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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxRelTRS
(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 286 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) SlicingProof [LOWER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 6 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 11.7 s]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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(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:

   rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n))
   rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

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

   rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y))
   rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y))
   rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST
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(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved termination of relative rules
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(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:

   rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n))
   rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

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

   rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y))
   rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y))
   rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST
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(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(4)
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:

   rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n))
   rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

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

   rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y))
   rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y))
   rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST
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(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Val/0
rewrite[Let][Let]/1
rewrite[Let][Let][Let]/0

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(6)
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:

   rewrite(Op(Val, y)) -> Op(rewrite(y), Val)
   rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
   rewrite(Val) -> Val
   second(Op(x, y)) -> y
   isOp(Val) -> False
   isOp(Op(x, y)) -> True
   first(Val) -> Val
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

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

   rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, a1, rewrite(y))
   rewrite[Let][Let](Op(x, y), a1, b1) -> rewrite[Let][Let][Let](a1, b1, rewrite(y))
   rewrite[Let][Let][Let](a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
rewrite(Op(Val, y)) -> Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
rewrite(Val) -> Val
second(Op(x, y)) -> y
isOp(Val) -> False
isOp(Op(x, y)) -> True
first(Val) -> Val
first(Op(x, y)) -> x
assrewrite(exp) -> rewrite(exp)
rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) -> rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op -> Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
rewrite[Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
second :: Val:Op -> Val:Op
isOp :: Val:Op -> False:True
False :: False:True
True :: False:True
first :: Val:Op -> Val:Op
assrewrite :: Val:Op -> Val:Op
rewrite[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rewrite[Let][Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat -> Val:Op

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rewrite
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(10)
Obligation:
Innermost TRS:
Rules:
rewrite(Op(Val, y)) -> Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
rewrite(Val) -> Val
second(Op(x, y)) -> y
isOp(Val) -> False
isOp(Op(x, y)) -> True
first(Val) -> Val
first(Op(x, y)) -> x
assrewrite(exp) -> rewrite(exp)
rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) -> rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op -> Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
rewrite[Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
second :: Val:Op -> Val:Op
isOp :: Val:Op -> False:True
False :: False:True
True :: False:True
first :: Val:Op -> Val:Op
assrewrite :: Val:Op -> Val:Op
rewrite[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rewrite[Let][Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat -> Val:Op


Generator Equations:
gen_Val:Op3_0(0) <=> Val
gen_Val:Op3_0(+(x, 1)) <=> Op(Val, gen_Val:Op3_0(x))


The following defined symbols remain to be analysed:
rewrite
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rewrite(gen_Val:Op3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
rewrite(gen_Val:Op3_0(+(1, 0)))

Induction Step:
rewrite(gen_Val:Op3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
Op(rewrite(gen_Val:Op3_0(+(1, n5_0))), Val) ->_IH
Op(*4_0, Val)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
rewrite(Op(Val, y)) -> Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x))
rewrite(Val) -> Val
second(Op(x, y)) -> y
isOp(Val) -> False
isOp(Op(x, y)) -> True
first(Val) -> Val
first(Op(x, y)) -> x
assrewrite(exp) -> rewrite(exp)
rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) -> rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op -> Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
rewrite[Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
second :: Val:Op -> Val:Op
isOp :: Val:Op -> False:True
False :: False:True
True :: False:True
first :: Val:Op -> Val:Op
assrewrite :: Val:Op -> Val:Op
rewrite[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rewrite[Let][Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat -> Val:Op


Generator Equations:
gen_Val:Op3_0(0) <=> Val
gen_Val:Op3_0(+(x, 1)) <=> Op(Val, gen_Val:Op3_0(x))


The following defined symbols remain to be analysed:
rewrite
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)