KILLED
proof of input_1Jyf2iMVE1.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 200 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 37 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 378 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 3988 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 1061 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (58) CdtProblem
    (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms]
    (62) CdtProblem
    (63) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 404 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) 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:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   log(0) -> logError
   log(s(x)) -> loop(s(x), s(0), 0)
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), s(z))

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:

   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   log(0') -> logError
   log(s(x)) -> loop(s(x), s(0'), 0')
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), s(z))

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:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   log(0) -> logError
   log(s(x)) -> loop(s(x), s(0), 0)
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), s(z))

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:

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   log(0) -> logError [1]
   log(s(x)) -> loop(s(x), s(0), 0) [1]
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, y, z) -> loop(x, double(y), s(z)) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   log(0) -> logError [1]
   log(s(x)) -> loop(s(x), s(0), 0) [1]
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, y, z) -> loop(x, double(y), s(z)) [1]

The TRS has the following type information:
le :: s:0:logError -> s:0:logError -> false:true
s :: s:0:logError -> s:0:logError
0 :: s:0:logError
false :: false:true
true :: false:true
double :: s:0:logError -> s:0:logError
log :: s:0:logError -> s:0:logError
logError :: s:0:logError
loop :: s:0:logError -> s:0:logError -> s:0:logError -> s:0:logError
if :: false:true -> s:0:logError -> s:0:logError -> s:0:logError -> s:0:logError

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   log_1
   loop_3
   if_4

(c) The following functions are completely defined:

   le_2
   double_1

Due to the following rules being added:

   le(v0, v1) -> null_le [0]
   double(v0) -> null_double [0]

And the following fresh constants:    null_le, null_double

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   log(0) -> logError [1]
   log(s(x)) -> loop(s(x), s(0), 0) [1]
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, y, z) -> loop(x, double(y), s(z)) [1]
   le(v0, v1) -> null_le [0]
   double(v0) -> null_double [0]

The TRS has the following type information:
le :: s:0:logError:null_double -> s:0:logError:null_double -> false:true:null_le
s :: s:0:logError:null_double -> s:0:logError:null_double
0 :: s:0:logError:null_double
false :: false:true:null_le
true :: false:true:null_le
double :: s:0:logError:null_double -> s:0:logError:null_double
log :: s:0:logError:null_double -> s:0:logError:null_double
logError :: s:0:logError:null_double
loop :: s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double
if :: false:true:null_le -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double
null_le :: false:true:null_le
null_double :: s:0:logError:null_double

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   log(0) -> logError [1]
   log(s(x)) -> loop(s(x), s(0), 0) [1]
   loop(0, s(y), z) -> if(true, 0, s(y), z) [2]
   loop(s(x'), s(y), z) -> if(le(x', y), s(x'), s(y), z) [2]
   loop(x, s(y), z) -> if(null_le, x, s(y), z) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, 0, z) -> loop(x, 0, s(z)) [2]
   if(false, x, s(x''), z) -> loop(x, s(s(double(x''))), s(z)) [2]
   if(false, x, y, z) -> loop(x, null_double, s(z)) [1]
   le(v0, v1) -> null_le [0]
   double(v0) -> null_double [0]

The TRS has the following type information:
le :: s:0:logError:null_double -> s:0:logError:null_double -> false:true:null_le
s :: s:0:logError:null_double -> s:0:logError:null_double
0 :: s:0:logError:null_double
false :: false:true:null_le
true :: false:true:null_le
double :: s:0:logError:null_double -> s:0:logError:null_double
log :: s:0:logError:null_double -> s:0:logError:null_double
logError :: s:0:logError:null_double
loop :: s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double
if :: false:true:null_le -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double
null_le :: false:true:null_le
null_double :: s:0:logError:null_double

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

   0 => 0
   false => 1
   true => 2
   logError => 1
   null_le => 0
   null_double => 0

