WORST_CASE(Omega(n^1),O(n^1))
proof of input_mBdmKsojOJ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 10 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), 13 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 355 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
            (32) 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(0) -> 0
   p(s(x)) -> x
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
   if(true, x, y) -> x
   if(false, x, y) -> y

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:
   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0) -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
S tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0) -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
K tuples:none
Defined Rule Symbols:   p_1, le_2, minus_2, if_3

Defined Pair Symbols:   P_1, LE_2, MINUS_2, IF_3

Compound Symbols:   c, c1, c2, c3, c4_1, c5, c6_2, c7_4, c8, c9


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   P(s(z0)) -> c1
   P(0) -> c
   IF(false, z0, z1) -> c9
   IF(true, z0, z1) -> c8
   MINUS(z0, 0) -> c5
   LE(s(z0), 0) -> c3
   LE(0, z0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
K tuples:none
Defined Rule Symbols:   p_1, le_2, minus_2, if_3

Defined Pair Symbols:   LE_2, MINUS_2

Compound Symbols:   c4_1, c6_2, c7_4


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))
S tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))
K tuples:none
Defined Rule Symbols:   p_1, le_2, minus_2, if_3

Defined Pair Symbols:   LE_2, MINUS_2

Compound Symbols:   c4_1, c6_1, c7_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   p(0) -> 0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
Tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))
S tuples:
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   LE_2, MINUS_2

Compound Symbols:   c4_1, c6_1, c7_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:

   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))

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:

   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, s(z1)) -> c6(LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(MINUS(z0, p(s(z1))))
   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, 3]
transitions: 
s0(0) -> 0
c40(0) -> 0
c60(0) -> 0
c70(0) -> 0
LE0(0, 0) -> 1
MINUS0(0, 0) -> 2
p0(0) -> 3
LE1(0, 0) -> 4
c41(4) -> 1
s1(0) -> 6
LE1(0, 6) -> 5
c61(5) -> 2
s1(0) -> 9
p1(9) -> 8
MINUS1(0, 8) -> 7
c71(7) -> 2
c41(4) -> 4
c41(4) -> 5
c61(5) -> 7
c71(7) -> 7
0 -> 3
0 -> 8

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

(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(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0) -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
S tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0) -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
K tuples:none
Defined Rule Symbols:   p_1, le_2, minus_2, if_3

Defined Pair Symbols:   P_1, LE_2, MINUS_2, IF_3

Compound Symbols:   c, c1, c2, c3, c4_1, c5, c6_2, c7_4, c8, c9


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

(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(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0) -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0, p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0, p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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(0') -> c
   P(s(z0)) -> c1
   LE(0', z0) -> c2
   LE(s(z0), 0') -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, 0') -> c5
   MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
   MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   p(0') -> 0'
   p(s(z0)) -> z0
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0') -> z0
   minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

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

(22)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, 0') -> c5
MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
le :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c:c1 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_0':s7_10 :: Nat -> 0':s
gen_c2:c3:c48_10 :: Nat -> c2:c3:c4
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, MINUS, le, minus

They will be analysed ascendingly in the following order:
LE < MINUS
le < MINUS
minus < MINUS
le < minus

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

(24)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, 0') -> c5
MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
le :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c:c1 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_0':s7_10 :: Nat -> 0':s
gen_c2:c3:c48_10 :: Nat -> c2:c3:c4
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_0':s7_10(0) <=> 0'
gen_0':s7_10(+(x, 1)) <=> s(gen_0':s7_10(x))
gen_c2:c3:c48_10(0) <=> c2
gen_c2:c3:c48_10(+(x, 1)) <=> c4(gen_c2:c3:c48_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(c8, c, gen_c5:c6:c79_10(x), c)


The following defined symbols remain to be analysed:
LE, MINUS, le, minus

They will be analysed ascendingly in the following order:
LE < MINUS
le < MINUS
minus < MINUS
le < minus

