KILLED
proof of input_UKnojHQUMe.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [UPPER BOUND(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) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (12) TRS for Loop Detection
(13) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(14) CpxTRS
(15) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(16) CdtProblem
    (17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxTRS
        (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) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (30) CpxTypedWeightedCompleteTrs
                (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (32) CpxRNTS
                (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (34) CpxRNTS
                (35) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (36) CpxRNTS
                (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (38) CpxRNTS
                (39) IntTrsBoundProof [UPPER BOUND(ID), 19.7 s]
                (40) CpxRNTS
                (41) IntTrsBoundProof [UPPER BOUND(ID), 6587 ms]
                (42) CpxRNTS
            (43) CompletionProof [UPPER BOUND(ID), 0 ms]
            (44) CpxTypedWeightedCompleteTrs
                (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (46) CpxRNTS
    (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms]
    (48) CdtProblem
        (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
        (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CdtProblem
        (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
        (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
(69) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(70) CpxWeightedTrs
    (71) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CpxTypedWeightedTrs
    (73) CompletionProof [UPPER BOUND(ID), 0 ms]
    (74) CpxTypedWeightedCompleteTrs
        (75) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CpxTypedWeightedCompleteTrs
        (77) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (78) CpxRNTS
        (79) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CpxRNTS
        (81) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CpxRNTS
        (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (84) CpxRNTS
        (85) IntTrsBoundProof [UPPER BOUND(ID), 20.1 s]
        (86) CpxRNTS
        (87) IntTrsBoundProof [UPPER BOUND(ID), 6610 ms]
        (88) CpxRNTS
    (89) CompletionProof [UPPER BOUND(ID), 0 ms]
    (90) CpxTypedWeightedCompleteTrs
        (91) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (92) 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(c(s(x), y)) -> f(c(x, s(y)))
   f(c(s(x), s(y))) -> g(c(x, y))
   g(c(x, s(y))) -> g(c(s(x), y))
   g(c(s(x), s(y))) -> f(c(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(c(s(x), y)) -> f(c(x, s(y)))
   f(c(s(x), s(y))) -> g(c(x, y))
   g(c(x, s(y))) -> g(c(s(x), y))
   g(c(s(x), s(y))) -> f(c(x, y))

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(c(s(z0), z1)) -> f(c(z0, s(z1)))
   f(c(s(z0), s(z1))) -> g(c(z0, z1))
   g(c(z0, s(z1))) -> g(c(s(z0), z1))
   g(c(s(z0), s(z1))) -> f(c(z0, z1))
Tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
S tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1


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

(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(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))

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

   f(c(s(z0), z1)) -> f(c(z0, s(z1)))
   f(c(s(z0), s(z1))) -> g(c(z0, z1))
   g(c(z0, s(z1))) -> g(c(s(z0), z1))
   g(c(s(z0), s(z1))) -> f(c(z0, 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(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
f(c(s(z0), z1)) -> f(c(z0, s(z1)))
f(c(s(z0), s(z1))) -> g(c(z0, z1))
g(c(z0, s(z1))) -> g(c(s(z0), z1))
g(c(s(z0), s(z1))) -> f(c(z0, z1))

Types:
F :: c -> c1:c2
c :: s -> s -> c
s :: s -> s
c1 :: c1:c2 -> c1:c2
c2 :: c3:c4 -> c1:c2
G :: c -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c1:c2 -> c3:c4
f :: c -> f:g
g :: c -> f:g
hole_c1:c21_5 :: c1:c2
hole_c2_5 :: c
hole_s3_5 :: s
hole_c3:c44_5 :: c3:c4
hole_f:g5_5 :: f:g
gen_c1:c26_5 :: Nat -> c1:c2
gen_s7_5 :: Nat -> s
gen_c3:c48_5 :: Nat -> c3:c4

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

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

(12)
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(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))

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

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

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

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

(14)
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(c(s(x), y)) -> f(c(x, s(y)))
   f(c(s(x), s(y))) -> g(c(x, y))
   g(c(x, s(y))) -> g(c(s(x), y))
   g(c(s(x), s(y))) -> f(c(x, y))

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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(c(s(z0), z1)) -> f(c(z0, s(z1)))
   f(c(s(z0), s(z1))) -> g(c(z0, z1))
   g(c(z0, s(z1))) -> g(c(s(z0), z1))
   g(c(s(z0), s(z1))) -> f(c(z0, z1))
Tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
S tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1


