WORST_CASE(Omega(n^1),O(n^1))
proof of input_BazT9QciQL.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 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, 16 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), 19 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 93.4 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), 24.1 s]
            (30) BOUNDS(1, INF)


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

(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:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

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:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7
S tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME_1, PRIME1_2, DIVP_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6_1, c7


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
Removed 6 trailing nodes:
   PRIME(0) -> c
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   DIVP(z0, z1) -> c7
   PRIME(s(0)) -> c1
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
S tuples:
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c6_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
S tuples:
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c6_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:

   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))

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:

   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))

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

(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
c60(0) -> 0
PRIME10(0, 0) -> 1
s1(0) -> 3
PRIME11(0, 3) -> 2
c61(2) -> 1
c61(2) -> 2

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

(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:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7
S tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME_1, PRIME1_2, DIVP_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6_1, c7


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

(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:

   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7

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

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)

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:

   PRIME(0') -> c
   PRIME(s(0')) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0') -> c3
   PRIME1(z0, s(0')) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
   PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c7

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

   prime(0') -> false
   prime(s(0')) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0') -> false
   prime1(z0, s(0')) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> ='(rem(z0, z1), 0')

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

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

(20)
Obligation:
Innermost TRS:
Rules:
PRIME(0') -> c
PRIME(s(0')) -> c1
PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
PRIME1(z0, 0') -> c3
PRIME1(z0, s(0')) -> c4
PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
DIVP(z0, z1) -> c7
prime(0') -> false
prime(s(0')) -> false
prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
prime1(z0, 0') -> false
prime1(z0, s(0')) -> true
prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
divp(z0, z1) -> ='(rem(z0, z1), 0')

Types:
PRIME :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c3:c4:c5:c6 -> c:c1:c2
PRIME1 :: 0':s -> 0':s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c7 -> c3:c4:c5:c6
DIVP :: 0':s -> 0':s -> c7
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c7 :: c7
prime :: 0':s -> false:true:and
false :: false:true:and
prime1 :: 0':s -> 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: 0':s -> 0':s -> ='
=' :: rem -> 0':s -> ='
rem :: 0':s -> 0':s -> rem
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c3:c4:c5:c63_8 :: c3:c4:c5:c6
hole_c74_8 :: c7
hole_false:true:and5_8 :: false:true:and
hole_not6_8 :: not
hole_='7_8 :: ='
hole_rem8_8 :: rem
gen_0':s9_8 :: Nat -> 0':s
gen_c3:c4:c5:c610_8 :: Nat -> c3:c4:c5:c6
gen_false:true:and11_8 :: Nat -> false:true:and

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PRIME1, prime1
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
PRIME(0') -> c
PRIME(s(0')) -> c1
PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
PRIME1(z0, 0') -> c3
PRIME1(z0, s(0')) -> c4
PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
DIVP(z0, z1) -> c7
prime(0') -> false
prime(s(0')) -> false
prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
prime1(z0, 0') -> false
prime1(z0, s(0')) -> true
prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
divp(z0, z1) -> ='(rem(z0, z1), 0')

Types:
PRIME :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c3:c4:c5:c6 -> c:c1:c2
PRIME1 :: 0':s -> 0':s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c7 -> c3:c4:c5:c6
DIVP :: 0':s -> 0':s -> c7
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c7 :: c7
prime :: 0':s -> false:true:and
false :: false:true:and
prime1 :: 0':s -> 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: 0':s -> 0':s -> ='
=' :: rem -> 0':s -> ='
rem :: 0':s -> 0':s -> rem
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c3:c4:c5:c63_8 :: c3:c4:c5:c6
hole_c74_8 :: c7
hole_false:true:and5_8 :: false:true:and
hole_not6_8 :: not
hole_='7_8 :: ='
hole_rem8_8 :: rem
gen_0':s9_8 :: Nat -> 0':s
gen_c3:c4:c5:c610_8 :: Nat -> c3:c4:c5:c6
gen_false:true:and11_8 :: Nat -> false:true:and


Generator Equations:
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c3:c4:c5:c610_8(0) <=> c3
gen_c3:c4:c5:c610_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c610_8(x))
gen_false:true:and11_8(0) <=> false
gen_false:true:and11_8(+(x, 1)) <=> and(not(='(rem(0', 0'), 0')), gen_false:true:and11_8(x))


The following defined symbols remain to be analysed:
PRIME1, prime1
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PRIME1(gen_0':s9_8(a), gen_0':s9_8(+(2, n13_8))) -> *12_8, rt in Omega(n13_8)

Induction Base:
PRIME1(gen_0':s9_8(a), gen_0':s9_8(+(2, 0)))

Induction Step:
PRIME1(gen_0':s9_8(a), gen_0':s9_8(+(2, +(n13_8, 1)))) ->_R^Omega(1)
c6(PRIME1(gen_0':s9_8(a), s(gen_0':s9_8(+(1, n13_8))))) ->_IH
c6(*12_8)