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0
   if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 2 }-> loop(x, 0, 1 + z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(x, 0, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(x, 1 + (1 + double(x'')), 1 + z) :|: z >= 0, z1 = 1 + x'', z2 = z, x >= 0, z'' = x, z' = 1, x'' >= 0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   log(z') -{ 1 }-> loop(1 + x, 1 + 0, 0) :|: z' = 1 + x, x >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(x', y), 1 + x', 1 + y, z) :|: z1 = z, z >= 0, z' = 1 + x', x' >= 0, y >= 0, z'' = 1 + y
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + y, z) :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, x, 1 + y, z) :|: z1 = z, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y


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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0


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

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

   { double }
   { le }
   { if, loop }
   { log }

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {double}, {le}, {if,loop}, {log}

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {double}, {le}, {if,loop}, {log}

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(21) 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'

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(23) 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'

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(33) 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: ?

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

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(35) 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: ?

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: z' >= 0
   double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0
   le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0

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

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

(37) 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:

   le(v0, v1) -> null_le [0]
   double(v0) -> null_double [0]
   log(v0) -> null_log [0]
   loop(v0, v1, v2) -> null_loop [0]
   if(v0, v1, v2, v3) -> null_if [0]

And the following fresh constants:    null_le, null_double, null_log, null_loop, null_if

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

(38)
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:

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   log(0) -> logError [1]
   log(s(x)) -> loop(s(x), s(0), 0) [1]
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, y, z) -> loop(x, double(y), s(z)) [1]
   le(v0, v1) -> null_le [0]
   double(v0) -> null_double [0]
   log(v0) -> null_log [0]
   loop(v0, v1, v2) -> null_loop [0]
   if(v0, v1, v2, v3) -> null_if [0]

The TRS has the following type information:
le :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> false:true:null_le
s :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if
0 :: s:0:logError:null_double:null_log:null_loop:null_if
false :: false:true:null_le
true :: false:true:null_le
double :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if
log :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if
logError :: s:0:logError:null_double:null_log:null_loop:null_if
loop :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if
if :: false:true:null_le -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if
null_le :: false:true:null_le
null_double :: s:0:logError:null_double:null_log:null_loop:null_if
null_log :: s:0:logError:null_double:null_log:null_loop:null_if
null_loop :: s:0:logError:null_double:null_log:null_loop:null_if
null_if :: s:0:logError:null_double:null_log:null_loop:null_if

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

(39) 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
   false => 1
   true => 2
   logError => 1
   null_le => 0
   null_double => 0
   null_log => 0
   null_loop => 0
   null_if => 0

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

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

   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0
   if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 1 }-> loop(x, double(y), 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   log(z') -{ 1 }-> loop(1 + x, 1 + 0, 0) :|: z' = 1 + x, x >= 0
   log(z') -{ 1 }-> 1 :|: z' = 0
   log(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   loop(z', z'', z1) -{ 1 }-> if(le(x, 1 + y), x, 1 + y, z) :|: z1 = z, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y
   loop(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   log(0) -> logError
   log(s(z0)) -> loop(s(z0), s(0), 0)
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))
Tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0) -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
S tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0) -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   le_2, double_1, log_1, loop_3, if_4

Defined Pair Symbols:   LE_2, DOUBLE_1, LOG_1, LOOP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6_1, c7_2, c8, c9_2


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
Removed 5 trailing nodes:
   IF(true, z0, z1, z2) -> c8
   LE(0, z0) -> c1
   DOUBLE(0) -> c3
   LE(s(z0), 0) -> c
   LOG(0) -> c5

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   log(0) -> logError
   log(s(z0)) -> loop(s(z0), s(0), 0)
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   le_2, double_1, log_1, loop_3, if_4

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c9_2


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   log(0) -> logError
   log(s(z0)) -> loop(s(z0), s(0), 0)
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c9_2


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) by 
   LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1)))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
   LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1)))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
   LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1)))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) by 
   IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c9_2


