MAYBE
proof of input_IT4Z00M2un.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) 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:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

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:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0') -> 0'
   p(s(x)) -> x

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:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

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:

   f(s(x)) -> s(f(f(p(s(x))))) [1]
   f(0) -> 0 [1]
   p(s(x)) -> x [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:

   f(s(x)) -> s(f(f(p(s(x))))) [1]
   f(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
f :: s:0 -> s:0
s :: s:0 -> s:0
p :: s:0 -> s:0
0 :: s:0

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:

   f_1
   p_1

Due to the following rules being added:

   p(v0) -> 0 [0]

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:

   f(s(x)) -> s(f(f(p(s(x))))) [1]
   f(0) -> 0 [1]
   p(s(x)) -> x [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: s:0 -> s:0
s :: s:0 -> s:0
p :: s:0 -> s:0
0 :: s:0

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:

   f(s(x)) -> s(f(f(x))) [2]
   f(s(x)) -> s(f(f(0))) [1]
   f(0) -> 0 [1]
   p(s(x)) -> x [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: s:0 -> s:0
s :: s:0 -> s:0
p :: s:0 -> s:0
0 :: s:0

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

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

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

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 2 }-> 1 + f(f(x)) :|: x >= 0, z = 1 + x
   f(z) -{ 1 }-> 1 + f(f(0)) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(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:

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + f(f(0)) :|: z - 1 >= 0
   f(z) -{ 2 }-> 1 + f(f(z - 1)) :|: z - 1 >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

(17) 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:

   p(v0) -> null_p [0]
   f(v0) -> null_f [0]

And the following fresh constants:    null_p, null_f

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

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   f(s(x)) -> s(f(f(p(s(x))))) [1]
   f(0) -> 0 [1]
   p(s(x)) -> x [1]
   p(v0) -> null_p [0]
   f(v0) -> null_f [0]

The TRS has the following type information:
f :: s:0:null_p:null_f -> s:0:null_p:null_f
s :: s:0:null_p:null_f -> s:0:null_p:null_f
p :: s:0:null_p:null_f -> s:0:null_p:null_f
0 :: s:0:null_p:null_f
null_p :: s:0:null_p:null_f
null_f :: s:0:null_p:null_f

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

(19) 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
   null_p => 0
   null_f => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
S tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_3, c1, c2


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(0) -> c1
   P(s(z0)) -> c2

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
S tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_3


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

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))))
S tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0)))) by 
   F(s(z0)) -> c(F(f(z0)), F(p(s(z0))))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(f(z0)), F(p(s(z0))))
S tuples:
   F(s(z0)) -> c(F(f(z0)), F(p(s(z0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c(F(f(z0)), F(p(s(z0)))) by 
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(0)) -> c(F(0), F(p(s(0))))
   F(s(x0)) -> c(F(p(s(x0))))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(0)) -> c(F(0), F(p(s(0))))
   F(s(x0)) -> c(F(p(s(x0))))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(0)) -> c(F(0), F(p(s(0))))
   F(s(x0)) -> c(F(p(s(x0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(0)) -> c(F(p(s(0))))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(0)) -> c(F(p(s(0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0)) -> c(F(p(s(0)))) by 
   F(s(0)) -> c(F(0))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(0)) -> c(F(0))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(0)) -> c(F(0))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0)) -> c(F(0))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0)))))
   F(s(x0)) -> c(F(p(s(x0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(37) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) by F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0)))
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0)))
S tuples:
   F(s(x0)) -> c(F(p(s(x0))))
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0)))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c_2


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

(39) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c(F(p(s(x0)))) by F(s(z0)) -> c(F(z0))
----------------------------------------

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0)))
   F(s(z0)) -> c(F(z0))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0)))
   F(s(z0)) -> c(F(z0))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(41) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) by F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(z0))
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
S tuples:
   F(s(z0)) -> c(F(z0))
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c_2


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(z0)) -> c(F(z0)) by 
   F(s(s(y0))) -> c(F(s(y0)))
   F(s(s(s(y0)))) -> c(F(s(s(y0))))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
   F(s(s(y0))) -> c(F(s(y0)))
   F(s(s(s(y0)))) -> c(F(s(s(y0))))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
   F(s(s(y0))) -> c(F(s(y0)))
   F(s(s(s(y0)))) -> c(F(s(s(y0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(s(y0))) -> c(F(s(y0))) by 
   F(s(s(s(y0)))) -> c(F(s(s(y0))))
   F(s(s(s(s(y0))))) -> c(F(s(s(s(y0)))))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
   F(s(s(s(y0)))) -> c(F(s(s(y0))))
   F(s(s(s(s(y0))))) -> c(F(s(s(s(y0)))))
S tuples:
   F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0)))
   F(s(s(s(y0)))) -> c(F(s(s(y0))))
   F(s(s(s(s(y0))))) -> c(F(s(s(s(y0)))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1