/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 109 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 480 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 52 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 74 ms]
            (30) typed CpxTrs


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

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


The TRS R consists of the following rules:

   active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(prod(0, X)) -> mark(0)
   active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(zero(0)) -> mark(true)
   active(zero(s(X))) -> mark(false)
   active(p(s(X))) -> mark(X)
   active(fact(X)) -> fact(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(zero(X)) -> zero(active(X))
   active(s(X)) -> s(active(X))
   active(prod(X1, X2)) -> prod(active(X1), X2)
   active(prod(X1, X2)) -> prod(X1, active(X2))
   active(p(X)) -> p(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   fact(mark(X)) -> mark(fact(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   zero(mark(X)) -> mark(zero(X))
   s(mark(X)) -> mark(s(X))
   prod(mark(X1), X2) -> mark(prod(X1, X2))
   prod(X1, mark(X2)) -> mark(prod(X1, X2))
   p(mark(X)) -> mark(p(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   proper(fact(X)) -> fact(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(zero(X)) -> zero(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
   proper(p(X)) -> p(proper(X))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   fact(ok(X)) -> ok(fact(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   zero(ok(X)) -> ok(zero(X))
   s(ok(X)) -> ok(s(X))
   prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
   p(ok(X)) -> ok(p(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))

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

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


The TRS R consists of the following rules:

   fact(mark(X)) -> mark(fact(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   zero(mark(X)) -> mark(zero(X))
   s(mark(X)) -> mark(s(X))
   prod(mark(X1), X2) -> mark(prod(X1, X2))
   prod(X1, mark(X2)) -> mark(prod(X1, X2))
   p(mark(X)) -> mark(p(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   fact(ok(X)) -> ok(fact(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   zero(ok(X)) -> ok(zero(X))
   s(ok(X)) -> ok(s(X))
   prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
   p(ok(X)) -> ok(p(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   fact(mark(X)) -> mark(fact(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   zero(mark(X)) -> mark(zero(X))
   s(mark(X)) -> mark(s(X))
   prod(mark(X1), X2) -> mark(prod(X1, X2))
   prod(X1, mark(X2)) -> mark(prod(X1, X2))
   p(mark(X)) -> mark(p(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   fact(ok(X)) -> ok(fact(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   zero(ok(X)) -> ok(zero(X))
   s(ok(X)) -> ok(s(X))
   prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
   p(ok(X)) -> ok(p(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(5) 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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
true0() -> 0
false0() -> 0
active0(0) -> 0
fact0(0) -> 1
if0(0, 0, 0) -> 2
zero0(0) -> 3
s0(0) -> 4
prod0(0, 0) -> 5
p0(0) -> 6
add0(0, 0) -> 7
proper0(0) -> 8
top0(0) -> 9
fact1(0) -> 10
mark1(10) -> 1
if1(0, 0, 0) -> 11
mark1(11) -> 2
zero1(0) -> 12
mark1(12) -> 3
s1(0) -> 13
mark1(13) -> 4
prod1(0, 0) -> 14
mark1(14) -> 5
p1(0) -> 15
mark1(15) -> 6
add1(0, 0) -> 16
mark1(16) -> 7
01() -> 17
ok1(17) -> 8
true1() -> 18
ok1(18) -> 8
false1() -> 19
ok1(19) -> 8
fact1(0) -> 20
ok1(20) -> 1
if1(0, 0, 0) -> 21
ok1(21) -> 2
zero1(0) -> 22
ok1(22) -> 3
s1(0) -> 23
ok1(23) -> 4
prod1(0, 0) -> 24
ok1(24) -> 5
p1(0) -> 25
ok1(25) -> 6
add1(0, 0) -> 26
ok1(26) -> 7
proper1(0) -> 27
top1(27) -> 9
active1(0) -> 28
top1(28) -> 9
mark1(10) -> 10
mark1(10) -> 20
mark1(11) -> 11
mark1(11) -> 21
mark1(12) -> 12
mark1(12) -> 22
mark1(13) -> 13
mark1(13) -> 23
mark1(14) -> 14
mark1(14) -> 24
mark1(15) -> 15
mark1(15) -> 25
mark1(16) -> 16
mark1(16) -> 26
ok1(17) -> 27
ok1(18) -> 27
ok1(19) -> 27
ok1(20) -> 10
ok1(20) -> 20
ok1(21) -> 11
ok1(21) -> 21
ok1(22) -> 12
ok1(22) -> 22
ok1(23) -> 13
ok1(23) -> 23
ok1(24) -> 14
ok1(24) -> 24
ok1(25) -> 15
ok1(25) -> 25
ok1(26) -> 16
ok1(26) -> 26
active2(17) -> 29
top2(29) -> 9
active2(18) -> 29
active2(19) -> 29

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

(6)
BOUNDS(1, n^1)

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

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

(8)
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:

   active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
   active(add(0', X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(prod(0', X)) -> mark(0')
   active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(zero(0')) -> mark(true)
   active(zero(s(X))) -> mark(false)
   active(p(s(X))) -> mark(X)
   active(fact(X)) -> fact(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(zero(X)) -> zero(active(X))
   active(s(X)) -> s(active(X))
   active(prod(X1, X2)) -> prod(active(X1), X2)
   active(prod(X1, X2)) -> prod(X1, active(X2))
   active(p(X)) -> p(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   fact(mark(X)) -> mark(fact(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   zero(mark(X)) -> mark(zero(X))
   s(mark(X)) -> mark(s(X))
   prod(mark(X1), X2) -> mark(prod(X1, X2))
   prod(X1, mark(X2)) -> mark(prod(X1, X2))
   p(mark(X)) -> mark(p(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   proper(fact(X)) -> fact(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(zero(X)) -> zero(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0') -> ok(0')
   proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
   proper(p(X)) -> p(proper(X))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   fact(ok(X)) -> ok(fact(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   zero(ok(X)) -> ok(zero(X))
   s(ok(X)) -> ok(s(X))
   prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
   p(ok(X)) -> ok(p(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(10)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, if, zero, s, prod, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
if < active
zero < active
s < active
prod < active
fact < active
p < active
add < active
active < top
if < proper
zero < proper
s < proper
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
if, active, zero, s, prod, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
if < active
zero < active
s < active
prod < active
fact < active
p < active
add < active
active < top
if < proper
zero < proper
s < proper
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)

