KILLED
proof of input_Texo5DXy9Q.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) 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) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 268 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 20 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 2030 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 77 ms]
    (40) CdtProblem
    (41) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem


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

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


The TRS R consists of the following rules:

   int(0, 0) -> .(0, nil)
   int(0, s(y)) -> .(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

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

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

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


The TRS R consists of the following rules:

   int(0', 0') -> .(0', nil)
   int(0', s(y)) -> .(0', int(s(0'), s(y)))
   int(s(x), 0') -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

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

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

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


The TRS R consists of the following rules:

   int(0, 0) -> .(0, nil)
   int(0, s(y)) -> .(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

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

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
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:

   int(0, 0) -> .(0, nil) [1]
   int(0, s(y)) -> .(0, int(s(0), s(y))) [1]
   int(s(x), 0) -> nil [1]
   int(s(x), s(y)) -> int_list(int(x, y)) [1]
   int_list(nil) -> nil [1]
   int_list(.(x, y)) -> .(s(x), int_list(y)) [1]

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

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

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

   int(0, 0) -> .(0, nil) [1]
   int(0, s(y)) -> .(0, int(s(0), s(y))) [1]
   int(s(x), 0) -> nil [1]
   int(s(x), s(y)) -> int_list(int(x, y)) [1]
   int_list(nil) -> nil [1]
   int_list(.(x, y)) -> .(s(x), int_list(y)) [1]

The TRS has the following type information:
int :: 0:s -> 0:s -> nil:.
0 :: 0:s
. :: 0:s -> nil:. -> nil:.
nil :: nil:.
s :: 0:s -> 0:s
int_list :: nil:. -> nil:.

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:

   int_2
   int_list_1

Due to the following rules being added:
none

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:

   int(0, 0) -> .(0, nil) [1]
   int(0, s(y)) -> .(0, int(s(0), s(y))) [1]
   int(s(x), 0) -> nil [1]
   int(s(x), s(y)) -> int_list(int(x, y)) [1]
   int_list(nil) -> nil [1]
   int_list(.(x, y)) -> .(s(x), int_list(y)) [1]

The TRS has the following type information:
int :: 0:s -> 0:s -> nil:.
0 :: 0:s
. :: 0:s -> nil:. -> nil:.
nil :: nil:.
s :: 0:s -> 0:s
int_list :: nil:. -> nil:.

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:

   int(0, 0) -> .(0, nil) [1]
   int(0, s(y)) -> .(0, int(s(0), s(y))) [1]
   int(s(x), 0) -> nil [1]
   int(s(0), s(0)) -> int_list(.(0, nil)) [2]
   int(s(0), s(s(y'))) -> int_list(.(0, int(s(0), s(y')))) [2]
   int(s(s(x')), s(0)) -> int_list(nil) [2]
   int(s(s(x'')), s(s(y''))) -> int_list(int_list(int(x'', y''))) [2]
   int_list(nil) -> nil [1]
   int_list(.(x, y)) -> .(s(x), int_list(y)) [1]

The TRS has the following type information:
int :: 0:s -> 0:s -> nil:.
0 :: 0:s
. :: 0:s -> nil:. -> nil:.
nil :: nil:.
s :: 0:s -> 0:s
int_list :: nil:. -> nil:.

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
   nil => 0

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

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

   int(z, z') -{ 2 }-> int_list(int_list(int(x'', y''))) :|: z' = 1 + (1 + y''), y'' >= 0, x'' >= 0, z = 1 + (1 + x'')
   int(z, z') -{ 2 }-> int_list(0) :|: x' >= 0, z = 1 + (1 + x'), z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + y')) :|: z' = 1 + (1 + y'), z = 1 + 0, y' >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + y) :|: z' = 1 + y, y >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0


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

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

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

   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0


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

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

   { int_list }
   { int }

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

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

   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int_list}, {int}

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

(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:

