/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 247 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 1661 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   if(true, x, y) -> true
   if(false, x, y) -> false
   pow(x, y) -> f(x, y, s(0))
   f(x, 0, z) -> z
   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))

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

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

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


The TRS R consists of the following rules:

   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(*'(x, y), x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   odd(0') -> false
   odd(s(0')) -> true
   odd(s(s(x))) -> odd(x)
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   if(true, x, y) -> true
   if(false, x, y) -> false
   pow(x, y) -> f(x, y, s(0'))
   f(x, 0', z) -> z
   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(x, z)), f(*'(x, x), half(s(y)), z))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
*'/0
+'/1

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

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


The TRS R consists of the following rules:

   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   *'(0') -> 0'
   *'(s(y)) -> +'(*'(y))
   if(true, x, y) -> x
   if(false, x, y) -> y
   odd(0') -> false
   odd(s(0')) -> true
   odd(s(s(x))) -> odd(x)
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   if(true, x, y) -> true
   if(false, x, y) -> false
   pow(x, y) -> f(x, y, s(0'))
   f(x, 0', z) -> z
   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

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

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

(6)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-, *', odd, half, f

They will be analysed ascendingly in the following order:
*' < f
odd < f
half < f

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

(8)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
-, *', odd, half, f

They will be analysed ascendingly in the following order:
*' < f
odd < f
half < f

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) -> gen_0':s:+':true:false2_0(0), rt in Omega(1 + n4_0)

Induction Base:
-(gen_0':s:+':true:false2_0(0), gen_0':s:+':true:false2_0(0)) ->_R^Omega(1)
gen_0':s:+':true:false2_0(0)

Induction Step:
-(gen_0':s:+':true:false2_0(+(n4_0, 1)), gen_0':s:+':true:false2_0(+(n4_0, 1))) ->_R^Omega(1)
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) ->_IH
gen_0':s:+':true:false2_0(0)

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
-, *', odd, half, f

They will be analysed ascendingly in the following order:
*' < f
odd < f
half < f

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Lemmas:
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) -> gen_0':s:+':true:false2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
*', odd, half, f

They will be analysed ascendingly in the following order:
*' < f
odd < f
half < f

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s:+':true:false2_0(+(1, n304_0))) -> *3_0, rt in Omega(n304_0)

Induction Base:
*'(gen_0':s:+':true:false2_0(+(1, 0)))

Induction Step:
*'(gen_0':s:+':true:false2_0(+(1, +(n304_0, 1)))) ->_R^Omega(1)
+'(*'(gen_0':s:+':true:false2_0(+(1, n304_0)))) ->_IH
+'(*3_0)

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

(16)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Lemmas:
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) -> gen_0':s:+':true:false2_0(0), rt in Omega(1 + n4_0)
*'(gen_0':s:+':true:false2_0(+(1, n304_0))) -> *3_0, rt in Omega(n304_0)


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
odd, half, f

They will be analysed ascendingly in the following order:
odd < f
half < f

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
odd(gen_0':s:+':true:false2_0(*(2, n1161_0))) -> false, rt in Omega(1 + n1161_0)

Induction Base:
odd(gen_0':s:+':true:false2_0(*(2, 0))) ->_R^Omega(1)
false

Induction Step:
odd(gen_0':s:+':true:false2_0(*(2, +(n1161_0, 1)))) ->_R^Omega(1)
odd(gen_0':s:+':true:false2_0(*(2, n1161_0))) ->_IH
false

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

(18)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Lemmas:
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) -> gen_0':s:+':true:false2_0(0), rt in Omega(1 + n4_0)
*'(gen_0':s:+':true:false2_0(+(1, n304_0))) -> *3_0, rt in Omega(n304_0)
odd(gen_0':s:+':true:false2_0(*(2, n1161_0))) -> false, rt in Omega(1 + n1161_0)


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
half, f

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s:+':true:false2_0(*(2, n1363_0))) -> gen_0':s:+':true:false2_0(n1363_0), rt in Omega(1 + n1363_0)

Induction Base:
half(gen_0':s:+':true:false2_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s:+':true:false2_0(*(2, +(n1363_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s:+':true:false2_0(*(2, n1363_0)))) ->_IH
s(gen_0':s:+':true:false2_0(c1364_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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
*'(0') -> 0'
*'(s(y)) -> +'(*'(y))
if(true, x, y) -> x
if(false, x, y) -> y
odd(0') -> false
odd(s(0')) -> true
odd(s(s(x))) -> odd(x)
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
if(true, x, y) -> true
if(false, x, y) -> false
pow(x, y) -> f(x, y, s(0'))
f(x, 0', z) -> z
f(x, s(y), z) -> if(odd(s(y)), f(x, y, *'(z)), f(*'(x), half(s(y)), z))

Types:
- :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
0' :: 0':s:+':true:false
s :: 0':s:+':true:false -> 0':s:+':true:false
*' :: 0':s:+':true:false -> 0':s:+':true:false
+' :: 0':s:+':true:false -> 0':s:+':true:false
if :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
true :: 0':s:+':true:false
false :: 0':s:+':true:false
odd :: 0':s:+':true:false -> 0':s:+':true:false
half :: 0':s:+':true:false -> 0':s:+':true:false
pow :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
f :: 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false -> 0':s:+':true:false
hole_0':s:+':true:false1_0 :: 0':s:+':true:false
gen_0':s:+':true:false2_0 :: Nat -> 0':s:+':true:false


Lemmas:
-(gen_0':s:+':true:false2_0(n4_0), gen_0':s:+':true:false2_0(n4_0)) -> gen_0':s:+':true:false2_0(0), rt in Omega(1 + n4_0)
*'(gen_0':s:+':true:false2_0(+(1, n304_0))) -> *3_0, rt in Omega(n304_0)
odd(gen_0':s:+':true:false2_0(*(2, n1161_0))) -> false, rt in Omega(1 + n1161_0)
half(gen_0':s:+':true:false2_0(*(2, n1363_0))) -> gen_0':s:+':true:false2_0(n1363_0), rt in Omega(1 + n1363_0)


Generator Equations:
gen_0':s:+':true:false2_0(0) <=> 0'
gen_0':s:+':true:false2_0(+(x, 1)) <=> s(gen_0':s:+':true:false2_0(x))


The following defined symbols remain to be analysed:
f