MAYBE
proof of input_SxENfVKKrx.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 9 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) 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), 462 ms]
                (46) CpxRNTS
                (47) IntTrsBoundProof [UPPER BOUND(ID), 66 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) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(58) CpxWeightedTrs
    (59) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxTypedWeightedTrs
    (61) CompletionProof [UPPER BOUND(ID), 0 ms]
    (62) CpxTypedWeightedCompleteTrs
        (63) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [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), 460 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
        (76) CpxRNTS
    (77) CompletionProof [UPPER BOUND(ID), 0 ms]
    (78) CpxTypedWeightedCompleteTrs
        (79) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (80) 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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0') -> cons(0', f(s(0')))
   f(s(0')) -> f(p(s(0')))
   p(s(0')) -> 0'

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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_1, c1_2, c2


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

(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(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2

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

   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0') -> c(F(s(0')))
   F(s(0')) -> c1(F(p(s(0'))), P(s(0')))
   P(s(0')) -> c2

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

   f(0') -> cons(0', f(s(0')))
   f(s(0')) -> f(p(s(0')))
   p(s(0')) -> 0'

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

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

(12)
Obligation:
Innermost TRS:
Rules:
F(0') -> c(F(s(0')))
F(s(0')) -> c1(F(p(s(0'))), P(s(0')))
P(s(0')) -> c2
f(0') -> cons(0', f(s(0')))
f(s(0')) -> f(p(s(0')))
p(s(0')) -> 0'

Types:
F :: 0':s -> c:c1
0' :: 0':s
c :: c:c1 -> c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c2 -> c:c1
p :: 0':s -> 0':s
P :: 0':s -> c2
c2 :: c2
f :: 0':s -> cons
cons :: 0':s -> cons -> cons
hole_c:c11_0 :: c:c1
hole_0':s2_0 :: 0':s
hole_c23_0 :: c2
hole_cons4_0 :: cons
gen_c:c15_0 :: Nat -> c:c1
gen_0':s6_0 :: Nat -> 0':s
gen_cons7_0 :: Nat -> cons

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(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, f
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
F(0') -> c(F(s(0')))
F(s(0')) -> c1(F(p(s(0'))), P(s(0')))
P(s(0')) -> c2
f(0') -> cons(0', f(s(0')))
f(s(0')) -> f(p(s(0')))
p(s(0')) -> 0'

Types:
F :: 0':s -> c:c1
0' :: 0':s
c :: c:c1 -> c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c2 -> c:c1
p :: 0':s -> 0':s
P :: 0':s -> c2
c2 :: c2
f :: 0':s -> cons
cons :: 0':s -> cons -> cons
hole_c:c11_0 :: c:c1
hole_0':s2_0 :: 0':s
hole_c23_0 :: c2
hole_cons4_0 :: cons
gen_c:c15_0 :: Nat -> c:c1
gen_0':s6_0 :: Nat -> 0':s
gen_cons7_0 :: Nat -> cons


Generator Equations:
gen_c:c15_0(0) <=> hole_c:c11_0
gen_c:c15_0(+(x, 1)) <=> c(gen_c:c15_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))
gen_cons7_0(0) <=> hole_cons4_0
gen_cons7_0(+(x, 1)) <=> cons(0', gen_cons7_0(x))


The following defined symbols remain to be analysed:
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(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2

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

   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_1, c1_2, c2


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   P(s(0)) -> c2

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_2


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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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

(23) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, 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 CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))

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

   p(s(0)) -> 0

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(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
   p(s(0)) -> 0

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(0) -> c(F(s(0))) [1]
   F(s(0)) -> c1(F(p(s(0)))) [1]
   p(s(0)) -> 0 [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:

   F(0) -> c(F(s(0))) [1]
   F(s(0)) -> c1(F(p(s(0)))) [1]
   p(s(0)) -> 0 [0]

The TRS has the following type information:
F :: 0:s -> c:c1
0 :: 0:s
c :: c:c1 -> c:c1
s :: 0:s -> 0:s
c1 :: c:c1 -> c:c1
p :: 0:s -> 0:s

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:

   p_1

Due to the following rules being added:

   p(v0) -> 0 [0]

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(0) -> c(F(s(0))) [1]
   F(s(0)) -> c1(F(p(s(0)))) [1]
   p(s(0)) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
F :: 0:s -> c:c1
0 :: 0:s
c :: c:c1 -> c:c1
s :: 0:s -> 0:s
c1 :: c:c1 -> c:c1
p :: 0:s -> 0:s
const :: c: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(0) -> c(F(s(0))) [1]
   F(s(0)) -> c1(F(0)) [1]
   F(s(0)) -> c1(F(0)) [1]
   p(s(0)) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
F :: 0:s -> c:c1
0 :: 0:s
c :: c:c1 -> c:c1
s :: 0:s -> 0:s
c1 :: c:c1 -> c:c1
p :: 0:s -> 0:s
const :: c: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:

   0 => 0
   const => 0

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

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

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


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

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


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

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

   { F }
   { p }

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

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

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

Function symbols to be analyzed: {F}, {p}

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

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

Function symbols to be analyzed: {F}, {p}

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

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

Function symbols to be analyzed: {F}, {p}
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 = 1 + 0
   F(z) -{ 1 }-> 1 + F(1 + 0) :|: z = 0
   p(z) -{ 0 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}, {p}
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:

   p(v0) -> null_p [0]
   F(v0) -> null_F [0]

And the following fresh constants:    null_p, null_F

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

(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(0) -> c(F(s(0))) [1]
   F(s(0)) -> c1(F(p(s(0)))) [1]
   p(s(0)) -> 0 [0]
   p(v0) -> null_p [0]
   F(v0) -> null_F [0]

The TRS has the following type information:
F :: 0:s:null_p -> c:c1:null_F
0 :: 0:s:null_p
c :: c:c1:null_F -> c:c1:null_F
s :: 0:s:null_p -> 0:s:null_p
c1 :: c:c1:null_F -> c:c1:null_F
p :: 0:s:null_p -> 0:s:null_p
null_p :: 0:s:null_p
null_F :: c:c1:null_F

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:

   0 => 0
   null_p => 0
   null_F => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0)) -> c1(F(p(s(0)))) by 
   F(s(0)) -> c1(F(0))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   p(s(0)) -> 0

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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

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

(58)
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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]

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

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

(60)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s

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

(61) 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:

   p_1

Due to the following rules being added:

   p(v0) -> 0 [0]

And the following fresh constants:    const

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

(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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s
const :: cons

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

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

(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:

   f(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(0) [2]
   f(s(0)) -> f(0) [1]
   p(s(0)) -> 0 [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s
const :: cons

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:

   0 => 0
   const => 0

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

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

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


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

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


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

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

   { f }
   { p }

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

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

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

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

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

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

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

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

(73) 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

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

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

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

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

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

(75) 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: ?

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

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

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

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

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

(77) 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]
   p(v0) -> null_p [0]

And the following fresh constants:    null_f, null_p

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

(78)
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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]
   f(v0) -> null_f [0]
   p(v0) -> null_p [0]

The TRS has the following type information:
f :: 0:s:null_p -> cons:null_f
0 :: 0:s:null_p
cons :: 0:s:null_p -> cons:null_f -> cons:null_f
s :: 0:s:null_p -> 0:s:null_p
p :: 0:s:null_p -> 0:s:null_p
null_f :: cons:null_f
null_p :: 0:s:null_p

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

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

   0 => 0
   null_f => 0
   null_p => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.