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:
PRIME(0') -> c
PRIME(s(0')) -> c1
PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
PRIME1(z0, 0') -> c3
PRIME1(z0, s(0')) -> c4
PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
DIVP(z0, z1) -> c7
prime(0') -> false
prime(s(0')) -> false
prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
prime1(z0, 0') -> false
prime1(z0, s(0')) -> true
prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
divp(z0, z1) -> ='(rem(z0, z1), 0')

Types:
PRIME :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c3:c4:c5:c6 -> c:c1:c2
PRIME1 :: 0':s -> 0':s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c7 -> c3:c4:c5:c6
DIVP :: 0':s -> 0':s -> c7
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c7 :: c7
prime :: 0':s -> false:true:and
false :: false:true:and
prime1 :: 0':s -> 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: 0':s -> 0':s -> ='
=' :: rem -> 0':s -> ='
rem :: 0':s -> 0':s -> rem
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c3:c4:c5:c63_8 :: c3:c4:c5:c6
hole_c74_8 :: c7
hole_false:true:and5_8 :: false:true:and
hole_not6_8 :: not
hole_='7_8 :: ='
hole_rem8_8 :: rem
gen_0':s9_8 :: Nat -> 0':s
gen_c3:c4:c5:c610_8 :: Nat -> c3:c4:c5:c6
gen_false:true:and11_8 :: Nat -> false:true:and


Generator Equations:
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c3:c4:c5:c610_8(0) <=> c3
gen_c3:c4:c5:c610_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c610_8(x))
gen_false:true:and11_8(0) <=> false
gen_false:true:and11_8(+(x, 1)) <=> and(not(='(rem(0', 0'), 0')), gen_false:true:and11_8(x))


The following defined symbols remain to be analysed:
PRIME1, prime1
----------------------------------------

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
PRIME(0') -> c
PRIME(s(0')) -> c1
PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
PRIME1(z0, 0') -> c3
PRIME1(z0, s(0')) -> c4
PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0))
PRIME1(z0, s(s(z1))) -> c6(PRIME1(z0, s(z1)))
DIVP(z0, z1) -> c7
prime(0') -> false
prime(s(0')) -> false
prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
prime1(z0, 0') -> false
prime1(z0, s(0')) -> true
prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
divp(z0, z1) -> ='(rem(z0, z1), 0')

Types:
PRIME :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c3:c4:c5:c6 -> c:c1:c2
PRIME1 :: 0':s -> 0':s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c7 -> c3:c4:c5:c6
DIVP :: 0':s -> 0':s -> c7
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c7 :: c7
prime :: 0':s -> false:true:and
false :: false:true:and
prime1 :: 0':s -> 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: 0':s -> 0':s -> ='
=' :: rem -> 0':s -> ='
rem :: 0':s -> 0':s -> rem
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c3:c4:c5:c63_8 :: c3:c4:c5:c6
hole_c74_8 :: c7
hole_false:true:and5_8 :: false:true:and
hole_not6_8 :: not
hole_='7_8 :: ='
hole_rem8_8 :: rem
gen_0':s9_8 :: Nat -> 0':s
gen_c3:c4:c5:c610_8 :: Nat -> c3:c4:c5:c6
gen_false:true:and11_8 :: Nat -> false:true:and


Lemmas:
PRIME1(gen_0':s9_8(a), gen_0':s9_8(+(2, n13_8))) -> *12_8, rt in Omega(n13_8)


Generator Equations:
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c3:c4:c5:c610_8(0) <=> c3
gen_c3:c4:c5:c610_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c610_8(x))
gen_false:true:and11_8(0) <=> false
gen_false:true:and11_8(+(x, 1)) <=> and(not(='(rem(0', 0'), 0')), gen_false:true:and11_8(x))


The following defined symbols remain to be analysed:
prime1
----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
prime1(gen_0':s9_8(a), gen_0':s9_8(+(1, n74700_8))) -> *12_8, rt in Omega(0)

Induction Base:
prime1(gen_0':s9_8(a), gen_0':s9_8(+(1, 0)))

Induction Step:
prime1(gen_0':s9_8(a), gen_0':s9_8(+(1, +(n74700_8, 1)))) ->_R^Omega(0)
and(not(divp(s(s(gen_0':s9_8(n74700_8))), gen_0':s9_8(a))), prime1(gen_0':s9_8(a), s(gen_0':s9_8(n74700_8)))) ->_R^Omega(0)
and(not(='(rem(s(s(gen_0':s9_8(n74700_8))), gen_0':s9_8(a)), 0')), prime1(gen_0':s9_8(a), s(gen_0':s9_8(n74700_8)))) ->_IH
and(not(='(rem(s(s(gen_0':s9_8(n74700_8))), gen_0':s9_8(a)), 0')), *12_8)

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

(30)
BOUNDS(1, INF)