/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^3), O(n^4))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, n^4).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 1039 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 346 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 572 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 196 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 584 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 557 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 9 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) FinalProof [FINISHED, 0 ms]
    (68) BOUNDS(1, n^4)
(69) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(70) CpxTRS
    (71) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) typed CpxTrs
    (73) OrderProof [LOWER BOUND(ID), 0 ms]
    (74) typed CpxTrs
    (75) RewriteLemmaProof [LOWER BOUND(ID), 229 ms]
    (76) BEST
        (77) proven lower bound
            (78) LowerBoundPropagationProof [FINISHED, 0 ms]
            (79) BOUNDS(n^1, INF)
        (80) typed CpxTrs
            (81) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
            (82) BEST
                (83) proven lower bound
                    (84) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (85) BOUNDS(n^3, INF)
                (86) typed CpxTrs
                    (87) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
                    (88) typed CpxTrs
                    (89) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
                    (90) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, n^4).


The TRS R consists of the following rules:

   p(s(x)) -> x
   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of plus: p
The following defined symbols can occur below the 1th argument of plus: p, plus, times, div, quot
The following defined symbols can occur below the 0th argument of p: p, plus, times
The following defined symbols can occur below the 0th argument of if: p, plus, eq, divides, div, times, quot
The following defined symbols can occur below the 1th argument of eq: p, plus, div, times, quot
The following defined symbols can occur below the 0th argument of times: div, quot

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
div(div(x, y), z) -> div(x, times(y, z))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^4).


The TRS R consists of the following rules:

   p(s(x)) -> x
   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)

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

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
times(s(x), [])
div(x, [])
divides(y, [])
divides([], x)
prime(s(s([])))
pr(x, s(s([])))
pr([], s(s(y)))


The defined contexts are:
plus(x0, [])
plus([], x1)
if([], x1, s(x2))
eq(x0, [])
times([], x1)
p(s([]))


[] just represents basic- or constructor-terms in the following defined contexts:
plus(x0, [])
plus([], x1)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^4).


The TRS R consists of the following rules:

   p(s(x)) -> x
   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^4).


The TRS R consists of the following rules:

   p(s(x)) -> x [1]
   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   plus(x, s(y)) -> s(plus(x, p(s(y)))) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   divides(y, x) -> eq(x, times(div(x, y), y)) [1]
   prime(s(s(x))) -> pr(s(s(x)), s(x)) [1]
   pr(x, s(0)) -> true [1]
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) [1]
   if(true, x, y) -> false [1]
   if(false, x, y) -> pr(x, y) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   p(s(x)) -> x [1]
   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   plus(x, s(y)) -> s(plus(x, p(s(y)))) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   divides(y, x) -> eq(x, times(div(x, y), y)) [1]
   prime(s(s(x))) -> pr(s(s(x)), s(x)) [1]
   pr(x, s(0)) -> true [1]
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) [1]
   if(true, x, y) -> false [1]
   if(false, x, y) -> pr(x, y) [1]

The TRS has the following type information:
p :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
0 :: s:0
times :: s:0 -> s:0 -> s:0
div :: s:0 -> s:0 -> s:0
quot :: s:0 -> s:0 -> s:0 -> s:0
eq :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
divides :: s:0 -> s:0 -> true:false
prime :: s:0 -> true:false
pr :: s:0 -> s:0 -> true:false
if :: true:false -> s:0 -> s:0 -> true:false

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

(9) 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:

   prime_1
   pr_2
   if_3

(c) The following functions are completely defined:

   p_1
   times_2
   div_2
   divides_2
   plus_2
   eq_2
   quot_3

Due to the following rules being added:

   p(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

And the following fresh constants: none

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

(10)
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:

   p(s(x)) -> x [1]
   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   plus(x, s(y)) -> s(plus(x, p(s(y)))) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   divides(y, x) -> eq(x, times(div(x, y), y)) [1]
   prime(s(s(x))) -> pr(s(s(x)), s(x)) [1]
   pr(x, s(0)) -> true [1]
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) [1]
   if(true, x, y) -> false [1]
   if(false, x, y) -> pr(x, y) [1]
   p(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
p :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
0 :: s:0
times :: s:0 -> s:0 -> s:0
div :: s:0 -> s:0 -> s:0
quot :: s:0 -> s:0 -> s:0 -> s:0
eq :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
divides :: s:0 -> s:0 -> true:false
prime :: s:0 -> true:false
pr :: s:0 -> s:0 -> true:false
if :: true:false -> s:0 -> s:0 -> true:false

