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:

active(p(0)) → mark(0)
active(p(s(X))) → mark(X)
active(leq(0, Y)) → mark(true)
active(leq(s(X), 0)) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

Heuristically decided to analyse the following defined symbols:
active, leq, if, s, diff, p, proper, top

They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
leq < proper
if < proper
s < proper
diff < proper
p < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

The following defined symbols remain to be analysed:
leq, active, if, s, diff, p, proper, top

They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
leq < proper
if < proper
s < proper
diff < proper
p < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

The following defined symbols remain to be analysed:
if, active, s, diff, p, proper, top

They will be analysed ascendingly in the following order:
if < active
s < active
diff < active
p < active
active < top
if < proper
s < proper
diff < proper
p < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

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

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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

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

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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

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

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

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

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

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

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

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

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

The following defined symbols remain to be analysed:
top

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

No more defined symbols left to analyse.