KILLED
proof of input_0SNIqNEPXY.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (30) CpxRNTS
            (31) CompletionProof [UPPER BOUND(ID), 0 ms]
            (32) CpxTypedWeightedCompleteTrs
                (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (34) CpxTypedWeightedCompleteTrs
                (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (36) CpxRNTS
                (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (38) CpxRNTS
    (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
        (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (42) CdtProblem
        (43) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CdtProblem
        (45) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (46) CdtProblem
        (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CdtProblem
        (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
(51) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(52) CpxWeightedTrs
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxTypedWeightedTrs
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxRNTS
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (66) 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(g(X), Y) -> f(X, f(g(X), Y))

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), Y) -> f(X, f(g(X), Y))

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(g(X), Y) -> f(X, f(g(X), Y))

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(g(z0), z1) -> f(z0, f(g(z0), z1))
Tuples:
   F(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))
S tuples:
   F(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), 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(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))

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

   f(g(z0), z1) -> f(z0, f(g(z0), z1))

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), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))

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

   f(g(z0), z1) -> f(z0, f(g(z0), z1))

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

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

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

Types:
F :: g -> f -> c
g :: g -> g
c :: c -> c -> c
f :: g -> f -> f
hole_c1_2 :: c
hole_g2_2 :: g
hole_f3_2 :: f
gen_c4_2 :: Nat -> c
gen_g5_2 :: Nat -> g

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

(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), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))
f(g(z0), z1) -> f(z0, f(g(z0), z1))

Types:
F :: g -> f -> c
g :: g -> g
c :: c -> c -> c
f :: g -> f -> f
hole_c1_2 :: c
hole_g2_2 :: g
hole_f3_2 :: f
gen_c4_2 :: Nat -> c
gen_g5_2 :: Nat -> g


Generator Equations:
gen_c4_2(0) <=> hole_c1_2
gen_c4_2(+(x, 1)) <=> c(hole_c1_2, gen_c4_2(x))
gen_g5_2(0) <=> hole_g2_2
gen_g5_2(+(x, 1)) <=> g(gen_g5_2(x))


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

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

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

(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(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))

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

   f(g(z0), z1) -> f(z0, f(g(z0), z1))

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(g(z0), z1) -> f(z0, f(g(z0), z1))
Tuples:
   F(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))
S tuples:
   F(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), 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(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))

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

   f(g(z0), z1) -> f(z0, f(g(z0), z1))

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(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1))
   f(g(z0), z1) -> f(z0, f(g(z0), z1))

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

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

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

(27) 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, const

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

(28)
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), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1)) [1]
   f(g(z0), z1) -> f(z0, f(g(z0), z1)) [0]
   f(v0, v1) -> null_f [0]
   F(v0, v1) -> null_F [0]

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

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

(29) 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
   const => 0

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

(30)
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(z0, f(1 + z0, z1)) + F(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   f(z, z') -{ 0 }-> f(z0, f(1 + z0, z1)) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(31) 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) -> const2 [0]

And the following fresh constants:    const2, const, const1

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

(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(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1)) [1]
   f(g(z0), z1) -> f(z0, f(g(z0), z1)) [0]
   f(v0, v1) -> const2 [0]

The TRS has the following type information:
F :: g -> const2 -> c
g :: g -> g
c :: c -> c -> c
f :: g -> const2 -> const2
const2 :: const2
const :: c
const1 :: g

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

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

(34)
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), z1) -> c(F(z0, f(z0, f(g(z0), z1))), F(g(z0), z1)) [1]
   F(g(z0), z1) -> c(F(z0, const2), F(g(z0), z1)) [1]
   f(g(z0), z1) -> f(z0, f(z0, f(g(z0), z1))) [0]
   f(g(z0), z1) -> f(z0, const2) [0]
   f(v0, v1) -> const2 [0]

The TRS has the following type information:
F :: g -> const2 -> c
g :: g -> g
c :: c -> c -> c
f :: g -> const2 -> const2
const2 :: const2
const :: c
const1 :: g

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

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

   const2 => 0
   const => 0
   const1 => 0

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

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

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


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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(g(z0), z1) -> c(F(z0, f(g(z0), z1)), F(g(z0), z1)) by 
   F(g(z0), z1) -> c(F(z0, f(z0, f(g(z0), z1))), F(g(z0), z1))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(g(z0), z1) -> c(F(z0, f(z0, f(g(z0), z1))), F(g(z0), z1)) by 
   F(g(g(z0)), x1) -> c(F(g(z0), f(z0, f(g(z0), f(g(g(z0)), x1)))), F(g(g(z0)), x1))
   F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(g(z0), z1)))), F(g(z0), z1))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0), z1) -> f(z0, f(g(z0), z1))
Tuples:
   F(g(g(z0)), x1) -> c(F(g(z0), f(z0, f(g(z0), f(g(g(z0)), x1)))), F(g(g(z0)), x1))
   F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(g(z0), z1)))), F(g(z0), z1))
S tuples:
   F(g(g(z0)), x1) -> c(F(g(z0), f(z0, f(g(z0), f(g(g(z0)), x1)))), F(g(g(z0)), x1))
   F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(g(z0), z1)))), F(g(z0), z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(43) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(g(z0)), x1) -> c(F(g(z0), f(z0, f(g(z0), f(g(g(z0)), x1)))), F(g(g(z0)), x1)) by F(g(g(z0)), z1) -> c(F(g(z0), f(z0, f(z0, f(g(z0), f(g(g(z0)), z1))))), F(g(g(z0)), z1))
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(45) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(g(z0), z1)))), F(g(z0), z1)) by F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(z0, f(g(z0), z1))))), F(g(z0), z1))
----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(47) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(g(z0)), z1) -> c(F(g(z0), f(z0, f(z0, f(g(z0), f(g(g(z0)), z1))))), F(g(g(z0)), z1)) by F(g(g(z0)), z1) -> c(F(g(z0), f(z0, f(z0, f(z0, f(g(z0), f(g(g(z0)), z1)))))), F(g(g(z0)), z1))
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

(49) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(z0, f(g(z0), z1))))), F(g(z0), z1)) by F(g(z0), z1) -> c(F(z0, f(z0, f(z0, f(z0, f(z0, f(g(z0), z1)))))), F(g(z0), z1))
----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2

Compound Symbols:   c_2


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

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

(52)
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), Y) -> f(X, f(g(X), Y)) [1]

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

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

(54)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(X), Y) -> f(X, f(g(X), Y)) [1]

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

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

(55) 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) -> const [0]

And the following fresh constants:    const, const1

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

(56)
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), Y) -> f(X, f(g(X), Y)) [1]
   f(v0, v1) -> const [0]

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

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

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

(58)
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), Y) -> f(X, f(X, f(g(X), Y))) [2]
   f(g(X), Y) -> f(X, const) [1]
   f(v0, v1) -> const [0]

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

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

(59) 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
   const1 => 0

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(63) 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, const

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

(64)
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), Y) -> f(X, f(g(X), Y)) [1]
   f(v0, v1) -> null_f [0]

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

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

(65) 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
   const => 0

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.