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

(17) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(c(s(z0), z1)) -> f(c(z0, s(z1)))
   f(c(s(z0), s(z1))) -> g(c(z0, z1))
   g(c(z0, s(z1))) -> g(c(s(z0), z1))
   g(c(s(z0), s(z1))) -> f(c(z0, z1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
S tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1


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

(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 CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))

S is empty.
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(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(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(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) [1]
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) [1]
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) [1]
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) [1]

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(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) [1]
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) [1]
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) [1]
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) [1]

The TRS has the following type information:
F :: c -> c1:c2
c :: s -> s -> c
s :: s -> s
c1 :: c1:c2 -> c1:c2
c2 :: c3:c4 -> c1:c2
G :: c -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c1:c2 -> c3:c4

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

(27) 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
   G_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3

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

(28)
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(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) [1]
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) [1]
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) [1]
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) [1]

The TRS has the following type information:
F :: c -> c1:c2
c :: s -> s -> c
s :: s -> s
c1 :: c1:c2 -> c1:c2
c2 :: c3:c4 -> c1:c2
G :: c -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c1:c2 -> c3:c4
const :: c1:c2
const1 :: c
const2 :: s
const3 :: c3:c4

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

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

(30)
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(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) [1]
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) [1]
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) [1]
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) [1]

The TRS has the following type information:
F :: c -> c1:c2
c :: s -> s -> c
s :: s -> s
c1 :: c1:c2 -> c1:c2
c2 :: c3:c4 -> c1:c2
G :: c -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c1:c2 -> c3:c4
const :: c1:c2
const1 :: c
const2 :: s
const3 :: c3:c4

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

(31) 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
   const2 => 0
   const3 => 0

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0


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

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

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

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0


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

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

   { G, F }

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

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

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0

Function symbols to be analyzed: {G,F}

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

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

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

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0

Function symbols to be analyzed: {G,F}

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z) -{ 1 }-> 1 + G(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0
   F(z) -{ 1 }-> 1 + F(1 + z0 + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0
   G(z) -{ 1 }-> 1 + G(1 + (1 + z0) + z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1), z0 >= 0
   G(z) -{ 1 }-> 1 + F(1 + z0 + z1) :|: z1 >= 0, z = 1 + (1 + z0) + (1 + z1), z0 >= 0

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

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

(43) 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]
   G(v0) -> null_G [0]

And the following fresh constants:    null_F, null_G, const, const1

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

(44)
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(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) [1]
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) [1]
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) [1]
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) [1]
   F(v0) -> null_F [0]
   G(v0) -> null_G [0]

The TRS has the following type information:
F :: c -> c1:c2:null_F
c :: s -> s -> c
s :: s -> s
c1 :: c1:c2:null_F -> c1:c2:null_F
c2 :: c3:c4:null_G -> c1:c2:null_F
G :: c -> c3:c4:null_G
c3 :: c3:c4:null_G -> c3:c4:null_G
c4 :: c1:c2:null_F -> c3:c4:null_G
null_F :: c1:c2:null_F
null_G :: c3:c4:null_G
const :: c
const1 :: s

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

(45) 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_G => 0
   const => 0
   const1 => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = x_1
   POL(G(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
S tuples:
   F(c(s(z0), z1)) -> c1(F(c(z0, s(z1))))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
K tuples:
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(c(s(z0), z1)) -> c1(F(c(z0, s(z1)))) by 
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
S tuples:
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
K tuples:
   F(c(s(z0), s(z1))) -> c2(G(c(z0, z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c2_1, c3_1, c4_1, c1_1


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(c(s(z0), s(z1))) -> c2(G(c(z0, z1))) by 
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
S tuples:
   G(c(z0, s(z1))) -> c3(G(c(s(z0), z1)))
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
K tuples:
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c3_1, c4_1, c1_1, c2_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(c(z0, s(z1))) -> c3(G(c(s(z0), z1))) by 
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
S tuples:
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
K tuples:
   G(c(s(z0), s(z1))) -> c4(F(c(z0, z1)))
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c4_1, c1_1, c2_1, c3_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(c(s(z0), s(z1))) -> c4(F(c(z0, z1))) by 
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
S tuples:
   F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
K tuples:
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(c(s(s(y0)), z1)) -> c1(F(c(s(y0), s(z1)))) by 
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
S tuples:
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
K tuples:
   F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1))))
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c2_1, c3_1, c4_1, c1_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(c(s(z0), s(s(y1)))) -> c2(G(c(z0, s(y1)))) by 
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
S tuples:
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
K tuples:
   F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c2_1, c3_1, c4_1, c1_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(c(s(s(y0)), s(s(y1)))) -> c2(G(c(s(y0), s(y1)))) by 
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
S tuples:
   G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
K tuples:
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c3_1, c4_1, c1_1, c2_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(c(z0, s(s(y1)))) -> c3(G(c(s(z0), s(y1)))) by 
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
S tuples:
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
K tuples:
   G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1)))
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c4_1, c1_1, c2_1, c3_1


