/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 201 ms]
(4) CpxRelTRS
(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), 50 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), 68 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 944 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 425 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 313 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 166 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 198 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) FinalProof [FINISHED, 0 ms]
    (56) BOUNDS(1, n^2)
(57) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(58) TRS for Loop Detection
    (59) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (60) BEST
        (61) proven lower bound
            (62) LowerBoundPropagationProof [FINISHED, 0 ms]
            (63) BOUNDS(n^1, INF)
        (64) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   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)))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))


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

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


The TRS R consists of the following rules:

   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)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   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)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))

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^2).


The TRS R consists of the following rules:

   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]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

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:

   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]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

The TRS has the following type information:
div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
0 :: 0:s:cons_div:cons_quot
quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encArg :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_0 :: 0:s:cons_div:cons_quot
encode_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot

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:
none

(c) The following functions are completely defined:

   div_2
   quot_3
   encArg_1
   encode_div_2
   encode_0
   encode_quot_3
   encode_s_1

Due to the following rules being added:

   encArg(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_s(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:

   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]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_s(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
0 :: 0:s:cons_div:cons_quot
quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encArg :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_0 :: 0:s:cons_div:cons_quot
encode_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot

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:

   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]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_s(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
0 :: 0:s:cons_div:cons_quot
quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encArg :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
cons_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_div :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_0 :: 0:s:cons_div:cons_quot
encode_quot :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot
encode_s :: 0:s:cons_div:cons_quot -> 0:s:cons_div:cons_quot

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

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = 1 + x_1
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_div(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   encode_s(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_s(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1
   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


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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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


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

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

   { encode_0 }
   { div, quot }
   { encArg }
   { encode_div }
   { encode_quot }
   { encode_s }

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {encode_0}, {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {encode_0}, {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_0
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'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {encode_0}, {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], 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'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {div,quot}, {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: ?, size: O(n^1) [z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 0 }-> div(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_div
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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: ?, size: O(n^1) [z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], 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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']
  encode_quot: runtime: ?, size: O(n^1) [1 + z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 5*z' + 3*z'^2 + 3*z'' + 3*z''^2 + 2*z1 + 3*z1^2

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

(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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [5 + 5*z' + 3*z'^2 + 3*z'' + 3*z''^2 + 2*z1 + 3*z1^2], size: O(n^1) [1 + 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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [5 + 5*z' + 3*z'^2 + 3*z'' + 3*z''^2 + 2*z1 + 3*z1^2], size: O(n^1) [1 + z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [5 + 5*z' + 3*z'^2 + 3*z'' + 3*z''^2 + 2*z1 + 3*z1^2], size: O(n^1) [1 + z']
  encode_s: runtime: ?, size: O(n^1) [1 + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z' + 3*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
   encArg(z') -{ 3 + 3*s2 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, s4 >= 0, s4 <= s2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 5 + 3*s5 + s6 + 2*x_1 + 3*x_1^2 + 2*x_2 + 3*x_2^2 + 2*x_3 + 3*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= s5 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -4*z' + 3*z'^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z', z'') -{ 3 + 3*s9 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= z'', s11 >= 0, s11 <= s9, z' >= 0, z'' >= 0
   encode_div(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 5 + 3*s12 + s13 + 2*z' + 3*z'^2 + 2*z'' + 3*z''^2 + 2*z1 + 3*z1^2 }-> s15 :|: s12 >= 0, s12 <= z', s13 >= 0, s13 <= z'', s14 >= 0, s14 <= z1, s15 >= 0, s15 <= s12 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z' + 3*z'^2 }-> 1 + s16 :|: s16 >= 0, s16 <= z', z' >= 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 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  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']
  encArg: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [z']
  encode_div: runtime: O(n^2) [3 + 5*z' + 3*z'^2 + 2*z'' + 3*z''^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [5 + 5*z' + 3*z'^2 + 3*z'' + 3*z''^2 + 2*z1 + 3*z1^2], size: O(n^1) [1 + z']
  encode_s: runtime: O(n^2) [2*z' + 3*z'^2], size: O(n^1) [1 + z']

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

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

(56)
BOUNDS(1, n^2)

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(57) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(58)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   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)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
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(59) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

quot(s(x), s(y), z) ->^+ quot(x, y, z)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




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(60)
Complex Obligation (BEST)

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

(61)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   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)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))

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

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

(63)
BOUNDS(n^1, INF)

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

(64)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   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)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST