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

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

0(#) → #
+(x, #) → x
+(#, x) → x
+(0(x), 0(y)) → 0(+(x, y))
+(0(x), 1(y)) → 1(+(x, y))
+(1(x), 0(y)) → 1(+(x, y))
+(1(x), 1(y)) → 0(+(+(x, y), 1(#)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +(+(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(1(x), 1(y)) →⁺ 0(+(+(x, y), 1(#)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x / 1(x), y / 1(y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
+', -, ge, min, max, bs, size, wb

They will be analysed ascendingly in the following order:
+' < size
- < wb
ge < bs
ge < wb
min < bs
max < bs
size < wb

(8) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
+', -, ge, min, max, bs, size, wb

They will be analysed ascendingly in the following order:
+' < size
- < wb
ge < bs
ge < wb
min < bs
max < bs
size < wb

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

Induction Base:
+'(gen_#:1:l:n3_2(+(1, 0)), gen_#:1:l:n3_2(+(1, 0)))

Induction Step:
+'(gen_#:1:l:n3_2(+(1, +(n5_2, 1))), gen_#:1:l:n3_2(+(1, +(n5_2, 1)))) →_R^Ω(1)
0(+'(+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))), 1(#))) →_IH
0(+'(*4_2, 1(#)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
-, ge, min, max, bs, size, wb

They will be analysed ascendingly in the following order:
- < wb
ge < bs
ge < wb
min < bs
max < bs
size < wb

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)

Induction Base:
-(gen_#:1:l:n3_2(0), gen_#:1:l:n3_2(0)) →_R^Ω(1)
gen_#:1:l:n3_2(0)

Induction Step:
-(gen_#:1:l:n3_2(+(n365471_2, 1)), gen_#:1:l:n3_2(+(n365471_2, 1))) →_R^Ω(1)
0(-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2))) →_IH
0(gen_#:1:l:n3_2(0)) →_R^Ω(1)
#

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
ge, min, max, bs, size, wb

They will be analysed ascendingly in the following order:
ge < bs
ge < wb
min < bs
max < bs
size < wb

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Induction Base:
ge(gen_#:1:l:n3_2(0), gen_#:1:l:n3_2(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_#:1:l:n3_2(+(n367605_2, 1)), gen_#:1:l:n3_2(+(n367605_2, 1))) →_R^Ω(1)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) →_IH
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
min, max, bs, size, wb

They will be analysed ascendingly in the following order:
min < bs
max < bs
size < wb

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol min.

(19) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
max, bs, size, wb

They will be analysed ascendingly in the following order:
max < bs
size < wb

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol max.

(21) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
bs, size, wb

They will be analysed ascendingly in the following order:
size < wb

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol bs.

(23) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
size, wb

They will be analysed ascendingly in the following order:
size < wb

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol size.

(25) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

The following defined symbols remain to be analysed:
wb

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol wb.

(27) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)
ge(gen_#:1:l:n3_2(n367605_2), gen_#:1:l:n3_2(n367605_2)) → true, rt ∈ Ω(1 + n367605₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)
-(gen_#:1:l:n3_2(n365471_2), gen_#:1:l:n3_2(n365471_2)) → gen_#:1:l:n3_2(0), rt ∈ Ω(1 + n365471₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, #) → x
-(#, x) → #
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(false) → true
not(true) → false
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 1(x)) → false
ge(#, 0(x)) → ge(#, x)
val(l(x)) → x
val(n(x, y, z)) → x
min(l(x)) → x
min(n(x, y, z)) → min(y)
max(l(x)) → x
max(n(x, y, z)) → max(z)
bs(l(x)) → true
bs(n(x, y, z)) → and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) → 1(#)
size(n(x, y, z)) → +'(+'(size(x), size(y)), 1(#))
wb(l(x)) → true
wb(n(x, y, z)) → and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Types:
0 :: #:1:l:n → #:1:l:n
# :: #:1:l:n
+' :: #:1:l:n → #:1:l:n → #:1:l:n
1 :: #:1:l:n → #:1:l:n
- :: #:1:l:n → #:1:l:n → #:1:l:n
not :: false:true → false:true
false :: false:true
true :: false:true
and :: false:true → false:true → false:true
if :: false:true → false:true → false:true → false:true
ge :: #:1:l:n → #:1:l:n → false:true
val :: #:1:l:n → #:1:l:n
l :: #:1:l:n → #:1:l:n
n :: #:1:l:n → #:1:l:n → #:1:l:n → #:1:l:n
min :: #:1:l:n → #:1:l:n
max :: #:1:l:n → #:1:l:n
bs :: #:1:l:n → false:true
size :: #:1:l:n → #:1:l:n
wb :: #:1:l:n → false:true
hole_#:1:l:n1_2 :: #:1:l:n
hole_false:true2_2 :: false:true
gen_#:1:l:n3_2 :: Nat → #:1:l:n

Lemmas:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

Generator Equations:
gen_#:1:l:n3_2(0) ⇔ #
gen_#:1:l:n3_2(+(x, 1)) ⇔ 1(gen_#:1:l:n3_2(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_#:1:l:n3_2(+(1, n5_2)), gen_#:1:l:n3_2(+(1, n5_2))) → *4_2, rt ∈ Ω(n5₂)

(38) BOUNDS(n^1, INF)