WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 4 ms]
(2) CpxRelTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 333 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 47 ms]
        (14) typed CpxTrs


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

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


The TRS R consists of the following rules:

   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))

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

   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))

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

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

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


The TRS R consists of the following rules:

   P(0') -> c
   P(s(z0)) -> c1
   LEQ(0', z0) -> c2
   LEQ(s(z0), 0') -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))

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

   p(0') -> 0'
   p(s(z0)) -> z0
   leq(0', z0) -> true
   leq(s(z0), 0') -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

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

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

(4)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

(6)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)

Induction Base:
LEQ(gen_0':s7_9(0), gen_0':s7_9(0)) ->_R^Omega(1)
c2

Induction Step:
LEQ(gen_0':s7_9(+(n11_9, 1)), gen_0':s7_9(+(n11_9, 1))) ->_R^Omega(1)
c4(LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9))) ->_IH
c4(gen_c2:c3:c48_9(c12_9))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Lemmas:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
leq, DIFF, diff

They will be analysed ascendingly in the following order:
leq < DIFF
diff < DIFF
leq < diff

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) -> true, rt in Omega(0)

Induction Base:
leq(gen_0':s7_9(0), gen_0':s7_9(0)) ->_R^Omega(0)
true

Induction Step:
leq(gen_0':s7_9(+(n590_9, 1)), gen_0':s7_9(+(n590_9, 1))) ->_R^Omega(0)
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) ->_IH
true

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

(14)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Lemmas:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
diff, DIFF

They will be analysed ascendingly in the following order:
diff < DIFF