WORST_CASE(Omega(n^1),O(n^1))
proof of input_P0odRfkyp3.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTRS
    (13) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (14) BOUNDS(1, n^1)
(15) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
    (17) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 16 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 288 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs


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

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


The TRS R consists of the following rules:

   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(x))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   p_1, fac_1

Defined Pair Symbols:   P_1, FAC_1

Compound Symbols:   c, c1, c2_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   P(s(z0)) -> c
   FAC(0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   p_1, fac_1

Defined Pair Symbols:   FAC_1

Compound Symbols:   c2_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))))
S tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, fac_1

Defined Pair Symbols:   FAC_1

Compound Symbols:   c2_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
Tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))))
S tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   FAC_1

Compound Symbols:   c2_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FAC(s(z0)) -> c2(FAC(p(s(z0))))

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

   p(s(z0)) -> z0

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

(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FAC(s(z0)) -> c2(FAC(p(s(z0))))
   p(s(z0)) -> z0

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

(13) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
s0(0) -> 0
c20(0) -> 0
FAC0(0) -> 1
p0(0) -> 2
s1(0) -> 5
p1(5) -> 4
FAC1(4) -> 3
c21(3) -> 1
c21(3) -> 3
0 -> 2
0 -> 4

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

(14)
BOUNDS(1, n^1)

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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   p_1, fac_1

Defined Pair Symbols:   P_1, FAC_1

Compound Symbols:   c, c1, c2_2


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

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

(18)
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:

   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))

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

   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

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

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

(20)
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:

   P(s(z0)) -> c
   FAC(0') -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))

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

   p(s(z0)) -> z0
   fac(0') -> s(0')
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

Rewrite Strategy: INNERMOST
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(21) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FAC(0') -> c1
FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
p(s(z0)) -> z0
fac(0') -> s(0')
fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

Types:
P :: s:0':times -> c
s :: s:0':times -> s:0':times
c :: c
FAC :: s:0':times -> c1:c2
0' :: s:0':times
c1 :: c1:c2
c2 :: c1:c2 -> c -> c1:c2
p :: s:0':times -> s:0':times
fac :: s:0':times -> s:0':times
times :: s:0':times -> s:0':times -> s:0':times
hole_c1_3 :: c
hole_s:0':times2_3 :: s:0':times
hole_c1:c23_3 :: c1:c2
gen_s:0':times4_3 :: Nat -> s:0':times
gen_c1:c25_3 :: Nat -> c1:c2

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(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FAC, fac
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FAC(0') -> c1
FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
p(s(z0)) -> z0
fac(0') -> s(0')
fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

Types:
P :: s:0':times -> c
s :: s:0':times -> s:0':times
c :: c
FAC :: s:0':times -> c1:c2
0' :: s:0':times
c1 :: c1:c2
c2 :: c1:c2 -> c -> c1:c2
p :: s:0':times -> s:0':times
fac :: s:0':times -> s:0':times
times :: s:0':times -> s:0':times -> s:0':times
hole_c1_3 :: c
hole_s:0':times2_3 :: s:0':times
hole_c1:c23_3 :: c1:c2
gen_s:0':times4_3 :: Nat -> s:0':times
gen_c1:c25_3 :: Nat -> c1:c2


Generator Equations:
gen_s:0':times4_3(0) <=> 0'
gen_s:0':times4_3(+(x, 1)) <=> s(gen_s:0':times4_3(x))
gen_c1:c25_3(0) <=> c1
gen_c1:c25_3(+(x, 1)) <=> c2(gen_c1:c25_3(x), c)


The following defined symbols remain to be analysed:
FAC, fac
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(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FAC(gen_s:0':times4_3(n7_3)) -> gen_c1:c25_3(n7_3), rt in Omega(1 + n7_3)

Induction Base:
FAC(gen_s:0':times4_3(0)) ->_R^Omega(1)
c1

Induction Step:
FAC(gen_s:0':times4_3(+(n7_3, 1))) ->_R^Omega(1)
c2(FAC(p(s(gen_s:0':times4_3(n7_3)))), P(s(gen_s:0':times4_3(n7_3)))) ->_R^Omega(0)
c2(FAC(gen_s:0':times4_3(n7_3)), P(s(gen_s:0':times4_3(n7_3)))) ->_IH
c2(gen_c1:c25_3(c8_3), P(s(gen_s:0':times4_3(n7_3)))) ->_R^Omega(1)
c2(gen_c1:c25_3(n7_3), c)

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
P(s(z0)) -> c
FAC(0') -> c1
FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
p(s(z0)) -> z0
fac(0') -> s(0')
fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

Types:
P :: s:0':times -> c
s :: s:0':times -> s:0':times
c :: c
FAC :: s:0':times -> c1:c2
0' :: s:0':times
c1 :: c1:c2
c2 :: c1:c2 -> c -> c1:c2
p :: s:0':times -> s:0':times
fac :: s:0':times -> s:0':times
times :: s:0':times -> s:0':times -> s:0':times
hole_c1_3 :: c
hole_s:0':times2_3 :: s:0':times
hole_c1:c23_3 :: c1:c2
gen_s:0':times4_3 :: Nat -> s:0':times
gen_c1:c25_3 :: Nat -> c1:c2


Generator Equations:
gen_s:0':times4_3(0) <=> 0'
gen_s:0':times4_3(+(x, 1)) <=> s(gen_s:0':times4_3(x))
gen_c1:c25_3(0) <=> c1
gen_c1:c25_3(+(x, 1)) <=> c2(gen_c1:c25_3(x), c)


The following defined symbols remain to be analysed:
FAC, fac
----------------------------------------

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
P(s(z0)) -> c
FAC(0') -> c1
FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
p(s(z0)) -> z0
fac(0') -> s(0')
fac(s(z0)) -> times(s(z0), fac(p(s(z0))))

Types:
P :: s:0':times -> c
s :: s:0':times -> s:0':times
c :: c
FAC :: s:0':times -> c1:c2
0' :: s:0':times
c1 :: c1:c2
c2 :: c1:c2 -> c -> c1:c2
p :: s:0':times -> s:0':times
fac :: s:0':times -> s:0':times
times :: s:0':times -> s:0':times -> s:0':times
hole_c1_3 :: c
hole_s:0':times2_3 :: s:0':times
hole_c1:c23_3 :: c1:c2
gen_s:0':times4_3 :: Nat -> s:0':times
gen_c1:c25_3 :: Nat -> c1:c2


Lemmas:
FAC(gen_s:0':times4_3(n7_3)) -> gen_c1:c25_3(n7_3), rt in Omega(1 + n7_3)


Generator Equations:
gen_s:0':times4_3(0) <=> 0'
gen_s:0':times4_3(+(x, 1)) <=> s(gen_s:0':times4_3(x))
gen_c1:c25_3(0) <=> c1
gen_c1:c25_3(+(x, 1)) <=> c2(gen_c1:c25_3(x), c)


The following defined symbols remain to be analysed:
fac