KILLED
proof of input_nVAD9VvYjs.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (8) TRS for Loop Detection
    (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), 0 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 472 ms]
        (16) typed CpxTrs
(17) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(18) CpxTRS
(19) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
        (41) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (42) CpxTRS
        (43) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CpxWeightedTrs
            (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (46) CpxTypedWeightedTrs
            (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), 0 ms]
                (54) CpxRNTS
            (55) CompletionProof [UPPER BOUND(ID), 0 ms]
            (56) CpxTypedWeightedCompleteTrs
                (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (58) CpxRNTS
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
        (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) 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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(X))

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(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
S tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c1


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

(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(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1

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

   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))

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

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

(8)
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(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1

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

   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(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(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1

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

   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
F(g(z0)) -> c(F(f(z0)), F(z0))
F(h(z0)) -> c1
f(g(z0)) -> g(f(f(z0)))
f(h(z0)) -> h(g(z0))

Types:
F :: g:h -> c:c1
g :: g:h -> g:h
c :: c:c1 -> c:c1 -> c:c1
f :: g:h -> g:h
h :: g:h -> g:h
c1 :: c:c1
hole_c:c11_2 :: c:c1
hole_g:h2_2 :: g:h
gen_c:c13_2 :: Nat -> c:c1
gen_g:h4_2 :: Nat -> g:h

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

(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(g(z0)) -> c(F(f(z0)), F(z0))
F(h(z0)) -> c1
f(g(z0)) -> g(f(f(z0)))
f(h(z0)) -> h(g(z0))

Types:
F :: g:h -> c:c1
g :: g:h -> g:h
c :: c:c1 -> c:c1 -> c:c1
f :: g:h -> g:h
h :: g:h -> g:h
c1 :: c:c1
hole_c:c11_2 :: c:c1
hole_g:h2_2 :: g:h
gen_c:c13_2 :: Nat -> c:c1
gen_g:h4_2 :: Nat -> g:h


Generator Equations:
gen_c:c13_2(0) <=> c1
gen_c:c13_2(+(x, 1)) <=> c(c1, gen_c:c13_2(x))
gen_g:h4_2(0) <=> hole_g:h2_2
gen_g:h4_2(+(x, 1)) <=> g(gen_g:h4_2(x))


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

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_g:h4_2(+(1, n6_2))) -> *5_2, rt in Omega(0)

Induction Base:
f(gen_g:h4_2(+(1, 0)))

Induction Step:
f(gen_g:h4_2(+(1, +(n6_2, 1)))) ->_R^Omega(0)
g(f(f(gen_g:h4_2(+(1, n6_2))))) ->_IH
g(f(*5_2))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
F(g(z0)) -> c(F(f(z0)), F(z0))
F(h(z0)) -> c1
f(g(z0)) -> g(f(f(z0)))
f(h(z0)) -> h(g(z0))

Types:
F :: g:h -> c:c1
g :: g:h -> g:h
c :: c:c1 -> c:c1 -> c:c1
f :: g:h -> g:h
h :: g:h -> g:h
c1 :: c:c1
hole_c:c11_2 :: c:c1
hole_g:h2_2 :: g:h
gen_c:c13_2 :: Nat -> c:c1
gen_g:h4_2 :: Nat -> g:h


Lemmas:
f(gen_g:h4_2(+(1, n6_2))) -> *5_2, rt in Omega(0)


Generator Equations:
gen_c:c13_2(0) <=> c1
gen_c:c13_2(+(x, 1)) <=> c(c1, gen_c:c13_2(x))
gen_g:h4_2(0) <=> hole_g:h2_2
gen_g:h4_2(+(x, 1)) <=> g(gen_g:h4_2(x))


The following defined symbols remain to be analysed:
F
----------------------------------------

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

(18)
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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(X))

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

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

(20)
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(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]

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

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

(23) 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

Due to the following rules being added:

   f(v0) -> const [0]

And the following fresh constants:    const

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

(24)
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(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> const [0]

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

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

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

(26)
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(g(g(X'))) -> g(f(g(f(f(X'))))) [2]
   f(g(h(X''))) -> g(f(h(g(X'')))) [2]
   f(g(X)) -> g(f(const)) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> const [0]

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

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

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

   const => 0

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

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

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


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

(31) 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) -> null_f [0]

