WORST_CASE(Omega(n^1),O(n^1))
proof of input_XM12dfRRzN.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsMatchBoundsProof [FINISHED, 4 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), 247 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
            (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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sum1_1

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c, c1_1, c2, c3_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUM1(0) -> c2
   SUM(0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sum1_1

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_1


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

(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:

   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))

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

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

(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:

   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))

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

(11) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[53, 54, 55, 56]
{(53,54,[SUM_1|0, SUM1_1|0]), (53,55,[c1_1|1]), (53,56,[c3_1|1]), (54,54,[s_1|0, c1_1|0, c3_1|0]), (55,54,[SUM_1|1]), (55,55,[c1_1|1]), (56,54,[SUM1_1|1]), (56,56,[c3_1|1])}"
----------------------------------------

(12)
BOUNDS(1, n^1)

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sum1_1

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c, c1_1, c2, c3_1


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

(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:

   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))

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

   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))

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

(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:

   SUM(0') -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0') -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))

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

   sum(0') -> 0'
   sum(s(z0)) -> +'(sum(z0), s(z0))
   sum1(0') -> 0'
   sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

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

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

(20)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SUM(z0))
SUM1(0') -> c2
SUM1(s(z0)) -> c3(SUM1(z0))
sum(0') -> 0'
sum(s(z0)) -> +'(sum(z0), s(z0))
sum1(0') -> 0'
sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

Types:
SUM :: 0':s:+' -> c:c1
0' :: 0':s:+'
c :: c:c1
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1 -> c:c1
SUM1 :: 0':s:+' -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
sum :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_c:c11_4 :: c:c1
hole_0':s:+'2_4 :: 0':s:+'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_0':s:+'5_4 :: Nat -> 0':s:+'
gen_c2:c36_4 :: Nat -> c2:c3

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SUM, SUM1, sum, sum1
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SUM(z0))
SUM1(0') -> c2
SUM1(s(z0)) -> c3(SUM1(z0))
sum(0') -> 0'
sum(s(z0)) -> +'(sum(z0), s(z0))
sum1(0') -> 0'
sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

Types:
SUM :: 0':s:+' -> c:c1
0' :: 0':s:+'
c :: c:c1
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1 -> c:c1
SUM1 :: 0':s:+' -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
sum :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_c:c11_4 :: c:c1
hole_0':s:+'2_4 :: 0':s:+'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_0':s:+'5_4 :: Nat -> 0':s:+'
gen_c2:c36_4 :: Nat -> c2:c3


Generator Equations:
gen_c:c14_4(0) <=> c
gen_c:c14_4(+(x, 1)) <=> c1(gen_c:c14_4(x))
gen_0':s:+'5_4(0) <=> 0'
gen_0':s:+'5_4(+(x, 1)) <=> s(gen_0':s:+'5_4(x))
gen_c2:c36_4(0) <=> c2
gen_c2:c36_4(+(x, 1)) <=> c3(gen_c2:c36_4(x))


The following defined symbols remain to be analysed:
SUM, SUM1, sum, sum1
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SUM(gen_0':s:+'5_4(n8_4)) -> gen_c:c14_4(n8_4), rt in Omega(1 + n8_4)

Induction Base:
SUM(gen_0':s:+'5_4(0)) ->_R^Omega(1)
c

Induction Step:
SUM(gen_0':s:+'5_4(+(n8_4, 1))) ->_R^Omega(1)
c1(SUM(gen_0':s:+'5_4(n8_4))) ->_IH
c1(gen_c:c14_4(c9_4))

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

(24)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SUM(z0))
SUM1(0') -> c2
SUM1(s(z0)) -> c3(SUM1(z0))
sum(0') -> 0'
sum(s(z0)) -> +'(sum(z0), s(z0))
sum1(0') -> 0'
sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

Types:
SUM :: 0':s:+' -> c:c1
0' :: 0':s:+'
c :: c:c1
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1 -> c:c1
SUM1 :: 0':s:+' -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
sum :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_c:c11_4 :: c:c1
hole_0':s:+'2_4 :: 0':s:+'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_0':s:+'5_4 :: Nat -> 0':s:+'
gen_c2:c36_4 :: Nat -> c2:c3


