WORST_CASE(NON_POLY,?)
proof of input_CM5OQQ8IU1.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 12 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 1730 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [FINISHED, 274 ms]
        (18) BOUNDS(EXP, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   p(s(x)) -> x
   fact(0) -> s(0)
   fact(s(x)) -> *(s(x), fact(p(s(x))))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   P(s(z0)) -> c
   FACT(0) -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0, z0) -> c3
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   P(s(z0)) -> c
   FACT(0) -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0, z0) -> c3
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   P_1, FACT_1, *'_2, +'_2

Compound Symbols:   c, c1, c2_3, c3, c4_2, c5, c6_1


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

(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   P(s(z0)) -> c
   FACT(0) -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0, z0) -> c3
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))

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

   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   P(s(z0)) -> c
   FACT(0') -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0', z0) -> c3
   *'(s(z0), z1) -> c4(+'(*'(z0, z1), z1), *'(z0, z1))
   +'(z0, 0') -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))

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

   p(s(z0)) -> z0
   fact(0') -> s(0')
   fact(s(z0)) -> *'(s(z0), fact(p(s(z0))))
   *'(0', z0) -> 0'
   *'(s(z0), z1) -> +'(*'(z0, z1), z1)
   +'(z0, 0') -> z0
   +'(z0, s(z1)) -> s(+'(z0, z1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FACT(0') -> c1
FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
*'(0', z0) -> c3
*'(s(z0), z1) -> c4(+'(*'(z0, z1), z1), *'(z0, z1))
+'(z0, 0') -> c5
+'(z0, s(z1)) -> c6(+'(z0, z1))
p(s(z0)) -> z0
fact(0') -> s(0')
fact(s(z0)) -> *'(s(z0), fact(p(s(z0))))
*'(0', z0) -> 0'
*'(s(z0), z1) -> +'(*'(z0, z1), z1)
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
P :: s:0':c3:c4:c5:c6 -> c
s :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c :: c
FACT :: s:0':c3:c4:c5:c6 -> c1:c2
0' :: s:0':c3:c4:c5:c6
c1 :: c1:c2
c2 :: s:0':c3:c4:c5:c6 -> c1:c2 -> c -> c1:c2
*' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
fact :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
p :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c3 :: s:0':c3:c4:c5:c6
c4 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
+' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c5 :: s:0':c3:c4:c5:c6
c6 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
hole_c1_7 :: c
hole_s:0':c3:c4:c5:c62_7 :: s:0':c3:c4:c5:c6
hole_c1:c23_7 :: c1:c2
gen_s:0':c3:c4:c5:c64_7 :: Nat -> s:0':c3:c4:c5:c6
gen_c1:c25_7 :: Nat -> c1:c2

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FACT, *', fact, +'

They will be analysed ascendingly in the following order:
*' < FACT
fact < FACT
*' < fact
+' < *'

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

(10)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FACT(0') -> c1
FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
*'(0', z0) -> c3
*'(s(z0), z1) -> c4(+'(*'(z0, z1), z1), *'(z0, z1))
+'(z0, 0') -> c5
+'(z0, s(z1)) -> c6(+'(z0, z1))
p(s(z0)) -> z0
fact(0') -> s(0')
fact(s(z0)) -> *'(s(z0), fact(p(s(z0))))
*'(0', z0) -> 0'
*'(s(z0), z1) -> +'(*'(z0, z1), z1)
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
P :: s:0':c3:c4:c5:c6 -> c
s :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c :: c
FACT :: s:0':c3:c4:c5:c6 -> c1:c2
0' :: s:0':c3:c4:c5:c6
c1 :: c1:c2
c2 :: s:0':c3:c4:c5:c6 -> c1:c2 -> c -> c1:c2
*' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
fact :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
p :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c3 :: s:0':c3:c4:c5:c6
c4 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
+' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c5 :: s:0':c3:c4:c5:c6
c6 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
hole_c1_7 :: c
hole_s:0':c3:c4:c5:c62_7 :: s:0':c3:c4:c5:c6
hole_c1:c23_7 :: c1:c2
gen_s:0':c3:c4:c5:c64_7 :: Nat -> s:0':c3:c4:c5:c6
gen_c1:c25_7 :: Nat -> c1:c2


Generator Equations:
gen_s:0':c3:c4:c5:c64_7(0) <=> 0'
gen_s:0':c3:c4:c5:c64_7(+(x, 1)) <=> s(gen_s:0':c3:c4:c5:c64_7(x))
gen_c1:c25_7(0) <=> c1
gen_c1:c25_7(+(x, 1)) <=> c2(0', gen_c1:c25_7(x), c)


The following defined symbols remain to be analysed:
+', FACT, *', fact

They will be analysed ascendingly in the following order:
*' < FACT
fact < FACT
*' < fact
+' < *'

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_s:0':c3:c4:c5:c64_7(a), gen_s:0':c3:c4:c5:c64_7(+(1, n7_7))) -> *6_7, rt in Omega(n7_7)

Induction Base:
+'(gen_s:0':c3:c4:c5:c64_7(a), gen_s:0':c3:c4:c5:c64_7(+(1, 0)))

Induction Step:
+'(gen_s:0':c3:c4:c5:c64_7(a), gen_s:0':c3:c4:c5:c64_7(+(1, +(n7_7, 1)))) ->_R^Omega(1)
c6(+'(gen_s:0':c3:c4:c5:c64_7(a), gen_s:0':c3:c4:c5:c64_7(+(1, n7_7)))) ->_IH
c6(*6_7)

