/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^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) InliningProof [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), 154 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 229 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 270 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 384 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 227 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^1)
(45) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(46) CpxTRS
    (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) typed CpxTrs
    (49) OrderProof [LOWER BOUND(ID), 0 ms]
    (50) typed CpxTrs
    (51) RewriteLemmaProof [LOWER BOUND(ID), 232 ms]
    (52) BEST
        (53) proven lower bound
            (54) LowerBoundPropagationProof [FINISHED, 0 ms]
            (55) BOUNDS(n^1, INF)
        (56) typed CpxTrs
            (57) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
            (58) typed CpxTrs


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

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

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

The duplicating contexts are:
div(s([]), s(Y))
div(s(X), s([]))


The defined contexts are:
if([], s(x1), 0)
if(x0, s([]), 0)
div([], s(x1))
geq([], x1)
minus([], x1)


[] just represents basic- or constructor-terms in the following defined contexts:
if([], s(x1), 0)
div([], s(x1))


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

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

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

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

(7) 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:
none

(c) The following functions are completely defined:

   geq_2
   div_2
   minus_2
   if_3

Due to the following rules being added:

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

And the following fresh constants: none

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

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

(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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(0), s(0)) -> if(true, s(div(0, s(0))), 0) [3]
   div(s(X), s(0)) -> if(true, s(div(0, s(0))), 0) [2]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [3]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [2]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

(11) 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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(13) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(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') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { minus }
   { if }
   { geq }
   { div }

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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

(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') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

(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') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + 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') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(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') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z'

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: O(n^1) [5 + 2*z'], size: O(1) [1]

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

(44)
BOUNDS(1, n^1)

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

(46)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus(0', Y) -> 0'
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0') -> true
   geq(0', s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0', s(Y)) -> 0'
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

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

(48)
Obligation:
TRS:
Rules:
minus(0', Y) -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0') -> true
geq(0', s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) -> X
if(false, X, Y) -> Y

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
geq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
div :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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(49) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, geq, div

They will be analysed ascendingly in the following order:
minus < div
geq < div

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(50)
Obligation:
TRS:
Rules:
minus(0', Y) -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0') -> true
geq(0', s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) -> X
if(false, X, Y) -> Y

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
geq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
div :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, geq, div

They will be analysed ascendingly in the following order:
minus < div
geq < div

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(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)

Induction Base:
minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
gen_0':s3_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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(52)
Complex Obligation (BEST)

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(53)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
minus(0', Y) -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0') -> true
geq(0', s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) -> X
if(false, X, Y) -> Y

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
geq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
div :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, geq, div

They will be analysed ascendingly in the following order:
minus < div
geq < div

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(54) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(55)
BOUNDS(n^1, INF)

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

(56)
Obligation:
TRS:
Rules:
minus(0', Y) -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0') -> true
geq(0', s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) -> X
if(false, X, Y) -> Y

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
geq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
div :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
geq, div

They will be analysed ascendingly in the following order:
geq < div

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
geq(gen_0':s3_0(n200_0), gen_0':s3_0(n200_0)) -> true, rt in Omega(1 + n200_0)

Induction Base:
geq(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
geq(gen_0':s3_0(+(n200_0, 1)), gen_0':s3_0(+(n200_0, 1))) ->_R^Omega(1)
geq(gen_0':s3_0(n200_0), gen_0':s3_0(n200_0)) ->_IH
true

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

(58)
Obligation:
TRS:
Rules:
minus(0', Y) -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0') -> true
geq(0', s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) -> X
if(false, X, Y) -> Y

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
geq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
div :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
geq(gen_0':s3_0(n200_0), gen_0':s3_0(n200_0)) -> true, rt in Omega(1 + n200_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
div