MAYBE
proof of input_MPxDAtiIKs.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 9 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) 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) CpxTRS
        (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), 219 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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
(55) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(56) CpxWeightedTrs
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxTypedWeightedTrs
    (59) CompletionProof [UPPER BOUND(ID), 0 ms]
    (60) CpxTypedWeightedCompleteTrs
        (61) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxTypedWeightedCompleteTrs
        (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
        (65) InliningProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CpxRNTS
        (69) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CpxRNTS
        (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 173 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 30 ms]
        (76) CpxRNTS
        (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (78) CpxRNTS
        (79) IntTrsBoundProof [UPPER BOUND(ID), 141 ms]
        (80) CpxRNTS
        (81) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
        (82) CpxRNTS
    (83) CompletionProof [UPPER BOUND(ID), 0 ms]
    (84) CpxTypedWeightedCompleteTrs
        (85) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (86) CpxRNTS


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> 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(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
S tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1, IF_3

Compound Symbols:   c1_2, c2, c3


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

(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(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3

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

   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3

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

   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c1(IF(z0, c, f(true)), F(true))
IF(true, z0, z1) -> c2
IF(false, z0, z1) -> c3
f(z0) -> if(z0, c, f(true))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
F :: true:false -> c1
c1 :: c2:c3 -> c1 -> c1
IF :: true:false -> c -> c -> c2:c3
c :: c
f :: true:false -> c
true :: true:false
c2 :: c2:c3
false :: true:false
c3 :: c2:c3
if :: true:false -> c -> c -> c
hole_c11_4 :: c1
hole_true:false2_4 :: true:false
hole_c2:c33_4 :: c2:c3
hole_c4_4 :: c
gen_c15_4 :: Nat -> c1

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

(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(z0) -> c1(IF(z0, c, f(true)), F(true))
IF(true, z0, z1) -> c2
IF(false, z0, z1) -> c3
f(z0) -> if(z0, c, f(true))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
F :: true:false -> c1
c1 :: c2:c3 -> c1 -> c1
IF :: true:false -> c -> c -> c2:c3
c :: c
f :: true:false -> c
true :: true:false
c2 :: c2:c3
false :: true:false
c3 :: c2:c3
if :: true:false -> c -> c -> c
hole_c11_4 :: c1
hole_true:false2_4 :: true:false
hole_c2:c33_4 :: c2:c3
hole_c4_4 :: c
gen_c15_4 :: Nat -> c1


Generator Equations:
gen_c15_4(0) <=> hole_c11_4
gen_c15_4(+(x, 1)) <=> c1(c2, gen_c15_4(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(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3

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

   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
S tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1, IF_3

Compound Symbols:   c1_2, c2, c3


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IF(false, z0, z1) -> c3
   IF(true, z0, z1) -> c2

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
S tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1

Compound Symbols:   c1_2


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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(F(true))
S tuples:
   F(z0) -> c1(F(true))
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

(23) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0) -> c1(F(true))
S tuples:
   F(z0) -> c1(F(true))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c1_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 CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c1(F(true))

S is empty.
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:

   F(z0) -> c1(F(true))

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:

   F(z0) -> c1(F(true)) [1]

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:

   F(z0) -> c1(F(true)) [1]

The TRS has the following type information:
F :: true -> c1
c1 :: c1 -> c1
true :: true

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:

   F_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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(z0) -> c1(F(true)) [1]

The TRS has the following type information:
F :: true -> c1
c1 :: c1 -> c1
true :: true
const :: c1

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:

   F(z0) -> c1(F(true)) [1]

The TRS has the following type information:
F :: true -> c1
c1 :: c1 -> c1
true :: true
const :: c1

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:

   true => 0
   const => 0

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

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

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

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


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

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

   { F }

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

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

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

Function symbols to be analyzed: {F}

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

(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:

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

Function symbols to be analyzed: {F}

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(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:
none

And the following fresh constants:    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:

   F(z0) -> c1(F(true)) [1]

The TRS has the following type information:
F :: true -> c1
c1 :: c1 -> c1
true :: true
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:

   true => 0
   const => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

(53) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0) -> c1(F(true)) by 
   F(true) -> c1(F(true))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(true) -> c1(F(true))
S tuples:
   F(true) -> c1(F(true))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

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

(56)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

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

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

(58)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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

(59) 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
   if_3

Due to the following rules being added:
none

And the following fresh constants: none

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

(60)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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

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

(62)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> if(X, c, if(true, c, f(true))) [2]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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

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

   c => 0
   true => 1
   false => 0

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(65) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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


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

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

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


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

   { if }
   { f }

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

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

Function symbols to be analyzed: {if}, {f}

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

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

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

Function symbols to be analyzed: {if}, {f}

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(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z' + z''

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

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

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

Function symbols to be analyzed: {if}, {f}
Previous analysis results are:
  if: runtime: ?, size: O(n^1) [z' + z'']

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(75) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

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

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

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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(79) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  f: runtime: ?, size: O(1) [0]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  f: runtime: INF, size: O(1) [0]

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(83) 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: none

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(84)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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

   c => 0
   true => 1
   false => 0

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.