WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 24 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 15 ms]
    (18) CdtProblem
    (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) BOUNDS(1, 1)
(21) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxTRS
    (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) typed CpxTrs
    (25) OrderProof [LOWER BOUND(ID), 0 ms]
    (26) typed CpxTrs
    (27) RewriteLemmaProof [LOWER BOUND(ID), 274 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (34) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   leq(0, y) -> true
   leq(s(x), 0) -> false
   leq(s(x), s(y)) -> leq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
mod(s(x), s([]))
mod(s([]), s(y))


The defined contexts are:
if([], x1, s(x2))
if(x0, [], s(x2))
mod([], s(x1))
leq(x0, [])
-(s([]), s(x1))
if(x0, x1, s([]))
-([], x1)


[] just represents basic- or constructor-terms in the following defined contexts:
if([], x1, s(x2))
mod([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   leq(0, y) -> true
   leq(s(x), 0) -> false
   leq(s(x), s(y)) -> leq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

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

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, IF_3, -'_2, MOD_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8, c9_4


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   MOD(0, z0) -> c7
   LEQ(s(z0), 0) -> c1
   IF(true, z0, z1) -> c3
   LEQ(0, z0) -> c
   MOD(s(z0), 0) -> c8
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_4


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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) by 
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(LEQ(x_1, x_2)) = 0
   POL(MOD(x_1, x_2)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(s(x_1)) = [1] + x_1

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
K tuples:
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = [2]
   POL(LEQ(x_1, x_2)) = x_2
   POL(MOD(x_1, x_2)) = x_1^2
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(s(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
K tuples:
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = [2]x_2
   POL(0) = [2]
   POL(LEQ(x_1, x_2)) = 0
   POL(MOD(x_1, x_2)) = x_1*x_2
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(s(x_1)) = [2] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:none
K tuples:
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_3


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

(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(20)
BOUNDS(1, 1)

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

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

(22)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   leq(0', y) -> true
   leq(s(x), 0') -> false
   leq(s(x), s(y)) -> leq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   mod(0', y) -> 0'
   mod(s(x), 0') -> 0'
   mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

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

(23) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(24)
Obligation:
TRS:
Rules:
leq(0', y) -> true
leq(s(x), 0') -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
if :: true:false -> 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(25) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
leq, -, mod

They will be analysed ascendingly in the following order:
leq < mod
- < mod

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

(26)
Obligation:
TRS:
Rules:
leq(0', y) -> true
leq(s(x), 0') -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
if :: true:false -> 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
leq, -, mod

They will be analysed ascendingly in the following order:
leq < mod
- < mod

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
leq(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
leq(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
Complex Obligation (BEST)

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(29)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
leq(0', y) -> true
leq(s(x), 0') -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
if :: true:false -> 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
leq, -, mod

They will be analysed ascendingly in the following order:
leq < mod
- < mod

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(30) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(31)
BOUNDS(n^1, INF)

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(32)
Obligation:
TRS:
Rules:
leq(0', y) -> true
leq(s(x), 0') -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
if :: true:false -> 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, mod

They will be analysed ascendingly in the following order:
- < mod

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(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) -> gen_0':s3_0(0), rt in Omega(1 + n252_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n252_0, 1)), gen_0':s3_0(+(n252_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) ->_IH
gen_0':s3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(34)
Obligation:
TRS:
Rules:
leq(0', y) -> true
leq(s(x), 0') -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
if :: true:false -> 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
-(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) -> gen_0':s3_0(0), rt in Omega(1 + n252_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
mod