MAYBE
proof of input_P4ptHu8xZf.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 1 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) 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(h(x), y) -> h(f(y, f(x, h(a))))

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(h(x), y) -> h(f(y, f(x, h(a))))

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(h(x), y) -> h(f(y, f(x, h(a))))

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(h(x), y) -> h(f(y, f(x, h(a)))) [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(h(x), y) -> h(f(y, f(x, h(a)))) [1]

The TRS has the following type information:
f :: h:a -> h:a -> h:a
h :: h:a -> h:a
a :: h:a

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

(9) 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:

   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_f

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

(10)
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(h(x), y) -> h(f(y, f(x, h(a)))) [1]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: h:a:null_f -> h:a:null_f -> h:a:null_f
h :: h:a:null_f -> h:a:null_f
a :: h:a:null_f
null_f :: h:a:null_f

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

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

   a => 0
   null_f => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 1 }-> 1 + f(y, f(x, 1 + 0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

(13) 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_2

Due to the following rules being added:

   f(v0, v1) -> a [0]

And the following fresh constants: none

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

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   f(h(x), y) -> h(f(y, f(x, h(a)))) [1]
   f(v0, v1) -> a [0]

The TRS has the following type information:
f :: h:a -> h:a -> h:a
h :: h:a -> h:a
a :: h:a

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

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

(16)
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(h(h(x')), y) -> h(f(y, h(f(h(a), f(x', h(a)))))) [2]
   f(h(x), y) -> h(f(y, a)) [1]
   f(v0, v1) -> a [0]

The TRS has the following type information:
f :: h:a -> h:a -> h:a
h :: h:a -> h:a
a :: h:a

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

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

   a => 0

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

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

   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 1 }-> 1 + f(y, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z, z') -{ 2 }-> 1 + f(y, 1 + f(1 + 0, f(x', 1 + 0))) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y


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

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

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

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


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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(h(z0), z1) -> h(f(z1, f(z0, h(a))))
Tuples:
   F(h(z0), z1) -> c(F(z1, f(z0, h(a))), F(z0, h(a)))
S tuples:
   F(h(z0), z1) -> c(F(z1, f(z0, h(a))), F(z0, h(a)))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c_2