MAYBE
proof of input_fMccPQkHqI.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
        (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) typed CpxTrs
        (13) OrderProof [LOWER BOUND(ID), 8 ms]
        (14) typed CpxTrs
    (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (16) TRS for Loop Detection
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (22) CpxTRS
    (23) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxWeightedTrs
        (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedTrs
        (27) CompletionProof [UPPER BOUND(ID), 0 ms]
        (28) CpxTypedWeightedCompleteTrs
            (29) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (30) CpxTypedWeightedCompleteTrs
            (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (32) CpxRNTS
            (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxRNTS
        (35) CompletionProof [UPPER BOUND(ID), 0 ms]
        (36) CpxTypedWeightedCompleteTrs
            (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (38) CpxRNTS
(39) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(40) CpxWeightedTrs
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CpxTypedWeightedTrs
    (43) CompletionProof [UPPER BOUND(ID), 0 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
    (47) CompletionProof [UPPER BOUND(ID), 0 ms]
    (48) CpxTypedWeightedCompleteTrs
        (49) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CpxTypedWeightedCompleteTrs
        (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) SimplificationProof [BOTH BOUNDS(ID, ID), 2 ms]
        (54) CpxRNTS


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

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

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

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))

The (relative) TRS S consists of the following rules:

   f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))

The (relative) TRS S consists of the following rules:

   f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

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

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))
f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

Types:
F :: h:a -> h:a -> c
h :: h:a -> h:a
c :: c -> c -> c
f :: h:a -> h:a -> h:a
a :: h:a
hole_c1_2 :: c
hole_h:a2_2 :: h:a
gen_c3_2 :: Nat -> c
gen_h:a4_2 :: Nat -> h:a

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, f

They will be analysed ascendingly in the following order:
f < F

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

(14)
Obligation:
Innermost TRS:
Rules:
F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))
f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

Types:
F :: h:a -> h:a -> c
h :: h:a -> h:a
c :: c -> c -> c
f :: h:a -> h:a -> h:a
a :: h:a
hole_c1_2 :: c
hole_h:a2_2 :: h:a
gen_c3_2 :: Nat -> c
gen_h:a4_2 :: Nat -> h:a


Generator Equations:
gen_c3_2(0) <=> hole_c1_2
gen_c3_2(+(x, 1)) <=> c(hole_c1_2, gen_c3_2(x))
gen_h:a4_2(0) <=> a
gen_h:a4_2(+(x, 1)) <=> h(gen_h:a4_2(x))


The following defined symbols remain to be analysed:
f, F

They will be analysed ascendingly in the following order:
f < F

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(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))

The (relative) TRS S consists of the following rules:

   f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(19) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))

The (relative) TRS S consists of the following rules:

   f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

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

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

(22)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1))
   f(z0, h(z1)) -> h(f(f(h(a), z1), z0))

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

(23) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(24)
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(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1)) [1]
   f(z0, h(z1)) -> h(f(f(h(a), z1), z0)) [0]

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

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

(26)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1)) [1]
   f(z0, h(z1)) -> h(f(f(h(a), z1), z0)) [0]

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

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

(27) 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:

   F_2

(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:    const

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

(28)
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(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1)) [1]
   f(z0, h(z1)) -> h(f(f(h(a), z1), z0)) [0]
   f(v0, v1) -> a [0]

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

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

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

(30)
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(z0, h(h(z1'))) -> c(F(h(f(f(h(a), z1'), h(a))), z0), F(h(a), h(z1'))) [1]
   F(z0, h(z1)) -> c(F(a, z0), F(h(a), z1)) [1]
   f(z0, h(h(z1''))) -> h(f(h(f(f(h(a), z1''), h(a))), z0)) [0]
   f(z0, h(z1)) -> h(f(a, z0)) [0]
   f(v0, v1) -> a [0]

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

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

(31) 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
   const => 0

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

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

   F(z, z') -{ 1 }-> 1 + F(0, z0) + F(1 + 0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(1 + f(f(1 + 0, z1'), 1 + 0), z0) + F(1 + 0, 1 + z1') :|: z' = 1 + (1 + z1'), z = z0, z1' >= 0, z0 >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + f(0, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> 1 + f(1 + f(f(1 + 0, z1''), 1 + 0), z0) :|: z = z0, z' = 1 + (1 + z1''), z0 >= 0, z1'' >= 0


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

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

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

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


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

(35) 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]
   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_f, null_F

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

(36)
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(z0, h(z1)) -> c(F(f(h(a), z1), z0), F(h(a), z1)) [1]
   f(z0, h(z1)) -> h(f(f(h(a), z1), z0)) [0]
   f(v0, v1) -> null_f [0]
   F(v0, v1) -> null_F [0]

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

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

(37) 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
   null_F => 0

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

(38)
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(f(1 + 0, z1), z0) + F(1 + 0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + f(f(1 + 0, z1), z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1

Only complete derivations are relevant for the runtime complexity.

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

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

(40)
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(x, h(y)) -> h(f(f(h(a), y), x)) [1]

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

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

(42)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

(43) 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

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

(44)
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(x, h(y)) -> h(f(f(h(a), y), x)) [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
----------------------------------------

(45) 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

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

(46)
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(f(1 + 0, y), x) :|: z' = 1 + y, x >= 0, y >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

(47) 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

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

(48)
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(x, h(y)) -> h(f(f(h(a), y), x)) [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
----------------------------------------

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

(50)
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(x, h(h(y'))) -> h(f(h(f(f(h(a), y'), h(a))), x)) [2]
   f(x, h(y)) -> h(f(a, x)) [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
----------------------------------------

(51) 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

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

(52)
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(0, x) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   f(z, z') -{ 2 }-> 1 + f(1 + f(f(1 + 0, y'), 1 + 0), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x


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

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

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