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


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KILLED
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) typed CpxTrs
    (5) OrderProof [LOWER BOUND(ID), 0 ms]
    (6) typed CpxTrs
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
(9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(10) CpxTRS
(11) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(12) CpxRelTRS
    (13) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxWeightedTrs
        (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxTypedWeightedTrs
        (19) CompletionProof [UPPER BOUND(ID), 0 ms]
        (20) CpxTypedWeightedCompleteTrs
            (21) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (22) CpxTypedWeightedCompleteTrs
            (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (24) CpxRNTS
            (25) InliningProof [UPPER BOUND(ID), 257 ms]
            (26) CpxRNTS
            (27) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (28) CpxRNTS
            (29) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (30) CpxRNTS
            (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (32) CpxRNTS
            (33) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
            (34) CpxRNTS
            (35) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
            (36) CpxRNTS
            (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (38) CpxRNTS
            (39) IntTrsBoundProof [UPPER BOUND(ID), 3544 ms]
            (40) CpxRNTS
            (41) IntTrsBoundProof [UPPER BOUND(ID), 1174 ms]
            (42) CpxRNTS
        (43) CompletionProof [UPPER BOUND(ID), 0 ms]
        (44) CpxTypedWeightedCompleteTrs
            (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (46) CpxRNTS
    (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (48) CdtProblem
        (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
        (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CdtProblem
        (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
        (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
        (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem


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

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


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

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

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


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

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

(4)
Obligation:
TRS:
Rules:
f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
g(x) -> i(x)
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a

Types:
f :: i:a -> i:a -> i:a
i :: i:a -> i:a
g :: i:a -> i:a
a :: i:a
hole_i:a1_0 :: i:a
gen_i:a2_0 :: Nat -> i:a

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
----------------------------------------

(6)
Obligation:
TRS:
Rules:
f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
g(x) -> i(x)
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a

Types:
f :: i:a -> i:a -> i:a
i :: i:a -> i:a
g :: i:a -> i:a
a :: i:a
hole_i:a1_0 :: i:a
gen_i:a2_0 :: Nat -> i:a


Generator Equations:
gen_i:a2_0(0) <=> a
gen_i:a2_0(+(x, 1)) <=> i(gen_i:a2_0(x))


The following defined symbols remain to be analysed:
f
----------------------------------------

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

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(10)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

(11) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(12)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(c_g(x))) -> a

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

   g(x0) -> c_g(x0)

Rewrite Strategy: FULL
----------------------------------------

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

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(c_g(x))) -> a

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

   g(x0) -> c_g(x0)

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   g(x0) -> c_g(x0) [0]

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

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

(18)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   g(x0) -> c_g(x0) [0]

The TRS has the following type information:
f :: i:c_g:a -> i:c_g:a -> i:c_g:a
i :: i:c_g:a -> i:c_g:a
g :: i:c_g:a -> i:c_g:a
c_g :: i:c_g:a -> i:c_g:a
a :: i:c_g:a

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

(19) 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:

   f_2

(c) The following functions are completely defined:

   g_1

Due to the following rules being added:

   g(v0) -> a [0]

And the following fresh constants: none

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

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

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

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   g(x0) -> c_g(x0) [0]
   g(v0) -> a [0]

The TRS has the following type information:
f :: i:c_g:a -> i:c_g:a -> i:c_g:a
i :: i:c_g:a -> i:c_g:a
g :: i:c_g:a -> i:c_g:a
c_g :: i:c_g:a -> i:c_g:a
a :: i:c_g:a

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

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

(22)
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:

   f(x, x) -> f(i(x), g(i(x))) [2]
   f(x, x) -> f(i(x), g(c_g(x))) [1]
   f(x, x) -> f(i(x), g(a)) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   g(x0) -> c_g(x0) [0]
   g(v0) -> a [0]

The TRS has the following type information:
f :: i:c_g:a -> i:c_g:a -> i:c_g:a
i :: i:c_g:a -> i:c_g:a
g :: i:c_g:a -> i:c_g:a
c_g :: i:c_g:a -> i:c_g:a
a :: i:c_g:a