And the following fresh constants:    null_f

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

(32)
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(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> null_f [0]

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

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:

   null_f => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
S tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c1


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(h(z0)) -> c1

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
S tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

(40)
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(g(z0)) -> c(F(f(z0)), F(z0))

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

   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))

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

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

(42)
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(g(z0)) -> c(F(f(z0)), F(z0))
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))

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

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

(44)
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(g(z0)) -> c(F(f(z0)), F(z0)) [1]
   f(g(z0)) -> g(f(f(z0))) [0]
   f(h(z0)) -> h(g(z0)) [0]

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

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

(46)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

(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:

   F_1

(c) The following functions are completely defined:

   f_1

Due to the following rules being added:

   f(v0) -> const1 [0]

And the following fresh constants:    const1, const

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

(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(g(z0)) -> c(F(f(z0)), F(z0)) [1]
   f(g(z0)) -> g(f(f(z0))) [0]
   f(h(z0)) -> h(g(z0)) [0]
   f(v0) -> const1 [0]

The TRS has the following type information:
F :: g:h:const1 -> c
g :: g:h:const1 -> g:h:const1
c :: c -> c -> c
f :: g:h:const1 -> g:h:const1
h :: g:h:const1 -> g:h:const1
const1 :: g:h:const1
const :: c

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(g(g(z0'))) -> c(F(g(f(f(z0')))), F(g(z0'))) [1]
   F(g(h(z0''))) -> c(F(h(g(z0''))), F(h(z0''))) [1]
   F(g(z0)) -> c(F(const1), F(z0)) [1]
   f(g(g(z01))) -> g(f(g(f(f(z01))))) [0]
   f(g(h(z02))) -> g(f(h(g(z02)))) [0]
   f(g(z0)) -> g(f(const1)) [0]
   f(h(z0)) -> h(g(z0)) [0]
   f(v0) -> const1 [0]

The TRS has the following type information:
F :: g:h:const1 -> c
g :: g:h:const1 -> g:h:const1
c :: c -> c -> c
f :: g:h:const1 -> g:h:const1
h :: g:h:const1 -> g:h:const1
const1 :: g:h:const1
const :: c

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:

   const1 => 0
   const => 0

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

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

   F(z) -{ 1 }-> 1 + F(0) + F(z0) :|: z = 1 + z0, z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + f(f(z0'))) + F(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   F(z) -{ 1 }-> 1 + F(1 + (1 + z0'')) + F(1 + z0'') :|: z = 1 + (1 + z0''), z0'' >= 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 0 }-> 1 + f(0) :|: z = 1 + z0, z0 >= 0
   f(z) -{ 0 }-> 1 + f(1 + f(f(z01))) :|: z = 1 + (1 + z01), z01 >= 0
   f(z) -{ 0 }-> 1 + f(1 + (1 + z02)) :|: z = 1 + (1 + z02), z02 >= 0
   f(z) -{ 0 }-> 1 + (1 + z0) :|: z = 1 + z0, z0 >= 0


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


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(55) 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) -> null_f [0]
   F(v0) -> null_F [0]

And the following fresh constants:    null_f, null_F

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

(56)
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(g(z0)) -> c(F(f(z0)), F(z0)) [1]
   f(g(z0)) -> g(f(f(z0))) [0]
   f(h(z0)) -> h(g(z0)) [0]
   f(v0) -> null_f [0]
   F(v0) -> null_F [0]

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

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

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

   null_f => 0
   null_F => 0

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(g(z0)) -> c(F(f(z0)), F(z0)) by 
   F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0)))
   F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0)))
   F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0)))
S tuples:
   F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0)))
   F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0)))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0)))
S tuples:
   F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0)))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) by 
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0))))
S tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0))))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))))
S tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(g(g(h(z0)))) -> c(F(g(f(h(g(z0)))))) by 
   F(g(g(h(x0)))) -> c(F(g(h(g(g(x0))))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(x0)))) -> c(F(g(h(g(g(x0))))))
S tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
   F(g(g(h(x0)))) -> c(F(g(h(g(g(x0))))))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(g(g(h(x0)))) -> c(F(g(h(g(g(x0))))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
S tuples:
   F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0))))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) by F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0))))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0))))
S tuples:
   F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0))))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2