MAYBE
proof of input_oHfKjplOi6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 9 ms]
        (14) typed CpxTrs
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
        (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxTRS
        (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxWeightedTrs
            (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedTrs
            (33) CompletionProof [UPPER BOUND(ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
                (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (36) CpxTypedWeightedCompleteTrs
                (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (38) CpxRNTS
                (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxRNTS
                (41) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (42) CpxRNTS
                (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (44) CpxRNTS
                (45) IntTrsBoundProof [UPPER BOUND(ID), 561 ms]
                (46) CpxRNTS
                (47) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
                (48) CpxRNTS
            (49) CompletionProof [UPPER BOUND(ID), 0 ms]
            (50) CpxTypedWeightedCompleteTrs
                (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (52) CpxRNTS
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
        (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
(59) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(60) CpxWeightedTrs
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CpxTypedWeightedTrs
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) NarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (66) CpxTypedWeightedCompleteTrs
        (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CpxRNTS
        (71) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CpxRNTS
        (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 263 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
        (78) CpxRNTS
    (79) CompletionProof [UPPER BOUND(ID), 0 ms]
    (80) CpxTypedWeightedCompleteTrs
        (81) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (82) 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:

   g(X) -> h(X)
   c -> d
   h(d) -> g(c)

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:

   g(X) -> h(X)
   c -> d
   h(d) -> g(c)

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:
   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)
Tuples:
   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)
S tuples:
   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)
K tuples:none
Defined Rule Symbols:   g_1, c, h_1

Defined Pair Symbols:   G_1, C, H_1

Compound Symbols:   c1_1, c2, c3_2


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

(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:

   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)

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

   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)

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:

   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)

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

   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)

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:

   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)

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

   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
G(z0) -> c1(H(z0))
C -> c2
H(d) -> c3(G(c), C)
g(z0) -> h(z0)
c -> d
h(d) -> g(c)

Types:
G :: d -> c1
c1 :: c3 -> c1
H :: d -> c3
C :: c2
c2 :: c2
d :: d
c3 :: c1 -> c2 -> c3
c :: d
g :: d -> g:h
h :: d -> g:h
hole_c11_4 :: c1
hole_d2_4 :: d
hole_c33_4 :: c3
hole_c24_4 :: c2
hole_g:h5_4 :: g:h

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
G, H, g, h

They will be analysed ascendingly in the following order:
G = H
g = h

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

(14)
Obligation:
Innermost TRS:
Rules:
G(z0) -> c1(H(z0))
C -> c2
H(d) -> c3(G(c), C)
g(z0) -> h(z0)
c -> d
h(d) -> g(c)

Types:
G :: d -> c1
c1 :: c3 -> c1
H :: d -> c3
C :: c2
c2 :: c2
d :: d
c3 :: c1 -> c2 -> c3
c :: d
g :: d -> g:h
h :: d -> g:h
hole_c11_4 :: c1
hole_d2_4 :: d
hole_c33_4 :: c3
hole_c24_4 :: c2
hole_g:h5_4 :: g:h


Generator Equations:


The following defined symbols remain to be analysed:
h, G, H, g

They will be analysed ascendingly in the following order:
G = H
g = h

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

(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:

   g(X) -> h(X)
   c -> d
   h(d) -> g(c)

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:
   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)
Tuples:
   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)
S tuples:
   G(z0) -> c1(H(z0))
   C -> c2
   H(d) -> c3(G(c), C)
K tuples:none
Defined Rule Symbols:   g_1, c, h_1

Defined Pair Symbols:   G_1, C, H_1

Compound Symbols:   c1_1, c2, c3_2


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   C -> c2

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)
Tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c), C)
S tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c), C)
K tuples:none
Defined Rule Symbols:   g_1, c, h_1

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_1, c3_2


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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> h(z0)
   c -> d
   h(d) -> g(c)
Tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))
S tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))
K tuples:none
Defined Rule Symbols:   g_1, c, h_1

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_1, c3_1


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

(23) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(z0) -> h(z0)
   h(d) -> g(c)

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> d
Tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))
S tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))
K tuples:none
Defined Rule Symbols:   c

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_1, c3_1


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

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

(26)
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:

   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))

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

   c -> d

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

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

(28)
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:

   G(z0) -> c1(H(z0))
   H(d) -> c3(G(c))
   c -> d

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

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

(30)
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:

   G(z0) -> c1(H(z0)) [1]
   H(d) -> c3(G(c)) [1]
   c -> d [0]

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

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

(32)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   G(z0) -> c1(H(z0)) [1]
   H(d) -> c3(G(c)) [1]
   c -> d [0]

The TRS has the following type information:
G :: d -> c1
c1 :: c3 -> c1
H :: d -> c3
d :: d
c3 :: c1 -> c3
c :: d

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

(33) 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:

   G_1
   H_1

(c) The following functions are completely defined:

   c

Due to the following rules being added:

   c -> d [0]

And the following fresh constants:    const, const1

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

(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:

   G(z0) -> c1(H(z0)) [1]
   H(d) -> c3(G(c)) [1]
   c -> d [0]
   c -> d [0]

The TRS has the following type information:
G :: d -> c1
c1 :: c3 -> c1
H :: d -> c3
d :: d
c3 :: c1 -> c3
c :: d
const :: c1
const1 :: c3

