KILLED
proof of input_zwjJBTCuSC.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
        (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) CpxTypedWeightedCompleteTrs
        (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 102 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 329 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 4666 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 2116 ms]
        (44) CpxRNTS
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (52) CdtProblem
    (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) 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:

   1024 -> 1024_1(0)
   1024_1(x) -> if(lt(x, 10), x)
   if(true, x) -> double(1024_1(s(x)))
   if(false, x) -> s(0)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   10 -> double(s(double(s(s(0)))))

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:

   1024' -> 1024_1(0')
   1024_1(x) -> if(lt(x, 10'), x)
   if(true, x) -> double(1024_1(s(x)))
   if(false, x) -> s(0')
   lt(0', s(y)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   10' -> double(s(double(s(s(0')))))

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:

   1024 -> 1024_1(0)
   1024_1(x) -> if(lt(x, 10), x)
   if(true, x) -> double(1024_1(s(x)))
   if(false, x) -> s(0)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   10 -> double(s(double(s(s(0)))))

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:

   1024 -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, 10), x) [1]
   if(true, x) -> double(1024_1(s(x))) [1]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10 -> double(s(double(s(s(0))))) [1]

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

(7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

1024 => 1024'
10 => 10'

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

(8)
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:

   1024' -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, 10'), x) [1]
   if(true, x) -> double(1024_1(s(x))) [1]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10' -> double(s(double(s(s(0))))) [1]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   1024' -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, 10'), x) [1]
   if(true, x) -> double(1024_1(s(x))) [1]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10' -> double(s(double(s(s(0))))) [1]

The TRS has the following type information:
1024' :: 0:s
1024_1 :: 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
10' :: 0:s
true :: true:false
double :: 0:s -> 0:s
s :: 0:s -> 0:s
false :: true:false

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

(11) 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:

   1024'

(c) The following functions are completely defined:

   lt_2
   10'
   double_1
   1024_1_1
   if_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   1024' -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, 10'), x) [1]
   if(true, x) -> double(1024_1(s(x))) [1]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10' -> double(s(double(s(s(0))))) [1]

The TRS has the following type information:
1024' :: 0:s
1024_1 :: 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
10' :: 0:s
true :: true:false
double :: 0:s -> 0:s
s :: 0:s -> 0:s
false :: true:false

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

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

(14)
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:

   1024' -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, double(s(double(s(s(0)))))), x) [2]
   if(true, x) -> double(if(lt(s(x), 10'), s(x))) [2]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10' -> double(s(s(s(double(s(0)))))) [2]

The TRS has the following type information:
1024' :: 0:s
1024_1 :: 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
10' :: 0:s
true :: true:false
double :: 0:s -> 0:s
s :: 0:s -> 0:s
false :: true:false

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

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

   0 => 0
   true => 1
   false => 0

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(x, double(1 + double(1 + (1 + 0)))), x) :|: x >= 0, z = x
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   if(z, z') -{ 2 }-> double(if(lt(1 + x, 10'), 1 + x)) :|: z' = x, z = 1, x >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' = x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0


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

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

   { double }
   { lt }
   { 10' }
   { if }
   { 1024_1 }
   { 1024' }

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'}

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

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: ?, size: O(n^1) [2*z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: ?, size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: 10'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 10

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  10': runtime: ?, size: O(1) [10]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: 10'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 10

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  10': runtime: O(1) [10], size: O(1) [10]

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  10': runtime: O(1) [10], size: O(1) [10]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  10': runtime: O(1) [10], size: O(1) [10]
  if: runtime: ?, size: INF

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1)))
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {1024_1}, {1024'}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  10': runtime: O(1) [10], size: O(1) [10]
  if: runtime: INF, size: INF

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

(45) 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

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

(46)
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:

   1024' -> 1024_1(0) [1]
   1024_1(x) -> if(lt(x, 10'), x) [1]
   if(true, x) -> double(1024_1(s(x))) [1]
   if(false, x) -> s(0) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   10' -> double(s(double(s(s(0))))) [1]

The TRS has the following type information:
1024' :: 0:s
1024_1 :: 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
10' :: 0:s
true :: true:false
double :: 0:s -> 0:s
s :: 0:s -> 0:s
false :: true:false

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

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

   0 => 0
   true => 1
   false => 0

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

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

   10' -{ 1 }-> double(1 + double(1 + (1 + 0))) :|: 
   1024' -{ 1 }-> 1024_1(0) :|: 
   1024_1(z) -{ 1 }-> if(lt(x, 10'), x) :|: x >= 0, z = x
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   if(z, z') -{ 1 }-> double(1024_1(1 + x)) :|: z' = x, z = 1, x >= 0
   if(z, z') -{ 1 }-> 1 + 0 :|: z' = x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   1024 -> 1024_1(0)
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   10 -> double(s(double(s(s(0)))))
Tuples:
   1024' -> c(1024_1'(0))
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   IF(false, z0) -> c3
   LT(0, s(z0)) -> c4
   LT(z0, 0) -> c5
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0))))
S tuples:
   1024' -> c(1024_1'(0))
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   IF(false, z0) -> c3
   LT(0, s(z0)) -> c4
   LT(z0, 0) -> c5
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0))))
K tuples:none
Defined Rule Symbols:   1024, 1024_1_1, if_2, lt_2, double_1, 10

Defined Pair Symbols:   1024', 1024_1'_1, IF_2, LT_2, DOUBLE_1, 10'

Compound Symbols:   c_1, c1_3, c2_2, c3, c4, c5, c6_1, c7, c8_1, c9_2


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

(51) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   1024' -> c(1024_1'(0))
Removed 4 trailing nodes:
   DOUBLE(0) -> c7
   IF(false, z0) -> c3
   LT(0, s(z0)) -> c4
   LT(z0, 0) -> c5

