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


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


WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 47 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 45 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), 284 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), 80 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 120 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   sqr(0) -> 0
   sqr(s(x)) -> +(sqr(x), s(double(x)))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   sqr(s(x)) -> s(+(sqr(x), double(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:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sqr_1, double_1, +_2

Defined Pair Symbols:   SQR_1, DOUBLE_1, +'_2

Compound Symbols:   c, c1_3, c2_3, c3, c4_1, c5, c6_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   +'(z0, 0) -> c5
   SQR(0) -> c
   DOUBLE(0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sqr_1, double_1, +_2

Defined Pair Symbols:   SQR_1, DOUBLE_1, +'_2

Compound Symbols:   c1_3, c2_3, c4_1, c6_1


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

(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [3]
   POL(+'(x_1, x_2)) = 0
   POL(0) = [3]
   POL(DOUBLE(x_1)) = [3]
   POL(SQR(x_1)) = [2]x_1
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(double(x_1)) = 0
   POL(s(x_1)) = [2] + x_1
   POL(sqr(x_1)) = [3]

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
Defined Rule Symbols:   sqr_1, double_1, +_2

Defined Pair Symbols:   SQR_1, DOUBLE_1, +'_2

Compound Symbols:   c1_3, c2_3, c4_1, c6_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [2]
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(DOUBLE(x_1)) = x_1
   POL(SQR(x_1)) = x_1^2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(double(x_1)) = 0
   POL(s(x_1)) = [1] + x_1
   POL(sqr(x_1)) = 0

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
Defined Rule Symbols:   sqr_1, double_1, +_2

Defined Pair Symbols:   SQR_1, DOUBLE_1, +'_2

Compound Symbols:   c1_3, c2_3, c4_1, c6_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(z0, s(z1)) -> c6(+'(z0, z1))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
And the Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [2] + [2]x_2
   POL(+'(x_1, x_2)) = x_2
   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(SQR(x_1)) = x_1^2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(double(x_1)) = [2]x_1
   POL(s(x_1)) = [1] + x_1
   POL(sqr(x_1)) = [2] + [2]x_1^2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:none
K tuples:
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
Defined Rule Symbols:   sqr_1, double_1, +_2

Defined Pair Symbols:   SQR_1, DOUBLE_1, +'_2

Compound Symbols:   c1_3, c2_3, c4_1, c6_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^2, INF).


The TRS R consists of the following rules:

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

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

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

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

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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:
sqr, +', double

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

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

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

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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:
+', sqr, double

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

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

(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:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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:
+', sqr, double

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

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

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

(23)
BOUNDS(n^1, INF)

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

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

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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, sqr

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_0)

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

Induction Step:
double(gen_0':s2_0(+(n513_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s2_0(n513_0)))) ->_IH
s(s(gen_0':s2_0(*(2, c514_0))))

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

(26)
Obligation:
Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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)
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_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:
sqr
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s2_0(n777_0)) -> gen_0':s2_0(*(n777_0, n777_0)), rt in Omega(1 + n777_0 + n777_0^2)

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

Induction Step:
sqr(gen_0':s2_0(+(n777_0, 1))) ->_R^Omega(1)
+'(sqr(gen_0':s2_0(n777_0)), s(double(gen_0':s2_0(n777_0)))) ->_IH
+'(gen_0':s2_0(*(c778_0, c778_0)), s(double(gen_0':s2_0(n777_0)))) ->_L^Omega(1 + n777_0)
+'(gen_0':s2_0(*(n777_0, n777_0)), s(gen_0':s2_0(*(2, n777_0)))) ->_L^Omega(2 + 2*n777_0)
gen_0':s2_0(+(+(*(2, n777_0), 1), *(n777_0, n777_0)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(28)
Obligation:
Proved the lower bound n^2 for the following obligation:

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

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 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)
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_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:
sqr
----------------------------------------

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

(30)
BOUNDS(n^2, INF)