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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 354 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 1 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), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 229 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) 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:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(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:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

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

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

(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:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

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

(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Val/0
rw[Let][Let]/0
rw[Let][Let][Let]/0

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

(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:

   rw(Val, c) -> Op(Val, rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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:

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

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

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

(8)
Obligation:
Innermost TRS:
Rules:
rw(Val, c) -> Op(Val, rewrite(c))
rewrite(Op(x, y)) -> rw(x, y)
rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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)
rw[Let](Op(x, y), c, a1) -> rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) -> rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) -> rw(a1, Op(b1, c1))

Types:
rw :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
rewrite :: Val:Op -> Val:Op
rw[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
rw[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rw[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

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rw, rewrite

They will be analysed ascendingly in the following order:
rw = rewrite

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

(10)
Obligation:
Innermost TRS:
Rules:
rw(Val, c) -> Op(Val, rewrite(c))
rewrite(Op(x, y)) -> rw(x, y)
rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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)
rw[Let](Op(x, y), c, a1) -> rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) -> rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) -> rw(a1, Op(b1, c1))

Types:
rw :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
rewrite :: Val:Op -> Val:Op
rw[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
rw[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rw[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, rw

They will be analysed ascendingly in the following order:
rw = rewrite

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rewrite(gen_Val:Op3_0(n5_0)) -> gen_Val:Op3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
rewrite(gen_Val:Op3_0(0)) ->_R^Omega(1)
Val

Induction Step:
rewrite(gen_Val:Op3_0(+(n5_0, 1))) ->_R^Omega(1)
rw(Val, gen_Val:Op3_0(n5_0)) ->_R^Omega(1)
Op(Val, rewrite(gen_Val:Op3_0(n5_0))) ->_IH
Op(Val, gen_Val:Op3_0(c6_0))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
rw(Val, c) -> Op(Val, rewrite(c))
rewrite(Op(x, y)) -> rw(x, y)
rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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)
rw[Let](Op(x, y), c, a1) -> rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) -> rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) -> rw(a1, Op(b1, c1))

Types:
rw :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
rewrite :: Val:Op -> Val:Op
rw[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
rw[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rw[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, rw

They will be analysed ascendingly in the following order:
rw = rewrite

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
rw(Val, c) -> Op(Val, rewrite(c))
rewrite(Op(x, y)) -> rw(x, y)
rw(Op(x, y), c) -> rw[Let](Op(x, y), c, 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)
rw[Let](Op(x, y), c, a1) -> rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) -> rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) -> rw(a1, Op(b1, c1))

Types:
rw :: Val:Op -> Val:Op -> Val:Op
Val :: Val:Op
Op :: Val:Op -> Val:Op -> Val:Op
rewrite :: Val:Op -> Val:Op
rw[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
rw[Let][Let] :: Val:Op -> Val:Op -> Val:Op -> Val:Op
rw[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


Lemmas:
rewrite(gen_Val:Op3_0(n5_0)) -> gen_Val:Op3_0(n5_0), rt in Omega(1 + n5_0)


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:
rw

They will be analysed ascendingly in the following order:
rw = rewrite