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:
P(s(z0)) -> c
FACT(0') -> c1
FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
*'(0', z0) -> c3
*'(s(z0), z1) -> c4(+'(*'(z0, z1), z1), *'(z0, z1))
+'(z0, 0') -> c5
+'(z0, s(z1)) -> c6(+'(z0, z1))
p(s(z0)) -> z0
fact(0') -> s(0')
fact(s(z0)) -> *'(s(z0), fact(p(s(z0))))
*'(0', z0) -> 0'
*'(s(z0), z1) -> +'(*'(z0, z1), z1)
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
P :: s:0':c3:c4:c5:c6 -> c
s :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c :: c
FACT :: s:0':c3:c4:c5:c6 -> c1:c2
0' :: s:0':c3:c4:c5:c6
c1 :: c1:c2
c2 :: s:0':c3:c4:c5:c6 -> c1:c2 -> c -> c1:c2
*' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
fact :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
p :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c3 :: s:0':c3:c4:c5:c6
c4 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
+' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c5 :: s:0':c3:c4:c5:c6
c6 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
hole_c1_7 :: c
hole_s:0':c3:c4:c5:c62_7 :: s:0':c3:c4:c5:c6
hole_c1:c23_7 :: c1:c2
gen_s:0':c3:c4:c5:c64_7 :: Nat -> s:0':c3:c4:c5:c6
gen_c1:c25_7 :: Nat -> c1:c2


Generator Equations:
gen_s:0':c3:c4:c5:c64_7(0) <=> 0'
gen_s:0':c3:c4:c5:c64_7(+(x, 1)) <=> s(gen_s:0':c3:c4:c5:c64_7(x))
gen_c1:c25_7(0) <=> c1
gen_c1:c25_7(+(x, 1)) <=> c2(0', gen_c1:c25_7(x), c)


The following defined symbols remain to be analysed:
+', FACT, *', fact

They will be analysed ascendingly in the following order:
*' < FACT
fact < FACT
*' < fact
+' < *'

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FACT(0') -> c1
FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
*'(0', z0) -> c3
*'(s(z0), z1) -> c4(+'(*'(z0, z1), z1), *'(z0, z1))
+'(z0, 0') -> c5
+'(z0, s(z1)) -> c6(+'(z0, z1))
p(s(z0)) -> z0
fact(0') -> s(0')
fact(s(z0)) -> *'(s(z0), fact(p(s(z0))))
*'(0', z0) -> 0'
*'(s(z0), z1) -> +'(*'(z0, z1), z1)
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
P :: s:0':c3:c4:c5:c6 -> c
s :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c :: c
FACT :: s:0':c3:c4:c5:c6 -> c1:c2
0' :: s:0':c3:c4:c5:c6
c1 :: c1:c2
c2 :: s:0':c3:c4:c5:c6 -> c1:c2 -> c -> c1:c2
*' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
fact :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
p :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c3 :: s:0':c3:c4:c5:c6
c4 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
+' :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
c5 :: s:0':c3:c4:c5:c6
c6 :: s:0':c3:c4:c5:c6 -> s:0':c3:c4:c5:c6
hole_c1_7 :: c
hole_s:0':c3:c4:c5:c62_7 :: s:0':c3:c4:c5:c6
hole_c1:c23_7 :: c1:c2
gen_s:0':c3:c4:c5:c64_7 :: Nat -> s:0':c3:c4:c5:c6
gen_c1:c25_7 :: Nat -> c1:c2


Lemmas:
+'(gen_s:0':c3:c4:c5:c64_7(a), gen_s:0':c3:c4:c5:c64_7(+(1, n7_7))) -> *6_7, rt in Omega(n7_7)


Generator Equations:
gen_s:0':c3:c4:c5:c64_7(0) <=> 0'
gen_s:0':c3:c4:c5:c64_7(+(x, 1)) <=> s(gen_s:0':c3:c4:c5:c64_7(x))
gen_c1:c25_7(0) <=> c1
gen_c1:c25_7(+(x, 1)) <=> c2(0', gen_c1:c25_7(x), c)


The following defined symbols remain to be analysed:
*', FACT, fact

They will be analysed ascendingly in the following order:
*' < FACT
fact < FACT
*' < fact

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

(17) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
*'(gen_s:0':c3:c4:c5:c64_7(+(1, n1865_7)), gen_s:0':c3:c4:c5:c64_7(b)) -> *6_7, rt in Omega(EXP)

Induction Base:
*'(gen_s:0':c3:c4:c5:c64_7(+(1, 0)), gen_s:0':c3:c4:c5:c64_7(b))

Induction Step:
*'(gen_s:0':c3:c4:c5:c64_7(+(1, +(n1865_7, 1))), gen_s:0':c3:c4:c5:c64_7(b)) ->_R^Omega(1)
c4(+'(*'(gen_s:0':c3:c4:c5:c64_7(+(1, n1865_7)), gen_s:0':c3:c4:c5:c64_7(b)), gen_s:0':c3:c4:c5:c64_7(b)), *'(gen_s:0':c3:c4:c5:c64_7(+(1, n1865_7)), gen_s:0':c3:c4:c5:c64_7(b))) ->_IH
c4(+'(*6_7, gen_s:0':c3:c4:c5:c64_7(b)), *'(gen_s:0':c3:c4:c5:c64_7(+(1, n1865_7)), gen_s:0':c3:c4:c5:c64_7(b))) ->_IH
c4(+'(*6_7, gen_s:0':c3:c4:c5:c64_7(b)), *6_7)

We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
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(18)
BOUNDS(EXP, INF)