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

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

(12)
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:

   p(s(x)) -> x [1]
   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [2]
   plus(s(x), y) -> s(plus(0, y)) [1]
   plus(x, s(y)) -> s(plus(x, y)) [2]
   plus(x, s(y)) -> s(plus(x, 0)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(0), y) -> plus(y, 0) [2]
   times(s(s(0)), y) -> plus(y, y) [2]
   times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   divides(y, 0) -> eq(0, times(0, y)) [2]
   divides(y, x) -> eq(x, times(quot(x, y, y), y)) [2]
   prime(s(s(x))) -> pr(s(s(x)), s(x)) [1]
   pr(x, s(0)) -> true [1]
   pr(x, s(s(y))) -> if(eq(x, times(div(x, s(s(y))), s(s(y)))), x, s(y)) [2]
   if(true, x, y) -> false [1]
   if(false, x, y) -> pr(x, y) [1]
   p(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
p :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
0 :: s:0
times :: s:0 -> s:0 -> s:0
div :: s:0 -> s:0 -> s:0
quot :: s:0 -> s:0 -> s:0 -> s:0
eq :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
divides :: s:0 -> s:0 -> true:false
prime :: s:0 -> true:false
pr :: s:0 -> s:0 -> true:false
if :: true:false -> s:0 -> s:0 -> true:false

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

(13) 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
   true => 1
   false => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 1 }-> quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(x, times(quot(x, y, y), y)) :|: y >= 0, x >= 0, z'' = x, z' = y
   divides(z', z'') -{ 2 }-> eq(0, times(0, y)) :|: z'' = 0, y >= 0, z' = y
   eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0
   eq(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(x, y) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   plus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(x, y) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   plus(z', z'') -{ 1 }-> 1 + plus(x, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   plus(z', z'') -{ 1 }-> 1 + plus(0, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   pr(z', z'') -{ 2 }-> if(eq(x, times(div(x, 1 + (1 + y)), 1 + (1 + y))), x, 1 + y) :|: z' = x, x >= 0, y >= 0, z'' = 1 + (1 + y)
   pr(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + x), 1 + x) :|: x >= 0, z' = 1 + (1 + x)
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   quot(z', z'', z1) -{ 1 }-> 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z
   times(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(y, y) :|: z' = 1 + (1 + 0), z'' = y, y >= 0
   times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0


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

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

   { eq }
   { div, quot }
   { plus }
   { p }
   { times }
   { if, pr }
   { divides }
   { prime }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {eq}, {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {eq}, {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {eq}, {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z'

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

(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:

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {div,quot}, {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: ?, size: O(n^1) [z']
  quot: runtime: ?, size: O(n^1) [1 + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 3*z' + z''

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 2 }-> eq(z'', times(quot(z'', z', z'), z')) :|: z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 2 }-> if(eq(z', times(div(z', 1 + (1 + (z'' - 2))), 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [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:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {plus}, {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z' + 2*z''

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', 0) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']

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

(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:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {p}, {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: ?, size: O(n^1) [z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']

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

(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:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}, {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: ?, size: O(n^2) [2*z'*z'' + 4*z'']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 7 + z' + 3*z'' }-> eq(z'', times(s2, z')) :|: s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   divides(z', z'') -{ 2 }-> eq(0, times(0, z')) :|: z'' = 0, z' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 5 + 3*z' }-> if(eq(z', times(s3, 1 + (1 + (z'' - 2)))), z', 1 + (z'' - 2)) :|: s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']