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

(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(c(s(s(s(y0))), s(z1))) -> c4(F(c(s(s(y0)), z1))) by 
   G(c(s(s(s(s(y0)))), s(z1))) -> c4(F(c(s(s(s(y0))), z1)))
   G(c(s(s(s(z0))), s(s(y1)))) -> c4(F(c(s(s(z0)), s(y1))))
   G(c(s(s(s(s(y0)))), s(s(y1)))) -> c4(F(c(s(s(s(y0))), s(y1))))
   G(c(s(s(s(z0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(z0)), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(y1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
   G(c(s(s(s(s(y0)))), s(z1))) -> c4(F(c(s(s(s(y0))), z1)))
   G(c(s(s(s(z0))), s(s(y1)))) -> c4(F(c(s(s(z0)), s(y1))))
   G(c(s(s(s(s(y0)))), s(s(y1)))) -> c4(F(c(s(s(s(y0))), s(y1))))
   G(c(s(s(s(z0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(z0)), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(y1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))
S tuples:
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
K tuples:
   G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1)))))
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(z1))) -> c4(F(c(s(s(s(y0))), z1)))
   G(c(s(s(s(z0))), s(s(y1)))) -> c4(F(c(s(s(z0)), s(y1))))
   G(c(s(s(s(s(y0)))), s(s(y1)))) -> c4(F(c(s(s(s(y0))), s(y1))))
   G(c(s(s(s(z0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(z0)), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(y1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c4_1, c1_1, c2_1, c3_1


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

(67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(c(s(s(y0)), s(s(s(y1))))) -> c4(F(c(s(y0), s(s(y1))))) by 
   G(c(s(s(s(s(y0)))), s(s(s(z1))))) -> c4(F(c(s(s(s(y0))), s(s(z1)))))
   G(c(s(s(s(y0))), s(s(s(z1))))) -> c4(F(c(s(s(y0)), s(s(z1)))))
   G(c(s(s(z0)), s(s(s(s(y1)))))) -> c4(F(c(s(z0), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(z1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(z1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))
   G(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(y0)), s(s(s(y1))))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
   G(c(s(s(s(s(y0)))), s(z1))) -> c4(F(c(s(s(s(y0))), z1)))
   G(c(s(s(s(z0))), s(s(y1)))) -> c4(F(c(s(s(z0)), s(y1))))
   G(c(s(s(s(s(y0)))), s(s(y1)))) -> c4(F(c(s(s(s(y0))), s(y1))))
   G(c(s(s(s(z0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(z0)), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(y1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(y0)))), s(s(s(z1))))) -> c4(F(c(s(s(s(y0))), s(s(z1)))))
   G(c(s(s(z0)), s(s(s(s(y1)))))) -> c4(F(c(s(z0), s(s(s(y1))))))
S tuples:
   F(c(s(s(s(y0))), z1)) -> c1(F(c(s(s(y0)), s(z1))))
   F(c(s(s(z0)), s(y1))) -> c1(F(c(s(z0), s(s(y1)))))
   F(c(s(s(s(y0))), s(y1))) -> c1(F(c(s(s(y0)), s(s(y1)))))
   G(c(z0, s(s(s(y1))))) -> c3(G(c(s(z0), s(s(y1)))))
   G(c(s(s(y0)), s(s(z1)))) -> c3(G(c(s(s(s(y0))), s(z1))))
   G(c(s(y0), s(s(s(s(y1)))))) -> c3(G(c(s(s(y0)), s(s(s(y1))))))
   G(c(s(s(y0)), s(s(s(s(y1)))))) -> c3(G(c(s(s(s(y0))), s(s(s(y1))))))
K tuples:
   G(c(s(s(s(y0))), s(s(s(y1))))) -> c4(F(c(s(s(y0)), s(s(y1)))))
   F(c(s(z0), s(s(s(y1))))) -> c2(G(c(z0, s(s(y1)))))
   F(c(s(s(s(s(y0)))), s(s(z1)))) -> c2(G(c(s(s(s(y0))), s(z1))))
   F(c(s(s(s(y0))), s(s(s(s(y1)))))) -> c2(G(c(s(s(y0)), s(s(s(y1))))))
   F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))) -> c2(G(c(s(s(s(y0))), s(s(s(y1))))))
   F(c(s(s(z0)), s(s(s(y1))))) -> c2(G(c(s(z0), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(z1))) -> c4(F(c(s(s(s(y0))), z1)))
   G(c(s(s(s(z0))), s(s(y1)))) -> c4(F(c(s(s(z0)), s(y1))))
   G(c(s(s(s(s(y0)))), s(s(y1)))) -> c4(F(c(s(s(s(y0))), s(y1))))
   G(c(s(s(s(z0))), s(s(s(s(y1)))))) -> c4(F(c(s(s(z0)), s(s(s(y1))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(y1))))) -> c4(F(c(s(s(s(s(y0)))), s(s(y1)))))
   G(c(s(s(s(s(y0)))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(y0))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(s(y0))))), s(s(s(s(s(y1))))))) -> c4(F(c(s(s(s(s(y0)))), s(s(s(s(y1)))))))
   G(c(s(s(s(s(y0)))), s(s(s(z1))))) -> c4(F(c(s(s(s(y0))), s(s(z1)))))
   G(c(s(s(z0)), s(s(s(s(y1)))))) -> c4(F(c(s(z0), s(s(s(y1))))))
Defined Rule Symbols:none

