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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 252 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 101 ms]
            (26) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   double(z0) -> +(z0, z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   +(s(z0), z1) -> s(+(z0, z1))
Tuples:
   DOUBLE(0) -> c
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, 0) -> c3
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:
   DOUBLE(0) -> c
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, 0) -> c3
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   double_1, +_2

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c, c1_1, c2_1, c3, c4_1, c5_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   DOUBLE(0) -> c
   +'(z0, 0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   double(z0) -> +(z0, z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   +(s(z0), z1) -> s(+(z0, z1))
Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   double_1, +_2

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   double(z0) -> +(z0, z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   +(s(z0), z1) -> s(+(z0, z1))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(s(z0), z1) -> c5(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+'(x_1, x_2)) = x_1
   POL(DOUBLE(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c4(+'(z0, z1))
K tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(s(z0), z1) -> c5(+'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(z0, s(z1)) -> c4(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+'(x_1, x_2)) = [1] + x_2
   POL(DOUBLE(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:none
K tuples:
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(s(z0), z1) -> c5(+'(z0, z1))
   +'(z0, s(z1)) -> c4(+'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1


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

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)

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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS 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)))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   +'(s(x), y) -> s(+'(x, y))
   double(x) -> +'(x, x)

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

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

(16)
Obligation:
Innermost TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(s(x), y) -> s(+'(x, y))
double(x) -> +'(x, x)

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
double, +'

They will be analysed ascendingly in the following order:
+' < double

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

(18)
Obligation:
Innermost TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(s(x), y) -> s(+'(x, y))
double(x) -> +'(x, x)

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
+', double

They will be analysed ascendingly in the following order:
+' < double

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_0(+(a, c5_0)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(s(x), y) -> s(+'(x, y))
double(x) -> +'(x, x)

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
+', double

They will be analysed ascendingly in the following order:
+' < double

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(s(x), y) -> s(+'(x, y))
double(x) -> +'(x, x)

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s2_0(n544_0)) -> gen_0':s2_0(*(2, n544_0)), rt in Omega(1 + n544_0)

Induction Base:
double(gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s2_0(+(n544_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s2_0(n544_0)))) ->_IH
s(s(gen_0':s2_0(*(2, c545_0))))

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

(26)
BOUNDS(1, INF)