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

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 16 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 4*z' + 6*z'' }-> if(s19, z', 1 + (z'' - 2)) :|: s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 16 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 4*z' + 6*z'' }-> if(s19, z', 1 + (z'' - 2)) :|: s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {if,pr}, {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: ?, size: O(1) [1]
  pr: runtime: ?, size: O(1) [1]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2

Computed RUNTIME bound using KoAT for: pr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> pr(z'', z1) :|: z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 16 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 4*z' + 6*z'' }-> if(s19, z', 1 + (z'' - 2)) :|: s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ 1 }-> pr(1 + (1 + (z' - 2)), 1 + (z' - 2)) :|: z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {divides}, {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]
  divides: runtime: ?, size: O(1) [1]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: divides
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 35 + 27*z' + 22*z'*z'' + 8*z'*z''^2 + 8*z''

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]
  divides: runtime: O(n^3) [35 + 27*z' + 22*z'*z'' + 8*z'*z''^2 + 8*z''], size: O(1) [1]

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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]
  divides: runtime: O(n^3) [35 + 27*z' + 22*z'*z'' + 8*z'*z''^2 + 8*z''], size: O(1) [1]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {prime}
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]
  divides: runtime: O(n^3) [35 + 27*z' + 22*z'*z'' + 8*z'*z''^2 + 8*z''], size: O(1) [1]
  prime: runtime: ?, size: O(1) [1]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: prime
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 12*z' + 16*z'^2 + 16*z'^4

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 6 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   divides(z', z'') -{ 13 + 6*z' }-> s15 :|: s14 >= 0, s14 <= 2 * (0 * z') + 4 * z', s15 >= 0, s15 <= 1, z'' = 0, z' >= 0
   divides(z', z'') -{ 18 + 4*s2 + 6*s2*z' + 8*s2^2*z' + 7*z' + 4*z'' }-> s17 :|: s16 >= 0, s16 <= 2 * (s2 * z') + 4 * z', s17 >= 0, s17 <= 1, s2 >= 0, s2 <= z'' + 1, z' >= 0, z'' >= 0
   eq(z', z'') -{ 3 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   if(z', z'', z1) -{ 9 + 6*z'' + 4*z''*z1 + 12*z''*z1^2 + 8*z''^2 + 16*z''^2*z1^2 + 22*z1 + 12*z1^2 }-> s22 :|: s22 >= 0, s22 <= 1, z'' >= 0, z1 >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z' = 1
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2*z' + 2*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + 2*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + z'', z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 + 2*z' + 2*z'' }-> 1 + s7 :|: s7 >= 0, s7 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0
   plus(z', z'') -{ 2 + 2*z' }-> 1 + s8 :|: s8 >= 0, s8 <= z' + 0, z' >= 0, z'' - 1 >= 0
   pr(z', z'') -{ 7 + 4*s3 + 6*s3*z'' + 8*s3^2*z'' + 2*z' + -2*z'*z'' + 12*z'*z''^2 + 8*z'^2 + -24*z'^2*z'' + 16*z'^2*z''^2 + 22*z'' + 12*z''^2 }-> s21 :|: s21 >= 0, s21 <= 1, s18 >= 0, s18 <= 2 * (s3 * (1 + (1 + (z'' - 2)))) + 4 * (1 + (1 + (z'' - 2))), s19 >= 0, s19 <= 1, s3 >= 0, s3 <= z', z' >= 0, z'' - 2 >= 0
   pr(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   prime(z') -{ -1 + 12*z' + 16*z'^2 + -20*z'^3 + 16*z'^4 }-> s20 :|: s20 >= 0, s20 <= 1, z' - 2 >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s1 :|: s1 >= 0, s1 <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + 4*z'' }-> s10 :|: s10 >= 0, s10 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 4 + 2*s11 + 2*s12 + 4*z' + -26*z'*z'' + 8*z'^2*z'' + 30*z'' }-> s13 :|: s11 >= 0, s11 <= 2 * ((z' - 2) * z'') + 4 * z'', s12 >= 0, s12 <= z'' + s11, s13 >= 0, s13 <= z'' + s12, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 3 + 2*z'' }-> s9 :|: s9 >= 0, s9 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + 2*z' + 2*z''], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  times: runtime: O(n^3) [8 + 4*z' + 6*z'*z'' + 8*z'^2*z'' + 6*z''], size: O(n^2) [2*z'*z'' + 4*z'']
  if: runtime: O(n^4) [19 + 8*z'' + 22*z''*z1 + 12*z''*z1^2 + 8*z''^2*z1 + 16*z''^2*z1^2 + 40*z1 + 12*z1^2], size: O(1) [1]
  pr: runtime: O(n^4) [8 + 6*z' + 4*z'*z'' + 12*z'*z''^2 + 8*z'^2 + 16*z'^2*z''^2 + 22*z'' + 12*z''^2], size: O(1) [1]
  divides: runtime: O(n^3) [35 + 27*z' + 22*z'*z'' + 8*z'*z''^2 + 8*z''], size: O(1) [1]
  prime: runtime: O(n^4) [12*z' + 16*z'^2 + 16*z'^4], size: O(1) [1]