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s7_10(n11_10), gen_0':s7_10(n11_10)) -> gen_c2:c3:c48_10(n11_10), rt in Omega(1 + n11_10)

Induction Base:
LE(gen_0':s7_10(0), gen_0':s7_10(0)) ->_R^Omega(1)
c2

Induction Step:
LE(gen_0':s7_10(+(n11_10, 1)), gen_0':s7_10(+(n11_10, 1))) ->_R^Omega(1)
c4(LE(gen_0':s7_10(n11_10), gen_0':s7_10(n11_10))) ->_IH
c4(gen_c2:c3:c48_10(c12_10))

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(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, 0') -> c5
MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
le :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c:c1 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_0':s7_10 :: Nat -> 0':s
gen_c2:c3:c48_10 :: Nat -> c2:c3:c4
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_0':s7_10(0) <=> 0'
gen_0':s7_10(+(x, 1)) <=> s(gen_0':s7_10(x))
gen_c2:c3:c48_10(0) <=> c2
gen_c2:c3:c48_10(+(x, 1)) <=> c4(gen_c2:c3:c48_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(c8, c, gen_c5:c6:c79_10(x), c)


The following defined symbols remain to be analysed:
LE, MINUS, le, minus

They will be analysed ascendingly in the following order:
LE < MINUS
le < MINUS
minus < MINUS
le < minus

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

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, 0') -> c5
MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
le :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c:c1 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_0':s7_10 :: Nat -> 0':s
gen_c2:c3:c48_10 :: Nat -> c2:c3:c4
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7


Lemmas:
LE(gen_0':s7_10(n11_10), gen_0':s7_10(n11_10)) -> gen_c2:c3:c48_10(n11_10), rt in Omega(1 + n11_10)


Generator Equations:
gen_0':s7_10(0) <=> 0'
gen_0':s7_10(+(x, 1)) <=> s(gen_0':s7_10(x))
gen_c2:c3:c48_10(0) <=> c2
gen_c2:c3:c48_10(+(x, 1)) <=> c4(gen_c2:c3:c48_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(c8, c, gen_c5:c6:c79_10(x), c)


The following defined symbols remain to be analysed:
le, MINUS, minus

They will be analysed ascendingly in the following order:
le < MINUS
minus < MINUS
le < minus

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s7_10(n610_10), gen_0':s7_10(n610_10)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s7_10(0), gen_0':s7_10(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s7_10(+(n610_10, 1)), gen_0':s7_10(+(n610_10, 1))) ->_R^Omega(0)
le(gen_0':s7_10(n610_10), gen_0':s7_10(n610_10)) ->_IH
true

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

(32)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, 0') -> c5
MINUS(z0, s(z1)) -> c6(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), LE(z0, s(z1)))
MINUS(z0, s(z1)) -> c7(IF(le(z0, s(z1)), 0', p(minus(z0, p(s(z1))))), P(minus(z0, p(s(z1)))), MINUS(z0, p(s(z1))), P(s(z1)))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(z0, s(z1)) -> if(le(z0, s(z1)), 0', p(minus(z0, p(s(z1)))))
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
le :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c:c1 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_0':s7_10 :: Nat -> 0':s
gen_c2:c3:c48_10 :: Nat -> c2:c3:c4
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7


Lemmas:
LE(gen_0':s7_10(n11_10), gen_0':s7_10(n11_10)) -> gen_c2:c3:c48_10(n11_10), rt in Omega(1 + n11_10)
le(gen_0':s7_10(n610_10), gen_0':s7_10(n610_10)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s7_10(0) <=> 0'
gen_0':s7_10(+(x, 1)) <=> s(gen_0':s7_10(x))
gen_c2:c3:c48_10(0) <=> c2
gen_c2:c3:c48_10(+(x, 1)) <=> c4(gen_c2:c3:c48_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(c8, c, gen_c5:c6:c79_10(x), c)


The following defined symbols remain to be analysed:
minus, MINUS

They will be analysed ascendingly in the following order:
minus < MINUS