   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int_list}, {int}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int_list}, {int}
Previous analysis results are:
  int_list: runtime: ?, size: O(n^1) [2*z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int}
Previous analysis results are:
  int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(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:

   int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int}
Previous analysis results are:
  int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: int
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

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

   int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int}
Previous analysis results are:
  int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  int: runtime: ?, size: EXP

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0
   int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0
   int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {int}
Previous analysis results are:
  int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  int: runtime: INF, size: EXP

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

(31) 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:
none

And the following fresh constants: none

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

(32)
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:

   int(0, 0) -> .(0, nil) [1]
   int(0, s(y)) -> .(0, int(s(0), s(y))) [1]
   int(s(x), 0) -> nil [1]
   int(s(x), s(y)) -> int_list(int(x, y)) [1]
   int_list(nil) -> nil [1]
   int_list(.(x, y)) -> .(s(x), int_list(y)) [1]

The TRS has the following type information:
int :: 0:s -> 0:s -> nil:.
0 :: 0:s
. :: 0:s -> nil:. -> nil:.
nil :: nil:.
s :: 0:s -> 0:s
int_list :: nil:. -> nil:.

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

(33) 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
   nil => 0

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

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

   int(z, z') -{ 1 }-> int_list(int(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   int(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + y) :|: z' = 1 + y, y >= 0, z = 0
   int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0
   int_list(z) -{ 1 }-> 0 :|: z = 0
   int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, 0) -> c
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), 0) -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
S tuples:
   INT(0, 0) -> c
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), 0) -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:none
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_1


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   INT_LIST(nil) -> c4
   INT(s(z0), 0) -> c2
   INT(0, 0) -> c

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
S tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:none
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c3_2, c5_1


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

(39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = [1] + x_1
   POL(0) = [1]
   POL(INT(x_1, x_2)) = x_2
   POL(INT_LIST(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(int(x_1, x_2)) = [1]
   POL(int_list(x_1)) = [1]
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
S tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c3_2, c5_1


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

(41) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c3_2, c5_1


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) by 
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0))
   INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0)))
   INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0))
   INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0))
   INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0)))
   INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0))
   INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1)))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_2


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

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0))
   INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0)))
   INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1)))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_2


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

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0)))
   INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1)))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_2, c3_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) by 
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1)))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_2, c3_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) by 
   INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0))))
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)), INT(s(s(z0)), s(0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0))))
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)), INT(s(s(z0)), s(0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_1, c3_2


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

(53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0))))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_1, c3_2


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

(55) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_1, c3_2, c_1


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

(57) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil))) by INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil))
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_1, c3_2, c_1


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c1_1, c5_1, c3_1, c3_2, c_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INT(0, s(z0)) -> c1(INT(s(0), s(z0))) by 
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
S tuples:
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_LIST_1, INT_2

Compound Symbols:   c5_1, c3_1, c3_2, c_1, c1_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) by 
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_1, c3_2, c_1, c1_1, c5_1


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

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   INT(0, s(0)) -> c1(INT(s(0), s(0)))
   INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)))
   INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c3_1, c_1, c1_1, c5_1


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) by INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil))))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0))))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil))))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c3_1, c_1, c1_1, c5_1


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) by INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(0), s(s(x0))) -> c3(INT(0, s(x0)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c3_1, c_1, c1_1, c5_1


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) by 
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c_1, c1_1, c5_1, c3_1


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

(73) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) by INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil)))
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil)))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c_1, c1_1, c5_1, c3_1


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))
S tuples:
   INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1)))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c_1, c1_1, c5_1, c3_1


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

(77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) by 
   INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0))))))
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))
   INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2))))
S tuples:
   INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2))))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c_1, c1_1, c3_1, c5_1


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) by INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int_list(int(0, z0))))))
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1)))
   INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0))))
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
   INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0))))
   INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2))))
   INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int_list(int(0, z0))))))
S tuples:
   INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2))))
K tuples:
   INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0))))
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c3_2, c_1, c1_1, c3_1, c5_1