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

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

(36)
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:

   G(z0) -> c1(H(z0)) [1]
   H(d) -> c3(G(d)) [1]
   H(d) -> c3(G(d)) [1]
   c -> d [0]
   c -> d [0]

The TRS has the following type information:
G :: d -> c1
c1 :: c3 -> c1
H :: d -> c3
d :: d
c3 :: c1 -> c3
c :: d
const :: c1
const1 :: c3

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:

   d => 0
   const => 0
   const1 => 0

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

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

   G(z) -{ 1 }-> 1 + H(z0) :|: z = z0, z0 >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 


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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   G(z) -{ 1 }-> 1 + H(z) :|: z >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 


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

(41) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { H, G }
   { c }

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   G(z) -{ 1 }-> 1 + H(z) :|: z >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {H,G}, {c}

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

(43) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   G(z) -{ 1 }-> 1 + H(z) :|: z >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {H,G}, {c}

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: H
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: G
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   G(z) -{ 1 }-> 1 + H(z) :|: z >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {H,G}, {c}
Previous analysis results are:
  H: runtime: ?, size: O(1) [0]
  G: runtime: ?, size: O(1) [1]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: H
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   G(z) -{ 1 }-> 1 + H(z) :|: z >= 0
   H(z) -{ 1 }-> 1 + G(0) :|: z = 0
   c -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {H,G}, {c}
Previous analysis results are:
  H: runtime: INF, size: O(1) [0]
  G: runtime: ?, size: O(1) [1]

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

(49) 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:

   c -> null_c [0]
   H(v0) -> null_H [0]

And the following fresh constants:    null_c, null_H, const

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

(50)
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:

   G(z0) -> c1(H(z0)) [1]
   H(d) -> c3(G(c)) [1]
   c -> d [0]
   c -> null_c [0]
   H(v0) -> null_H [0]

The TRS has the following type information:
G :: d:null_c -> c1
c1 :: c3:null_H -> c1
H :: d:null_c -> c3:null_H
d :: d:null_c
c3 :: c1 -> c3:null_H
c :: d:null_c
null_c :: d:null_c
null_H :: c3:null_H
const :: c1

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:

   d => 1
   null_c => 0
   null_H => 0
   const => 0

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

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

   G(z) -{ 1 }-> 1 + H(z0) :|: z = z0, z0 >= 0
   H(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   H(z) -{ 1 }-> 1 + G(c) :|: z = 1
   c -{ 0 }-> 1 :|: 
   c -{ 0 }-> 0 :|: 

Only complete derivations are relevant for the runtime complexity.

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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(d) -> c3(G(c)) by 
   H(d) -> c3(G(d))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> d
Tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(d))
S tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(d))
K tuples:none
Defined Rule Symbols:   c

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_1, c3_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   c -> d

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(d))
S tuples:
   G(z0) -> c1(H(z0))
   H(d) -> c3(G(d))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_1, c3_1


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

(57) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace G(z0) -> c1(H(z0)) by 
   G(d) -> c1(H(d))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(d) -> c3(G(d))
   G(d) -> c1(H(d))
S tuples:
   H(d) -> c3(G(d))
   G(d) -> c1(H(d))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c3_1, c1_1


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

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

(60)
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:

   g(X) -> h(X) [1]
   c -> d [1]
   h(d) -> g(c) [1]

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

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

(62)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(X) -> h(X) [1]
   c -> d [1]
   h(d) -> g(c) [1]

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

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

(63) 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:

   g_1
   h_1

(c) The following functions are completely defined:

   c

Due to the following rules being added:
none

And the following fresh constants:    const

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

(64)
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:

   g(X) -> h(X) [1]
   c -> d [1]
   h(d) -> g(c) [1]

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

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

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

(66)
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:

   g(X) -> h(X) [1]
   c -> d [1]
   h(d) -> g(d) [2]

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

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

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

   d => 0
   const => 0

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(X) :|: X >= 0, z = X
   h(z) -{ 2 }-> g(0) :|: z = 0


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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(z) :|: z >= 0
   h(z) -{ 2 }-> g(0) :|: z = 0


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

(71) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { h, g }
   { c }

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(z) :|: z >= 0
   h(z) -{ 2 }-> g(0) :|: z = 0

Function symbols to be analyzed: {h,g}, {c}

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

(73) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(z) :|: z >= 0
   h(z) -{ 2 }-> g(0) :|: z = 0

Function symbols to be analyzed: {h,g}, {c}

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: h
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(z) :|: z >= 0
   h(z) -{ 2 }-> g(0) :|: z = 0

Function symbols to be analyzed: {h,g}, {c}
Previous analysis results are:
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: h
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(z) :|: z >= 0
   h(z) -{ 2 }-> g(0) :|: z = 0

Function symbols to be analyzed: {h,g}, {c}
Previous analysis results are:
  h: runtime: INF, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(79) 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:
none

And the following fresh constants:    const

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

(80)
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:

   g(X) -> h(X) [1]
   c -> d [1]
   h(d) -> g(c) [1]

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

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

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

   d => 0
   const => 0

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   g(z) -{ 1 }-> h(X) :|: X >= 0, z = X
   h(z) -{ 1 }-> g(c) :|: z = 0

Only complete derivations are relevant for the runtime complexity.