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

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

(68)
BOUNDS(1, n^4)

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

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

(70)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   p(s(x)) -> x
   plus(x, 0') -> x
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0', y) -> 0'
   times(s(0'), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(0', s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))
   eq(0', 0') -> true
   eq(s(x), 0') -> false
   eq(0', s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0')) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)

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

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

(72)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'

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

(73) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, div, quot, eq, pr

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(74)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, eq, pr

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), p(s(gen_s:0'3_0(n5_0))))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) ->_IH
s(gen_s:0'3_0(+(a, c6_0)))

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

(76)
Complex Obligation (BEST)

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

(77)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, eq, pr

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

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

(79)
BOUNDS(n^1, INF)

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

(80)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
times, div, quot, eq, pr

They will be analysed ascendingly in the following order:
times < div
div = quot

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

(81) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0)

Induction Base:
times(gen_s:0'3_0(0), gen_s:0'3_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_s:0'3_0(+(n854_0, 1)), gen_s:0'3_0(b)) ->_R^Omega(1)
plus(gen_s:0'3_0(b), times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b))) ->_IH
plus(gen_s:0'3_0(b), gen_s:0'3_0(*(c855_0, b))) ->_L^Omega(1 + b*n854_0)
gen_s:0'3_0(+(*(n854_0, b), b))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
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(82)
Complex Obligation (BEST)

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(83)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
times, div, quot, eq, pr

They will be analysed ascendingly in the following order:
times < div
div = quot

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(84) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(85)
BOUNDS(n^3, INF)

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(86)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
eq, div, quot, pr

They will be analysed ascendingly in the following order:
div = quot

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(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0)

Induction Base:
eq(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_s:0'3_0(+(n1961_0, 1)), gen_s:0'3_0(+(n1961_0, 1))) ->_R^Omega(1)
eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(88)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0)
eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
pr, div, quot

They will be analysed ascendingly in the following order:
div = quot

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(89) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) -> gen_s:0'3_0(0), rt in Omega(1 + n2634_0)

Induction Base:
quot(gen_s:0'3_0(0), gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(c)) ->_R^Omega(1)
0'

Induction Step:
quot(gen_s:0'3_0(+(n2634_0, 1)), gen_s:0'3_0(+(1, +(n2634_0, 1))), gen_s:0'3_0(c)) ->_R^Omega(1)
quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) ->_IH
gen_s:0'3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(90)
Obligation:
TRS:
Rules:
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
0' :: s:0'
times :: s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0' -> s:0'
eq :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
pr :: s:0' -> s:0' -> true:false
if :: true:false -> s:0' -> s:0' -> true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0)
eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0)
quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) -> gen_s:0'3_0(0), rt in Omega(1 + n2634_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
div

They will be analysed ascendingly in the following order:
div = quot