WORST_CASE(Omega(n^1),O(n^1))
proof of input_11bISRCX2G.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 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), 14 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 305 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) 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)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(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)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1, SQR_1

Compound Symbols:   c, c1_1, c2_1, c3_1, c4


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)))
   SQR(z0) -> c4

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
S tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1

Compound Symbols:   c2_1, c3_1


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

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   SUM_1

Compound Symbols:   c2_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)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(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)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(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.

"[43, 44, 45, 46]
{(43,44,[SUM_1|0]), (43,45,[c2_1|1]), (43,46,[c3_1|1]), (44,44,[s_1|0, c2_1|0, c3_1|0]), (45,44,[SUM_1|1]), (45,45,[c2_1|1]), (45,46,[c3_1|1]), (46,44,[SUM_1|1]), (46,45,[c2_1|1]), (46,46,[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)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1, SQR_1

Compound Symbols:   c, c1_1, c2_1, c3_1, c4


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

(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(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4

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

   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(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(SQR(s(z0)))
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c3(SUM(z0))
   SQR(z0) -> c4

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

   sum(0') -> 0'
   sum(s(z0)) -> +'(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +'(*'(s(z0), s(z0)), sum(z0))
   sqr(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(SQR(s(z0)))
SUM(s(z0)) -> c2(SUM(z0))
SUM(s(z0)) -> c3(SUM(z0))
SQR(z0) -> c4
sum(0') -> 0'
sum(s(z0)) -> +'(sqr(s(z0)), sum(z0))
sum(s(z0)) -> +'(*'(s(z0), s(z0)), sum(z0))
sqr(z0) -> *'(z0, z0)

Types:
SUM :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c4 -> c:c1:c2:c3
SQR :: 0':s:+' -> c4
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
c4 :: c4
sum :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: 0':s:+' -> *'
*' :: 0':s:+' -> 0':s:+' -> *'
hole_c:c1:c2:c31_5 :: c:c1:c2:c3
hole_0':s:+'2_5 :: 0':s:+'
hole_c43_5 :: c4
hole_*'4_5 :: *'
gen_c:c1:c2:c35_5 :: Nat -> c:c1:c2:c3
gen_0':s:+'6_5 :: Nat -> 0':s:+'

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

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

(22)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SQR(s(z0)))
SUM(s(z0)) -> c2(SUM(z0))
SUM(s(z0)) -> c3(SUM(z0))
SQR(z0) -> c4
sum(0') -> 0'
sum(s(z0)) -> +'(sqr(s(z0)), sum(z0))
sum(s(z0)) -> +'(*'(s(z0), s(z0)), sum(z0))
sqr(z0) -> *'(z0, z0)

Types:
SUM :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c4 -> c:c1:c2:c3
SQR :: 0':s:+' -> c4
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
c4 :: c4
sum :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: 0':s:+' -> *'
*' :: 0':s:+' -> 0':s:+' -> *'
hole_c:c1:c2:c31_5 :: c:c1:c2:c3
hole_0':s:+'2_5 :: 0':s:+'
hole_c43_5 :: c4
hole_*'4_5 :: *'
gen_c:c1:c2:c35_5 :: Nat -> c:c1:c2:c3
gen_0':s:+'6_5 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c35_5(0) <=> c
gen_c:c1:c2:c35_5(+(x, 1)) <=> c2(gen_c:c1:c2:c35_5(x))
gen_0':s:+'6_5(0) <=> 0'
gen_0':s:+'6_5(+(x, 1)) <=> s(gen_0':s:+'6_5(x))


The following defined symbols remain to be analysed:
SUM, sum
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SUM(gen_0':s:+'6_5(n8_5)) -> gen_c:c1:c2:c35_5(n8_5), rt in Omega(1 + n8_5)

Induction Base:
SUM(gen_0':s:+'6_5(0)) ->_R^Omega(1)
c

Induction Step:
SUM(gen_0':s:+'6_5(+(n8_5, 1))) ->_R^Omega(1)
c2(SUM(gen_0':s:+'6_5(n8_5))) ->_IH
c2(gen_c:c1:c2:c35_5(c9_5))

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(SQR(s(z0)))
SUM(s(z0)) -> c2(SUM(z0))
SUM(s(z0)) -> c3(SUM(z0))
SQR(z0) -> c4
sum(0') -> 0'
sum(s(z0)) -> +'(sqr(s(z0)), sum(z0))
sum(s(z0)) -> +'(*'(s(z0), s(z0)), sum(z0))
sqr(z0) -> *'(z0, z0)

Types:
SUM :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c4 -> c:c1:c2:c3
SQR :: 0':s:+' -> c4
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
c4 :: c4
sum :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: 0':s:+' -> *'
*' :: 0':s:+' -> 0':s:+' -> *'
hole_c:c1:c2:c31_5 :: c:c1:c2:c3
hole_0':s:+'2_5 :: 0':s:+'
hole_c43_5 :: c4
hole_*'4_5 :: *'
gen_c:c1:c2:c35_5 :: Nat -> c:c1:c2:c3
gen_0':s:+'6_5 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c35_5(0) <=> c
gen_c:c1:c2:c35_5(+(x, 1)) <=> c2(gen_c:c1:c2:c35_5(x))
gen_0':s:+'6_5(0) <=> 0'
gen_0':s:+'6_5(+(x, 1)) <=> s(gen_0':s:+'6_5(x))


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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
SUM(0') -> c
SUM(s(z0)) -> c1(SQR(s(z0)))
SUM(s(z0)) -> c2(SUM(z0))
SUM(s(z0)) -> c3(SUM(z0))
SQR(z0) -> c4
sum(0') -> 0'
sum(s(z0)) -> +'(sqr(s(z0)), sum(z0))
sum(s(z0)) -> +'(*'(s(z0), s(z0)), sum(z0))
sqr(z0) -> *'(z0, z0)

Types:
SUM :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c4 -> c:c1:c2:c3
SQR :: 0':s:+' -> c4
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
c4 :: c4
sum :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: 0':s:+' -> *'
*' :: 0':s:+' -> 0':s:+' -> *'
hole_c:c1:c2:c31_5 :: c:c1:c2:c3
hole_0':s:+'2_5 :: 0':s:+'
hole_c43_5 :: c4
hole_*'4_5 :: *'
gen_c:c1:c2:c35_5 :: Nat -> c:c1:c2:c3
gen_0':s:+'6_5 :: Nat -> 0':s:+'


Lemmas:
SUM(gen_0':s:+'6_5(n8_5)) -> gen_c:c1:c2:c35_5(n8_5), rt in Omega(1 + n8_5)


Generator Equations:
gen_c:c1:c2:c35_5(0) <=> c
gen_c:c1:c2:c35_5(+(x, 1)) <=> c2(gen_c:c1:c2:c35_5(x))
gen_0':s:+'6_5(0) <=> 0'
gen_0':s:+'6_5(+(x, 1)) <=> s(gen_0':s:+'6_5(x))


The following defined symbols remain to be analysed:
sum