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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 8 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 30 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 13 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxTRS
    (19) SlicingProof [LOWER BOUND(ID), 0 ms]
    (20) CpxTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 658 ms]
    (26) proven lower bound
    (27) LowerBoundPropagationProof [FINISHED, 0 ms]
    (28) BOUNDS(n^1, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   *(x, *(y, z)) -> *(otimes(x, y), z)
   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
*(x, *(y, z)) -> *(otimes(x, y), z)

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

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


The TRS R consists of the following rules:

   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

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

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(1, z0) -> z0
   *(+(z0, z1), z2) -> oplus(*(z0, z2), *(z1, z2))
   *(z0, oplus(z1, z2)) -> oplus(*(z0, z1), *(z0, z2))
Tuples:
   *'(1, z0) -> c
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(1, z0) -> c
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c, c1_2, c2_2


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   *'(1, z0) -> c

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(1, z0) -> z0
   *(+(z0, z1), z2) -> oplus(*(z0, z2), *(z1, z2))
   *(z0, oplus(z1, z2)) -> oplus(*(z0, z1), *(z0, z2))
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(1, z0) -> z0
   *(+(z0, z1), z2) -> oplus(*(z0, z2), *(z1, z2))
   *(z0, oplus(z1, z2)) -> oplus(*(z0, z1), *(z0, z2))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(oplus(x_1, x_2)) = [2] + x_1 + x_2

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = [1] + [2]x_1 + x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(oplus(x_1, x_2)) = [1] + x_1 + x_2

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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

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

(16)
BOUNDS(1, 1)

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

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

(18)
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, *'(y, z)) -> *'(otimes(x, y), z)
   *'(1', y) -> y
   *'(+'(x, y), z) -> oplus(*'(x, z), *'(y, z))
   *'(x, oplus(y, z)) -> oplus(*'(x, y), *'(x, z))

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

(19) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
otimes/0
otimes/1

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

(20)
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, *'(y, z)) -> *'(otimes, z)
   *'(1', y) -> y
   *'(+'(x, y), z) -> oplus(*'(x, z), *'(y, z))
   *'(x, oplus(y, z)) -> oplus(*'(x, y), *'(x, z))

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

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

(22)
Obligation:
TRS:
Rules:
*'(x, *'(y, z)) -> *'(otimes, z)
*'(1', y) -> y
*'(+'(x, y), z) -> oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) -> oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' -> oplus -> oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' -> otimes:1':+' -> otimes:1':+'
oplus :: oplus -> oplus -> oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat -> oplus
gen_otimes:1':+'4_0 :: Nat -> otimes:1':+'

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
*'
----------------------------------------

(24)
Obligation:
TRS:
Rules:
*'(x, *'(y, z)) -> *'(otimes, z)
*'(1', y) -> y
*'(+'(x, y), z) -> oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) -> oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' -> oplus -> oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' -> otimes:1':+' -> otimes:1':+'
oplus :: oplus -> oplus -> oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat -> oplus
gen_otimes:1':+'4_0 :: Nat -> otimes:1':+'


Generator Equations:
gen_oplus3_0(0) <=> hole_oplus1_0
gen_oplus3_0(+(x, 1)) <=> oplus(hole_oplus1_0, gen_oplus3_0(x))
gen_otimes:1':+'4_0(0) <=> 1'
gen_otimes:1':+'4_0(+(x, 1)) <=> +'(1', gen_otimes:1':+'4_0(x))


The following defined symbols remain to be analysed:
*'
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) -> *5_0, rt in Omega(n6_0)

Induction Base:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(0))

Induction Step:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(+(n6_0, 1))) ->_R^Omega(1)
oplus(*'(gen_otimes:1':+'4_0(0), hole_oplus1_0), *'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0))) ->_R^Omega(1)
oplus(hole_oplus1_0, *'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0))) ->_IH
oplus(hole_oplus1_0, *5_0)

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

(26)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
*'(x, *'(y, z)) -> *'(otimes, z)
*'(1', y) -> y
*'(+'(x, y), z) -> oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) -> oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' -> oplus -> oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' -> otimes:1':+' -> otimes:1':+'
oplus :: oplus -> oplus -> oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat -> oplus
gen_otimes:1':+'4_0 :: Nat -> otimes:1':+'


Generator Equations:
gen_oplus3_0(0) <=> hole_oplus1_0
gen_oplus3_0(+(x, 1)) <=> oplus(hole_oplus1_0, gen_oplus3_0(x))
gen_otimes:1':+'4_0(0) <=> 1'
gen_otimes:1':+'4_0(+(x, 1)) <=> +'(1', gen_otimes:1':+'4_0(x))


The following defined symbols remain to be analysed:
*'
----------------------------------------

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

(28)
BOUNDS(n^1, INF)