Generator Equations:
gen_c:c14_4(0) <=> c
gen_c:c14_4(+(x, 1)) <=> c1(gen_c:c14_4(x))
gen_0':s:+'5_4(0) <=> 0'
gen_0':s:+'5_4(+(x, 1)) <=> s(gen_0':s:+'5_4(x))
gen_c2:c36_4(0) <=> c2
gen_c2:c36_4(+(x, 1)) <=> c3(gen_c2:c36_4(x))


The following defined symbols remain to be analysed:
SUM, SUM1, sum, sum1
----------------------------------------

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SUM(z0))
SUM1(0') -> c2
SUM1(s(z0)) -> c3(SUM1(z0))
sum(0') -> 0'
sum(s(z0)) -> +'(sum(z0), s(z0))
sum1(0') -> 0'
sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

Types:
SUM :: 0':s:+' -> c:c1
0' :: 0':s:+'
c :: c:c1
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1 -> c:c1
SUM1 :: 0':s:+' -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
sum :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_c:c11_4 :: c:c1
hole_0':s:+'2_4 :: 0':s:+'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_0':s:+'5_4 :: Nat -> 0':s:+'
gen_c2:c36_4 :: Nat -> c2:c3


Lemmas:
SUM(gen_0':s:+'5_4(n8_4)) -> gen_c:c14_4(n8_4), rt in Omega(1 + n8_4)


Generator Equations:
gen_c:c14_4(0) <=> c
gen_c:c14_4(+(x, 1)) <=> c1(gen_c:c14_4(x))
gen_0':s:+'5_4(0) <=> 0'
gen_0':s:+'5_4(+(x, 1)) <=> s(gen_0':s:+'5_4(x))
gen_c2:c36_4(0) <=> c2
gen_c2:c36_4(+(x, 1)) <=> c3(gen_c2:c36_4(x))


The following defined symbols remain to be analysed:
SUM1, sum, sum1
----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SUM1(gen_0':s:+'5_4(n216_4)) -> gen_c2:c36_4(n216_4), rt in Omega(1 + n216_4)

Induction Base:
SUM1(gen_0':s:+'5_4(0)) ->_R^Omega(1)
c2

Induction Step:
SUM1(gen_0':s:+'5_4(+(n216_4, 1))) ->_R^Omega(1)
c3(SUM1(gen_0':s:+'5_4(n216_4))) ->_IH
c3(gen_c2:c36_4(c217_4))

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

(30)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SUM(z0))
SUM1(0') -> c2
SUM1(s(z0)) -> c3(SUM1(z0))
sum(0') -> 0'
sum(s(z0)) -> +'(sum(z0), s(z0))
sum1(0') -> 0'
sum1(s(z0)) -> s(+'(sum1(z0), +'(z0, z0)))

Types:
SUM :: 0':s:+' -> c:c1
0' :: 0':s:+'
c :: c:c1
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1 -> c:c1
SUM1 :: 0':s:+' -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
sum :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_c:c11_4 :: c:c1
hole_0':s:+'2_4 :: 0':s:+'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_0':s:+'5_4 :: Nat -> 0':s:+'
gen_c2:c36_4 :: Nat -> c2:c3


Lemmas:
SUM(gen_0':s:+'5_4(n8_4)) -> gen_c:c14_4(n8_4), rt in Omega(1 + n8_4)
SUM1(gen_0':s:+'5_4(n216_4)) -> gen_c2:c36_4(n216_4), rt in Omega(1 + n216_4)


Generator Equations:
gen_c:c14_4(0) <=> c
gen_c:c14_4(+(x, 1)) <=> c1(gen_c:c14_4(x))
gen_0':s:+'5_4(0) <=> 0'
gen_0':s:+'5_4(+(x, 1)) <=> s(gen_0':s:+'5_4(x))
gen_c2:c36_4(0) <=> c2
gen_c2:c36_4(+(x, 1)) <=> c3(gen_c2:c36_4(x))


The following defined symbols remain to be analysed:
sum, sum1