Induction Base:
if(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))

Induction Step:
if(gen_0':mark:true:false:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) ->_R^Omega(1)
mark(if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))) ->_IH
mark(*4_0)

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
if, active, zero, s, prod, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
if < active
zero < active
s < active
prod < active
fact < active
p < active
add < active
active < top
if < proper
zero < proper
s < proper
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
zero, active, s, prod, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
zero < active
s < active
prod < active
fact < active
p < active
add < active
active < top
zero < proper
s < proper
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)

Induction Base:
zero(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
zero(gen_0':mark:true:false:ok3_0(+(1, +(n1783_0, 1)))) ->_R^Omega(1)
mark(zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0)))) ->_IH
mark(*4_0)

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

(20)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, prod, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
s < active
prod < active
fact < active
p < active
add < active
active < top
s < proper
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)

Induction Base:
s(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:true:false:ok3_0(+(1, +(n2390_0, 1)))) ->_R^Omega(1)
mark(s(gen_0':mark:true:false:ok3_0(+(1, n2390_0)))) ->_IH
mark(*4_0)

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

(22)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
prod, active, fact, p, add, proper, top

They will be analysed ascendingly in the following order:
prod < active
fact < active
p < active
add < active
active < top
prod < proper
fact < proper
p < proper
add < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n3098_0)

Induction Base:
prod(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))

Induction Step:
prod(gen_0':mark:true:false:ok3_0(+(1, +(n3098_0, 1))), gen_0':mark:true:false:ok3_0(b)) ->_R^Omega(1)
mark(prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b))) ->_IH
mark(*4_0)

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

(24)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)
prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n3098_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
fact, active, p, add, proper, top

They will be analysed ascendingly in the following order:
fact < active
p < active
add < active
active < top
fact < proper
p < proper
add < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fact(gen_0':mark:true:false:ok3_0(+(1, n5210_0))) -> *4_0, rt in Omega(n5210_0)

Induction Base:
fact(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
fact(gen_0':mark:true:false:ok3_0(+(1, +(n5210_0, 1)))) ->_R^Omega(1)
mark(fact(gen_0':mark:true:false:ok3_0(+(1, n5210_0)))) ->_IH
mark(*4_0)

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

(26)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)
prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n3098_0)
fact(gen_0':mark:true:false:ok3_0(+(1, n5210_0))) -> *4_0, rt in Omega(n5210_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
p, active, add, proper, top

They will be analysed ascendingly in the following order:
p < active
add < active
active < top
p < proper
add < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':mark:true:false:ok3_0(+(1, n6169_0))) -> *4_0, rt in Omega(n6169_0)

Induction Base:
p(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
p(gen_0':mark:true:false:ok3_0(+(1, +(n6169_0, 1)))) ->_R^Omega(1)
mark(p(gen_0':mark:true:false:ok3_0(+(1, n6169_0)))) ->_IH
mark(*4_0)

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

(28)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)
prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n3098_0)
fact(gen_0':mark:true:false:ok3_0(+(1, n5210_0))) -> *4_0, rt in Omega(n5210_0)
p(gen_0':mark:true:false:ok3_0(+(1, n6169_0))) -> *4_0, rt in Omega(n6169_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
add, active, proper, top

They will be analysed ascendingly in the following order:
add < active
active < top
add < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':mark:true:false:ok3_0(+(1, n7229_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n7229_0)

Induction Base:
add(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))

Induction Step:
add(gen_0':mark:true:false:ok3_0(+(1, +(n7229_0, 1))), gen_0':mark:true:false:ok3_0(b)) ->_R^Omega(1)
mark(add(gen_0':mark:true:false:ok3_0(+(1, n7229_0)), gen_0':mark:true:false:ok3_0(b))) ->_IH
mark(*4_0)

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

(30)
Obligation:
TRS:
Rules:
active(fact(X)) -> mark(if(zero(X), s(0'), prod(X, fact(p(X)))))
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0', X)) -> mark(0')
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0')) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
s(mark(X)) -> mark(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
p(mark(X)) -> mark(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
fact(ok(X)) -> ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(ok(X)) -> ok(zero(X))
s(ok(X)) -> ok(s(X))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(ok(X)) -> ok(p(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
fact :: 0':mark:true:false:ok -> 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
zero :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
prod :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
p :: 0':mark:true:false:ok -> 0':mark:true:false:ok
add :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n5_0)
zero(gen_0':mark:true:false:ok3_0(+(1, n1783_0))) -> *4_0, rt in Omega(n1783_0)
s(gen_0':mark:true:false:ok3_0(+(1, n2390_0))) -> *4_0, rt in Omega(n2390_0)
prod(gen_0':mark:true:false:ok3_0(+(1, n3098_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n3098_0)
fact(gen_0':mark:true:false:ok3_0(+(1, n5210_0))) -> *4_0, rt in Omega(n5210_0)
p(gen_0':mark:true:false:ok3_0(+(1, n6169_0))) -> *4_0, rt in Omega(n6169_0)
add(gen_0':mark:true:false:ok3_0(+(1, n7229_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n7229_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top