 :: c8:c9
hole_c10:c11:c12:c13:c14:c157_16 :: c10:c11:c12:c13:c14:c15
gen_c:c1:c28_16 :: Nat -> c:c1:c2
gen_0':s9_16 :: Nat -> 0':s
gen_c3:c4:c510_16 :: Nat -> c3:c4:c5
gen_c10:c11:c12:c13:c14:c1511_16 :: Nat -> c10:c11:c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s9_16(n13_16), gen_0':s9_16(n13_16)) -> gen_c:c1:c28_16(n13_16), rt in Omega(1 + n13_16)
LE(gen_0':s9_16(n647_16), gen_0':s9_16(n647_16)) -> gen_c3:c4:c510_16(n647_16), rt in Omega(1 + n647_16)
minus(gen_0':s9_16(n1356_16), gen_0':s9_16(n1356_16)) -> gen_0':s9_16(0), rt in Omega(0)
le(gen_0':s9_16(n2185_16), gen_0':s9_16(n2185_16)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_16(0) <=> c
gen_c:c1:c28_16(+(x, 1)) <=> c2(gen_c:c1:c28_16(x))
gen_0':s9_16(0) <=> 0'
gen_0':s9_16(+(x, 1)) <=> s(gen_0':s9_16(x))
gen_c3:c4:c510_16(0) <=> c3
gen_c3:c4:c510_16(+(x, 1)) <=> c5(gen_c3:c4:c510_16(x))
gen_c10:c11:c12:c13:c14:c1511_16(0) <=> c10
gen_c10:c11:c12:c13:c14:c1511_16(+(x, 1)) <=> c12(gen_c10:c11:c12:c13:c14:c1511_16(x), c)


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

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