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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 269 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 320 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 88 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 2932 ms]
        (22) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus_active(0, y) -> 0
   mark(0) -> 0
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0) -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0, s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0, s(y)) -> 0
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
   encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
   encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
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:

   minus_active(0, y) -> 0
   mark(0) -> 0
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0) -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0, s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0, s(y)) -> 0
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
   encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
   encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
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:

   minus_active(0, y) -> 0
   mark(0) -> 0
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0) -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0, s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0, s(y)) -> 0
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
   encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
   encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(6)
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:

   minus_active(0', y) -> 0'
   mark(0') -> 0'
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0') -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0', s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0', s(y)) -> 0'
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
   encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
   encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus_active, mark, ge_active, div_active, if_active, encArg

They will be analysed ascendingly in the following order:
minus_active < mark
minus_active < encArg
ge_active < mark
mark = div_active
mark = if_active
mark < encArg
ge_active < div_active
ge_active < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active


Generator Equations:
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(x))


The following defined symbols remain to be analysed:
minus_active, mark, ge_active, div_active, if_active, encArg

They will be analysed ascendingly in the following order:
minus_active < mark
minus_active < encArg
ge_active < mark
mark = div_active
mark = if_active
mark < encArg
ge_active < div_active
ge_active < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), rt in Omega(1 + n4_4)

Induction Base:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0)) ->_R^Omega(1)
0'

Induction Step:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n4_4, 1)), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n4_4, 1))) ->_R^Omega(1)
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4)) ->_IH
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0)

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active


Generator Equations:
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(x))


The following defined symbols remain to be analysed:
minus_active, mark, ge_active, div_active, if_active, encArg

They will be analysed ascendingly in the following order:
minus_active < mark
minus_active < encArg
ge_active < mark
mark = div_active
mark = if_active
mark < encArg
ge_active < div_active
ge_active < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active


Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(x))


The following defined symbols remain to be analysed:
ge_active, mark, div_active, if_active, encArg

They will be analysed ascendingly in the following order:
ge_active < mark
mark = div_active
mark = if_active
mark < encArg
ge_active < div_active
ge_active < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4)) -> true, rt in Omega(1 + n1376_4)

Induction Base:
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0)) ->_R^Omega(1)
true

Induction Step:
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n1376_4, 1)), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n1376_4, 1))) ->_R^Omega(1)
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4)) ->_IH
true

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

(18)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active


Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), rt in Omega(1 + n4_4)
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4)) -> true, rt in Omega(1 + n1376_4)


Generator Equations:
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(x))


The following defined symbols remain to be analysed:
div_active, mark, if_active, encArg

They will be analysed ascendingly in the following order:
mark = div_active
mark = if_active
mark < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n3987_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n3987_4), rt in Omega(1 + n3987_4)

Induction Base:
mark(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n3987_4, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n3987_4))) ->_IH
s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(c3988_4))

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

(20)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(false) -> false
encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2))
encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2))
encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))
encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2))
encode_true -> true
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_false -> false
encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
0' :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encArg :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
cons_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_0 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_mark :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_s :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_true :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_minus :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_false :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_ge :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_div_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
encode_if_active :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
hole_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active1_4 :: 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4 :: Nat -> 0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active


Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n4_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0), rt in Omega(1 + n4_4)
ge_active(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4), gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n1376_4)) -> true, rt in Omega(1 + n1376_4)
mark(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n3987_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n3987_4), rt in Omega(1 + n3987_4)


Generator Equations:
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(x))


The following defined symbols remain to be analysed:
div_active, if_active, encArg

They will be analysed ascendingly in the following order:
mark = div_active
mark = if_active
mark < encArg
div_active = if_active
div_active < encArg
if_active < encArg

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n7845_4)) -> gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n7845_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(+(n7845_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(n7845_4))) ->_IH
s(gen_0':s:true:minus:false:ge:div:if:cons_minus_active:cons_mark:cons_ge_active:cons_div_active:cons_if_active2_4(c7846_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)