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

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

   a => 0

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

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

   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(0)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 2 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

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

   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 2 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0
   f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0
   f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0


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

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

   { g }
   { f }

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}

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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  g: runtime: ?, size: O(n^1) [1 + z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [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:

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  f: runtime: ?, size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  f: runtime: INF, size: INF

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

(43) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   g(v0) -> null_g [0]

And the following fresh constants:    null_g

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

(44)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   g(x0) -> c_g(x0) [0]
   g(v0) -> null_g [0]

The TRS has the following type information:
f :: i:c_g:a:null_g -> i:c_g:a:null_g -> i:c_g:a:null_g
i :: i:c_g:a:null_g -> i:c_g:a:null_g
g :: i:c_g:a:null_g -> i:c_g:a:null_g
c_g :: i:c_g:a:null_g -> i:c_g:a:null_g
a :: i:c_g:a:null_g
null_g :: i:c_g:a:null_g

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

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

   a => 0
   null_g => 0

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

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

   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(g(x))) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(47) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   G(z0) -> c
   G(z0) -> c1
   F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c3
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c5
S tuples:
   G(z0) -> c1
   F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c3
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c5
K tuples:none
Defined Rule Symbols:   f_2, g_1

Defined Pair Symbols:   G_1, F_2

Compound Symbols:   c, c1, c2_3, c3, c4_1, c5


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   F(z0, z1) -> c3
   G(z0) -> c1
   G(z0) -> c
   F(i(z0), i(c_g(z0))) -> c5

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, i(z0)) -> c4(F(z0, z0))
S tuples:
   F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, i(z0)) -> c4(F(z0, z0))
K tuples:none
Defined Rule Symbols:   f_2, g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c2_3, c4_1


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

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(g(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(g(z0))))
K tuples:none
Defined Rule Symbols:   f_2, g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(g(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(g(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(g(z0)))) by 
   F(x0, x0) -> c2(F(i(x0), c_g(g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(g(x0))))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(x0, x0) -> c2(F(i(x0), c_g(g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(g(x0))))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(x0, x0) -> c2(F(i(x0), c_g(g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(g(x0))))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(x0, x0) -> c2(F(i(x0), c_g(g(x0))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(x0, x0) -> c2(F(i(x0), i(g(x0))))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(x0, x0) -> c2(F(i(x0), i(g(x0))))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(x0, x0) -> c2(F(i(x0), i(g(x0)))) by 
   F(z0, z0) -> c2(F(i(z0), i(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(z0, z0) -> c2(F(i(z0), i(c_g(z0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(c_g(z0))))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) by 
   F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(c_g(x0))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(c_g(x0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0))))
   F(x0, x0) -> c2(F(i(x0), i(c_g(x0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(x0, x0) -> c2(F(i(x0), i(c_g(x0))))
   F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), g(i(z0))))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(i(z0)))) by 
   F(x0, x0) -> c2(F(i(x0), c_g(i(x0))))
   F(x0, x0) -> c2(F(i(x0), i(i(x0))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(x0, x0) -> c2(F(i(x0), c_g(i(x0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(x0, x0) -> c2(F(i(x0), c_g(i(x0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(x0, x0) -> c2(F(i(x0), c_g(i(x0))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
S tuples:
   F(z0, i(z0)) -> c4(F(z0, z0))
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(73) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0, i(z0)) -> c4(F(z0, z0)) by 
   F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0)))
S tuples:
   F(z0, z0) -> c2(F(i(z0), i(i(z0))))
   F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c4_1


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

(75) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0, z0) -> c2(F(i(z0), i(i(z0)))) by 
   F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0)))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0)))
   F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0)))))
S tuples:
   F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0)))
   F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1


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

(77) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) by 
   F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0)))))
   F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0))))
S tuples:
   F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0)))))
   F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c4_1


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) by 
   F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0))))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0))))
   F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0))))))
S tuples:
   F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0))))
   F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c4_1, c2_1