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   1024 -> 1024_1(0)
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   10 -> double(s(double(s(s(0)))))
Tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0))))
S tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0))))
K tuples:none
Defined Rule Symbols:   1024, 1024_1_1, if_2, lt_2, double_1, 10

Defined Pair Symbols:   1024_1'_1, IF_2, LT_2, DOUBLE_1, 10'

Compound Symbols:   c1_3, c2_2, c6_1, c8_1, c9_2


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   1024 -> 1024_1(0)
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   10 -> double(s(double(s(s(0)))))
Tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
S tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
K tuples:none
Defined Rule Symbols:   1024, 1024_1_1, if_2, lt_2, double_1, 10

Defined Pair Symbols:   1024_1'_1, IF_2, LT_2, DOUBLE_1, 10'

Compound Symbols:   c1_3, c2_2, c6_1, c8_1, c_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   1024 -> 1024_1(0)

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
S tuples:
   1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10')
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   1024_1'_1, IF_2, LT_2, DOUBLE_1, 10'

Compound Symbols:   c1_3, c2_2, c6_1, c8_1, c_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') by 
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
S tuples:
   IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   IF_2, LT_2, DOUBLE_1, 10', 1024_1'_1

Compound Symbols:   c2_2, c6_1, c8_1, c_1, c1_3


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) by 
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(double(s(s(0))))))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c2_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 10' -> c(DOUBLE(s(double(s(s(0)))))) by 
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c2_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') by 
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(LT(x0, 10), 10')

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(LT(x0, 10), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(LT(x0, 10), 10')
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1

Compound Symbols:   c6_1, c8_1, c_1, c2_2, c1_3, c1_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
K tuples:none
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1

Compound Symbols:   c6_1, c8_1, c_1, c2_2, c1_3, c3_1


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

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
We considered the (Usable) Rules:none
And the Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(10) = 0
   POL(10') = 0
   POL(1024_1(x_1)) = [1] + x_1
   POL(1024_1'(x_1)) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(IF(x_1, x_2)) = [1]
   POL(LT(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = 0
   POL(false) = [1]
   POL(if(x_1, x_2)) = [1] + x_2
   POL(lt(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1

Compound Symbols:   c6_1, c8_1, c_1, c2_2, c1_3, c3_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) by 
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(double(s(0)))))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 10' -> c(DOUBLE(s(s(s(double(s(0))))))) by 
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(73) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') by 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') by 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(x0) -> c3(LT(x0, 10)) by 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(10')
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0)))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) by IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(10')
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(double(0))))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c1_3, c2_2


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) by 10' -> c(DOUBLE(s(s(s(s(s(0)))))))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(10')
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c1_3, c2_2


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') by 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10')
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c1_3, c2_2


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') by 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(x0) -> c3(10')
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(x0) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(87) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(x0) -> c3(10') by 
   1024_1'(s(x0)) -> c3(10')

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) by 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0))))))))
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(10')
   1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0))))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0)))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1

Compound Symbols:   c6_1, c8_1, c_1, c2_2, c1_3, c3_1


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) by IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(10')
   1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0))))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(93) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') by 
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(10')
   1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0))))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10')
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(95) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') by 
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c3(10')
   1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0))))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) by 
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c3(10')
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0)))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(99) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) by 
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c3(10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(x0)) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c1_3, c2_2


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(s(x0)) -> c3(10') by 
   1024_1'(s(s(x0))) -> c3(10')

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c3(10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') by 
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c1_3, c3_1, c2_2


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

(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') by 
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0))))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c2_2, c1_3


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

(107) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) by 
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1

Compound Symbols:   c6_1, c8_1, c_1, c2_2, c3_1, c1_3


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

(109) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) by 
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2

Compound Symbols:   c6_1, c8_1, c_1, c3_1, c1_3, c2_2


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LT(s(z0), s(z1)) -> c6(LT(z0, z1)) by 
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   DOUBLE_1, 10', 1024_1'_1, IF_2, LT_2

Compound Symbols:   c8_1, c_1, c3_1, c1_3, c2_2, c6_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(s(z0)) -> c8(DOUBLE(z0)) by 
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
S tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   10', 1024_1'_1, IF_2, LT_2, DOUBLE_1

Compound Symbols:   c_1, c3_1, c1_3, c2_2, c6_1, c8_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) by 
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
S tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   10', 1024_1'_1, IF_2, DOUBLE_1, LT_2

Compound Symbols:   c_1, c3_1, c1_3, c2_2, c8_1, c6_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) by 
   DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0))))
S tuples:
   10' -> c(DOUBLE(s(s(0))))
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0))))
K tuples:
   1024_1'(x0) -> c3(LT(x0, 10))
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   10', 1024_1'_1, IF_2, LT_2, DOUBLE_1

Compound Symbols:   c_1, c3_1, c1_3, c2_2, c6_1, c8_1


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

(119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   10' -> c(DOUBLE(s(s(0))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   10 -> double(s(double(s(s(0)))))
   double(s(z0)) -> s(s(double(z0)))
   double(0) -> 0
   1024_1(z0) -> if(lt(z0, 10), z0)
   if(true, z0) -> double(1024_1(s(z0)))
   if(false, z0) -> s(0)
Tuples:
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c3(10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0))))))))
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0))))
S tuples:
   10' -> c(DOUBLE(s(s(s(s(s(0)))))))
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10')
   IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0))))
K tuples:
   1024_1'(s(s(x0))) -> c3(10')
Defined Rule Symbols:   lt_2, 10, double_1, 1024_1_1, if_2

Defined Pair Symbols:   10', 1024_1'_1, IF_2, LT_2, DOUBLE_1

Compound Symbols:   c_1, c3_1, c1_3, c2_2, c6_1, c8_1