/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 208 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 25 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 24 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
        (22) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_double(x_1) -> double(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_double(x_1) -> double(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_double(x_1) -> double(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   if(0', y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_double(x_1) -> double(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

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

(8)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
double, half, -, encArg

They will be analysed ascendingly in the following order:
double < encArg
half < encArg
- < encArg

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

(10)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if


Generator Equations:
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0) <=> 0'
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(x, 1)) <=> s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(x))


The following defined symbols remain to be analysed:
double, half, -, encArg

They will be analysed ascendingly in the following order:
double < encArg
half < encArg
- < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n4_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n4_4)), rt in Omega(1 + n4_4)

Induction Base:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(n4_4, 1))) ->_R^Omega(1)
s(s(double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n4_4)))) ->_IH
s(s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, c5_4))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if


Generator Equations:
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0) <=> 0'
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(x, 1)) <=> s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(x))


The following defined symbols remain to be analysed:
double, half, -, encArg

They will be analysed ascendingly in the following order:
double < encArg
half < encArg
- < encArg

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

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if


Lemmas:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n4_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n4_4)), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0) <=> 0'
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(x, 1)) <=> s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(x))


The following defined symbols remain to be analysed:
half, -, encArg

They will be analysed ascendingly in the following order:
half < encArg
- < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n296_4))) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n296_4), rt in Omega(1 + n296_4)

Induction Base:
half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, +(n296_4, 1)))) ->_R^Omega(1)
s(half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n296_4)))) ->_IH
s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(c297_4))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if


Lemmas:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n4_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n4_4)), rt in Omega(1 + n4_4)
half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n296_4))) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n296_4), rt in Omega(1 + n296_4)


Generator Equations:
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0) <=> 0'
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(x, 1)) <=> s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(x))


The following defined symbols remain to be analysed:
-, encArg

They will be analysed ascendingly in the following order:
- < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4), gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0), rt in Omega(1 + n771_4)

Induction Base:
-(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0), gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0)) ->_R^Omega(1)
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0)

Induction Step:
-(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(n771_4, 1)), gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(n771_4, 1))) ->_R^Omega(1)
-(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4), gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4)) ->_IH
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_double(x_1)) -> double(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_double(x_1) -> double(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
0' :: 0':s:cons_double:cons_half:cons_-:cons_if
s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encArg :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
cons_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_double :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_0 :: 0':s:cons_double:cons_half:cons_-:cons_if
encode_s :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_half :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_- :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
encode_if :: 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if -> 0':s:cons_double:cons_half:cons_-:cons_if
hole_0':s:cons_double:cons_half:cons_-:cons_if1_4 :: 0':s:cons_double:cons_half:cons_-:cons_if
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4 :: Nat -> 0':s:cons_double:cons_half:cons_-:cons_if


Lemmas:
double(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n4_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n4_4)), rt in Omega(1 + n4_4)
half(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(*(2, n296_4))) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n296_4), rt in Omega(1 + n296_4)
-(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4), gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n771_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0), rt in Omega(1 + n771_4)


Generator Equations:
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0) <=> 0'
gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(x, 1)) <=> s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n1127_4)) -> gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n1127_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(+(n1127_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(n1127_4))) ->_IH
s(gen_0':s:cons_double:cons_half:cons_-:cons_if2_4(c1128_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
BOUNDS(1, INF)