KILLED
proof of input_xg6KS6Y8a0.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 11 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRelTRS
    (23) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTRS
    (25) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxWeightedTrs
        (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (28) CpxTypedWeightedTrs
        (29) CompletionProof [UPPER BOUND(ID), 0 ms]
        (30) CpxTypedWeightedCompleteTrs
            (31) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedCompleteTrs
            (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (34) CpxRNTS
            (35) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (36) CpxRNTS
        (37) CompletionProof [UPPER BOUND(ID), 0 ms]
        (38) CpxTypedWeightedCompleteTrs
            (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (40) CpxRNTS
(41) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(42) CpxWeightedTrs
    (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CpxTypedWeightedTrs
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CpxTypedWeightedCompleteTrs
        (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CpxRNTS
    (53) CompletionProof [UPPER BOUND(ID), 0 ms]
    (54) CpxTypedWeightedCompleteTrs
        (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (56) 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(h(x), y) -> h(f(y, f(x, h(f(a, 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(f(a, a)))))

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_3


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

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

(6)
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(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))), F(a, a))

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

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

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

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

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

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))), F(a, a))
f(h(z0), z1) -> h(f(z1, f(z0, h(f(a, a)))))

Types:
F :: h:a -> h:a -> c
h :: h:a -> h:a
c :: 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

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

(11) 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

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

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

Types:
F :: h:a -> h:a -> c
h :: h:a -> h:a
c :: 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, 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

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

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

(14)
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(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))), F(a, a))

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

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

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

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

(16)
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(f(a, a)))))

S is empty.
Rewrite Strategy: PARALLEL_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(h(z0), z1) -> h(f(z1, f(z0, h(f(a, a)))))
Tuples:
   F(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))), F(a, a))
S tuples:
   F(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))), F(a, a))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c_3


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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

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

(22)
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(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))))

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

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

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

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

(24)
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(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a))))
   f(h(z0), z1) -> h(f(z1, f(z0, h(f(a, a)))))

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

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

(26)
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(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a)))) [1]
   f(h(z0), z1) -> h(f(z1, f(z0, h(f(a, a))))) [0]

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

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

(28)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

(29) 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

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

(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(h(z0), z1) -> c(F(z1, f(z0, h(f(a, a)))), F(z0, h(f(a, a)))) [1]
   f(h(z0), z1) -> h(f(z1, f(z0, h(f(a, a))))) [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) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(32)
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(z0), z1) -> c(F(z1, f(z0, h(a))), F(z0, h(a))) [1]
   f(h(z0), z1) -> h(f(z1, f(z0, h(a)))) [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
----------------------------------------

(33) 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

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

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

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


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

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

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

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


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

(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:

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

And the following fresh constants:    null_f, null_F

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

(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:

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

(39) 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

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

(40)
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(z1, f(z0, 1 + f(0, 0))) + F(z0, 1 + f(0, 0)) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + f(z1, f(z0, 1 + f(0, 0))) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
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(f(a, a))))) [1]

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

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

(44)
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(f(a, 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
----------------------------------------

(45) 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

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

(46)
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(f(a, 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
----------------------------------------

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

(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(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
----------------------------------------

(49) 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

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

(50)
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


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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(53) 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

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

(54)
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(f(a, 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
----------------------------------------

(55) 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

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

(56)
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 + f(0, 0))) :|: x >= 0, y >= 0, z = 1 + x, z' = y

Only complete derivations are relevant for the runtime complexity.