MAYBE
proof of input_3NnPTIQEPR.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [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), 184 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 294 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 416 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 287 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 [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) 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:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

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

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

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


The TRS R consists of the following rules:

   p(0') -> 0'
   p(s(X)) -> X
   leq(0', Y) -> true
   leq(s(X), 0') -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   diff(X, Y) -> if(leq(X, Y), 0', s(diff(p(X), Y)))

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

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

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


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   p(0) -> 0 [1]
   p(s(X)) -> X [1]
   leq(0, Y) -> true [1]
   leq(s(X), 0) -> false [1]
   leq(s(X), s(Y)) -> leq(X, Y) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1]

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

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

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

   p(0) -> 0 [1]
   p(s(X)) -> X [1]
   leq(0, Y) -> true [1]
   leq(s(X), 0) -> false [1]
   leq(s(X), s(Y)) -> leq(X, Y) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1]

The TRS has the following type information:
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
leq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if :: true:false -> 0:s -> 0:s -> 0:s
diff :: 0:s -> 0:s -> 0:s

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   leq_2
   diff_2
   p_1
   if_3

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   p(0) -> 0 [1]
   p(s(X)) -> X [1]
   leq(0, Y) -> true [1]
   leq(s(X), 0) -> false [1]
   leq(s(X), s(Y)) -> leq(X, Y) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1]

The TRS has the following type information:
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
leq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if :: true:false -> 0:s -> 0:s -> 0:s
diff :: 0:s -> 0:s -> 0:s

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:

   p(0) -> 0 [1]
   p(s(X)) -> X [1]
   leq(0, Y) -> true [1]
   leq(s(X), 0) -> false [1]
   leq(s(X), s(Y)) -> leq(X, Y) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   diff(0, Y) -> if(true, 0, s(diff(0, Y))) [3]
   diff(s(X'), 0) -> if(false, 0, s(diff(X', 0))) [3]
   diff(s(X''), s(Y')) -> if(leq(X'', Y'), 0, s(diff(X'', s(Y')))) [3]

The TRS has the following type information:
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
leq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if :: true:false -> 0:s -> 0:s -> 0:s
diff :: 0:s -> 0:s -> 0:s

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

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

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

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

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

   diff(z, z') -{ 3 }-> if(leq(X'', Y'), 0, 1 + diff(X'', 1 + Y')) :|: z = 1 + X'', Y' >= 0, z' = 1 + Y', X'' >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, Y)) :|: z' = Y, Y >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(X', 0)) :|: X' >= 0, z = 1 + X', z' = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0
   p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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

   diff(z, z') -{ 3 }-> if(leq(X'', Y'), 0, 1 + diff(X'', 1 + Y')) :|: z = 1 + X'', Y' >= 0, z' = 1 + Y', X'' >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, Y)) :|: z' = Y, Y >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(X', 0)) :|: X' >= 0, z = 1 + X', z' = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0
   p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0
   p(z) -{ 1 }-> 0 :|: 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:

   diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { leq }
   { if }
   { p }
   { diff }

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

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

   diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {leq}, {if}, {p}, {diff}

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

(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:

   diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {leq}, {if}, {p}, {diff}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {leq}, {if}, {p}, {diff}
Previous analysis results are:
  leq: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {if}, {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(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:

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {if}, {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {if}, {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: ?, size: O(n^1) [z' + z'']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(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:

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(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:

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  diff: runtime: ?, size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0
   diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {diff}
Previous analysis results are:
  leq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  diff: runtime: INF, size: O(1) [0]

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

(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:

   p(0) -> 0 [1]
   p(s(X)) -> X [1]
   leq(0, Y) -> true [1]
   leq(s(X), 0) -> false [1]
   leq(s(X), s(Y)) -> leq(X, Y) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1]

The TRS has the following type information:
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
leq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if :: true:false -> 0:s -> 0:s -> 0:s
diff :: 0:s -> 0:s -> 0:s

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:

   diff(z, z') -{ 1 }-> if(leq(X, Y), 0, 1 + diff(p(X), Y)) :|: z' = Y, Y >= 0, X >= 0, z = X
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0
   leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0
   p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0
   p(z) -{ 1 }-> 0 :|: 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:
   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))
Tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
S tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:   p_1, leq_2, if_3, diff_2

Defined Pair Symbols:   P_1, LEQ_2, IF_3, DIFF_2

Compound Symbols:   c, c1, c2, c3, c4_1, c5, c6, c7_2, c8_3


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   IF(false, z0, z1) -> c6
   LEQ(0, z0) -> c2
   P(s(z0)) -> c1
   LEQ(s(z0), 0) -> c3
   P(0) -> c
   IF(true, z0, z1) -> c5

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))
Tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
S tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:   p_1, leq_2, if_3, diff_2

Defined Pair Symbols:   LEQ_2, DIFF_2

Compound Symbols:   c4_1, c7_2, c8_3


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

(53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))
Tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
S tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:none
Defined Rule Symbols:   p_1, leq_2, if_3, diff_2

Defined Pair Symbols:   LEQ_2, DIFF_2

Compound Symbols:   c4_1, c7_1, c8_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
S tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   LEQ_2, DIFF_2

Compound Symbols:   c4_1, c7_1, c8_1


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
We considered the (Usable) Rules:
   p(0) -> 0
   p(s(z0)) -> z0
And the Tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DIFF(x_1, x_2)) = [1] + x_1 + x_2
   POL(LEQ(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
S tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
Defined Rule Symbols:   p_1

Defined Pair Symbols:   LEQ_2, DIFF_2

Compound Symbols:   c4_1, c7_1, c8_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) by 
   LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
   LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1)))
S tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:
   DIFF(z0, z1) -> c7(LEQ(z0, z1))
   LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1)))
Defined Rule Symbols:   p_1

Defined Pair Symbols:   DIFF_2, LEQ_2

Compound Symbols:   c7_1, c8_1, c4_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DIFF(z0, z1) -> c7(LEQ(z0, z1)) by 
   DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
   LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1)))
   DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1))))
S tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:
   LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1)))
   DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   p_1

Defined Pair Symbols:   DIFF_2, LEQ_2

Compound Symbols:   c8_1, c4_1, c7_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) by 
   LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
   DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1))))
   LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1))))
S tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:
   DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1))))
   LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   p_1

Defined Pair Symbols:   DIFF_2, LEQ_2

Compound Symbols:   c8_1, c7_1, c4_1


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(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) by 
   DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1)))))

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(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
   LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1))))
   DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   DIFF(z0, z1) -> c8(DIFF(p(z0), z1))
K tuples:
   LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1))))
   DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   p_1

Defined Pair Symbols:   DIFF_2, LEQ_2

Compound Symbols:   c8_1, c4_1, c7_1