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c4_1, c1_1, c2_1, c3_1


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

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

(70)
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(c(s(x), y)) -> f(c(x, s(y))) [1]
   f(c(s(x), s(y))) -> g(c(x, y)) [1]
   g(c(x, s(y))) -> g(c(s(x), y)) [1]
   g(c(s(x), s(y))) -> f(c(x, y)) [1]

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

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

(72)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(c(s(x), y)) -> f(c(x, s(y))) [1]
   f(c(s(x), s(y))) -> g(c(x, y)) [1]
   g(c(x, s(y))) -> g(c(s(x), y)) [1]
   g(c(s(x), s(y))) -> f(c(x, y)) [1]

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

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

(73) 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
   g_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2

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

(74)
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(c(s(x), y)) -> f(c(x, s(y))) [1]
   f(c(s(x), s(y))) -> g(c(x, y)) [1]
   g(c(x, s(y))) -> g(c(s(x), y)) [1]
   g(c(s(x), s(y))) -> f(c(x, y)) [1]

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

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

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

(76)
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(c(s(x), y)) -> f(c(x, s(y))) [1]
   f(c(s(x), s(y))) -> g(c(x, y)) [1]
   g(c(x, s(y))) -> g(c(s(x), y)) [1]
   g(c(s(x), s(y))) -> f(c(x, y)) [1]

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

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

(77) 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
   const2 => 0

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

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

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)


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

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

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

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)


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

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

   { f, g }

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

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

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)

Function symbols to be analyzed: {f,g}

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

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)

Function symbols to be analyzed: {f,g}

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

(85) 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

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

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

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

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)

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

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

(87) 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: ?

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> g(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)
   f(z) -{ 1 }-> f(1 + x + (1 + y)) :|: z = 1 + (1 + x) + y, x >= 0, y >= 0
   g(z) -{ 1 }-> g(1 + (1 + x) + y) :|: z = 1 + x + (1 + y), x >= 0, y >= 0
   g(z) -{ 1 }-> f(1 + x + y) :|: x >= 0, y >= 0, z = 1 + (1 + x) + (1 + y)

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

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

(89) 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]
   g(v0) -> null_g [0]

And the following fresh constants:    null_f, null_g, const, const1

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

(90)
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(c(s(x), y)) -> f(c(x, s(y))) [1]
   f(c(s(x), s(y))) -> g(c(x, y)) [1]
   g(c(x, s(y))) -> g(c(s(x), y)) [1]
   g(c(s(x), s(y))) -> f(c(x, y)) [1]
   f(v0) -> null_f [0]
   g(v0) -> null_g [0]

The TRS has the following type information:
f :: c -> null_f:null_g
c :: s -> s -> c
s :: s -> s
g :: c -> null_f:null_g
null_f :: null_f:null_g
null_g :: null_f:null_g
const :: c
const1 :: s

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

(91) 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_g => 0
   const => 0
   const1 => 0

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.