WORST_CASE(Omega(n^1),O(n^1))
proof of input_lm1AwmiIUn.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTRS
    (9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTRS
    (11) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (12) BOUNDS(1, n^1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 93.2 s]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 1029 ms]
            (30) BOUNDS(1, INF)


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(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:

   bin(x, 0) -> s(0)
   bin(0, s(y)) -> 0
   bin(s(x), s(y)) -> +(bin(x, s(y)), bin(x, y))

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
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bin(z0, 0) -> s(0)
   bin(0, s(z0)) -> 0
   bin(s(z0), s(z1)) -> +(bin(z0, s(z1)), bin(z0, z1))
Tuples:
   BIN(z0, 0) -> c
   BIN(0, s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
S tuples:
   BIN(z0, 0) -> c
   BIN(0, s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
K tuples:none
Defined Rule Symbols:   bin_2

Defined Pair Symbols:   BIN_2

Compound Symbols:   c, c1, c2_1, c3_1


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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bin(z0, 0) -> s(0)
   bin(0, s(z0)) -> 0
   bin(s(z0), s(z1)) -> +(bin(z0, s(z1)), bin(z0, z1))
Tuples:
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
S tuples:
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
K tuples:none
Defined Rule Symbols:   bin_2

Defined Pair Symbols:   BIN_2

Compound Symbols:   c2_1, c3_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   bin(z0, 0) -> s(0)
   bin(0, s(z0)) -> 0
   bin(s(z0), s(z1)) -> +(bin(z0, s(z1)), bin(z0, z1))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
S tuples:
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   BIN_2

Compound Symbols:   c2_1, c3_1


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

(8)
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:

   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(10)
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:

   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))

S is empty.
Rewrite Strategy: INNERMOST
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(11) 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]
transitions: 
s0(0) -> 0
c20(0) -> 0
c30(0) -> 0
BIN0(0, 0) -> 1
s1(0) -> 3
BIN1(0, 3) -> 2
c21(2) -> 1
BIN1(0, 0) -> 4
c31(4) -> 1
c21(2) -> 4
c21(2) -> 2
c31(4) -> 4
c31(4) -> 2

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(12)
BOUNDS(1, n^1)

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bin(z0, 0) -> s(0)
   bin(0, s(z0)) -> 0
   bin(s(z0), s(z1)) -> +(bin(z0, s(z1)), bin(z0, z1))
Tuples:
   BIN(z0, 0) -> c
   BIN(0, s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
S tuples:
   BIN(z0, 0) -> c
   BIN(0, s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
K tuples:none
Defined Rule Symbols:   bin_2

Defined Pair Symbols:   BIN_2

Compound Symbols:   c, c1, c2_1, c3_1


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

(16)
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:

   BIN(z0, 0) -> c
   BIN(0, s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))

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

   bin(z0, 0) -> s(0)
   bin(0, s(z0)) -> 0
   bin(s(z0), s(z1)) -> +(bin(z0, s(z1)), bin(z0, z1))

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

(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:

   BIN(z0, 0') -> c
   BIN(0', s(z0)) -> c1
   BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
   BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))

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

   bin(z0, 0') -> s(0')
   bin(0', s(z0)) -> 0'
   bin(s(z0), s(z1)) -> +'(bin(z0, s(z1)), bin(z0, z1))

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

(20)
Obligation:
Innermost TRS:
Rules:
BIN(z0, 0') -> c
BIN(0', s(z0)) -> c1
BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
bin(z0, 0') -> s(0')
bin(0', s(z0)) -> 0'
bin(s(z0), s(z1)) -> +'(bin(z0, s(z1)), bin(z0, z1))

Types:
BIN :: 0':s:+' -> 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
bin :: 0':s:+' -> 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'

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(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
BIN, bin
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
BIN(z0, 0') -> c
BIN(0', s(z0)) -> c1
BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
bin(z0, 0') -> s(0')
bin(0', s(z0)) -> 0'
bin(s(z0), s(z1)) -> +'(bin(z0, s(z1)), bin(z0, z1))

Types:
BIN :: 0':s:+' -> 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
bin :: 0':s:+' -> 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
BIN, bin
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
BIN(gen_0':s:+'4_4(+(1, n6_4)), gen_0':s:+'4_4(1)) -> *5_4, rt in Omega(n6_4)

Induction Base:
BIN(gen_0':s:+'4_4(+(1, 0)), gen_0':s:+'4_4(1))

Induction Step:
BIN(gen_0':s:+'4_4(+(1, +(n6_4, 1))), gen_0':s:+'4_4(1)) ->_R^Omega(1)
c2(BIN(gen_0':s:+'4_4(+(1, n6_4)), s(gen_0':s:+'4_4(0)))) ->_IH
c2(*5_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Complex Obligation (BEST)

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(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
BIN(z0, 0') -> c
BIN(0', s(z0)) -> c1
BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
bin(z0, 0') -> s(0')
bin(0', s(z0)) -> 0'
bin(s(z0), s(z1)) -> +'(bin(z0, s(z1)), bin(z0, z1))

Types:
BIN :: 0':s:+' -> 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
bin :: 0':s:+' -> 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
BIN, bin
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(26) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(27)
BOUNDS(n^1, INF)

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(28)
Obligation:
Innermost TRS:
Rules:
BIN(z0, 0') -> c
BIN(0', s(z0)) -> c1
BIN(s(z0), s(z1)) -> c2(BIN(z0, s(z1)))
BIN(s(z0), s(z1)) -> c3(BIN(z0, z1))
bin(z0, 0') -> s(0')
bin(0', s(z0)) -> 0'
bin(s(z0), s(z1)) -> +'(bin(z0, s(z1)), bin(z0, z1))

Types:
BIN :: 0':s:+' -> 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
bin :: 0':s:+' -> 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Lemmas:
BIN(gen_0':s:+'4_4(+(1, n6_4)), gen_0':s:+'4_4(1)) -> *5_4, rt in Omega(n6_4)


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
bin
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(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
bin(gen_0':s:+'4_4(+(1, n74444_4)), gen_0':s:+'4_4(1)) -> *5_4, rt in Omega(0)

Induction Base:
bin(gen_0':s:+'4_4(+(1, 0)), gen_0':s:+'4_4(1))

Induction Step:
bin(gen_0':s:+'4_4(+(1, +(n74444_4, 1))), gen_0':s:+'4_4(1)) ->_R^Omega(0)
+'(bin(gen_0':s:+'4_4(+(1, n74444_4)), s(gen_0':s:+'4_4(0))), bin(gen_0':s:+'4_4(+(1, n74444_4)), gen_0':s:+'4_4(0))) ->_IH
+'(*5_4, bin(gen_0':s:+'4_4(+(1, n74444_4)), gen_0':s:+'4_4(0))) ->_R^Omega(0)
+'(*5_4, s(0'))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(30)
BOUNDS(1, INF)