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

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c9_2


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by 
   LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2, c7_1


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

(57) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2, c7_1


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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
K tuples:none
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2, c7_1


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
We considered the (Usable) Rules:
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   le(0, z0) -> true
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(double(x_1)) = [1] + x_1
   POL(false) = 0
   POL(le(x_1, x_2)) = 0
   POL(s(x_1)) = [1]
   POL(true) = 0

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_1, c9_2, c7_2, c7_1


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

(63) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) by 
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c7_1, c9_2


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

(65) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c7_1, c9_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) by 
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c7_1, c9_2, c9_1


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_3*x_4 + x_2*x_4 + x_3^2 + x_2*x_3 + x_2^2 + x_2^3 + x_2^2*x_3 + x_2^2*x_4 + x_2*x_4^2 + x_2*x_3*x_4 + x_2*x_3^2 + x_3^3 + x_3^2*x_4 + x_3*x_4^2 + x_4^3
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3)) = [1] + x_1 + x_1*x_3 + x_1^2 + x_1^3 + x_1^2*x_3 + x_1*x_3^2 + x_3^3
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(double(x_1)) = 0
   POL(false) = 0
   POL(le(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, DOUBLE_1, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_1, c7_2, c7_1, c9_2, c9_1


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by 
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LOOP_3, IF_4, LE_2

Compound Symbols:   c4_1, c7_2, c7_1, c9_2, c9_1, c2_1


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

(73) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) by 
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LOOP_3, IF_4, LE_2

Compound Symbols:   c4_1, c7_1, c9_2, c9_1, c2_1, c7_2


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

(75) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) by 
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
K tuples:
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, IF_4, LE_2, LOOP_3

Compound Symbols:   c4_1, c9_2, c9_1, c2_1, c7_2, c7_1


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

(77) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) by 
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
K tuples:
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, IF_4, LE_2, LOOP_3

Compound Symbols:   c4_1, c9_1, c2_1, c7_2, c7_1, c9_2


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) by 
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
K tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LE_2, LOOP_3, IF_4

Compound Symbols:   c4_1, c2_1, c7_2, c7_1, c9_2, c9_1


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

(81) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) by 
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
K tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LE_2, LOOP_3, IF_4

Compound Symbols:   c4_1, c2_1, c7_1, c9_2, c9_1, c7_2


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) by IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
K tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LE_2, LOOP_3, IF_4

Compound Symbols:   c4_1, c2_1, c7_1, c9_1, c7_2, c9_2


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) by IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
K tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LE_2, LOOP_3, IF_4

Compound Symbols:   c4_1, c2_1, c7_1, c9_1, c7_2, c9_2


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

(87) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) by 
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
K tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   DOUBLE_1, LE_2, IF_4, LOOP_3

Compound Symbols:   c4_1, c2_1, c9_1, c7_2, c9_2, c7_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(s(z0)) -> c4(DOUBLE(z0)) by 
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
K tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   LE_2, IF_4, LOOP_3, DOUBLE_1

Compound Symbols:   c2_1, c9_1, c7_2, c9_2, c7_1, c4_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by 
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0))))))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   IF_4, LOOP_3, DOUBLE_1, LE_2

Compound Symbols:   c9_1, c7_2, c9_2, c7_1, c4_1, c2_1


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(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) by 
   LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1))))))))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1))))))))))
S tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1))))))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   IF_4, LOOP_3, DOUBLE_1, LE_2

Compound Symbols:   c9_1, c7_2, c9_2, c4_1, c2_1, c7_1


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(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) by 
   DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1))))))))))
   DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0))))
S tuples:
   LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0))))))
   IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1))))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0))))
K tuples:
   IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1))))))
   LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1))))))))))
Defined Rule Symbols:   le_2, double_1

Defined Pair Symbols:   IF_4, LOOP_3, LE_2, DOUBLE_1

Compound Symbols:   c9_1, c7_2, c9_2, c2_1, c7_1, c4_1