KILLED
Runtime Complexity (full) proof of /tmp/tmpOfthm8/PEANO_complete_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
0 CpxTRS
↳1 DecreasingLoopProof (⇔, 5418 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 0 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 347 ms)
↳10 BEST
↳11 typed CpxTrs
↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
↳13 typed CpxTrs
↳14 RewriteLemmaProof (LOWER BOUND(ID), 53 ms)
↳15 BEST
↳16 typed CpxTrs
↳17 RewriteLemmaProof (LOWER BOUND(ID), 116 ms)
↳18 BEST
↳19 typed CpxTrs
↳20 RewriteLemmaProof (LOWER BOUND(ID), 88 ms)
↳21 BEST
↳22 typed CpxTrs
↳23 RewriteLemmaProof (LOWER BOUND(ID), 85 ms)
↳24 BEST
↳25 typed CpxTrs
↳26 RewriteLemmaProof (LOWER BOUND(ID), 151 ms)
↳27 BEST
↳28 typed CpxTrs
↳29 RewriteLemmaProof (LOWER BOUND(ID), 56 ms)
↳30 BEST
↳31 typed CpxTrs
↳32 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
↳33 typed CpxTrs
↳34 RewriteLemmaProof (LOWER BOUND(ID), 38 ms)
↳35 BEST
↳36 typed CpxTrs
↳37 RewriteLemmaProof (LOWER BOUND(ID), 18 ms)
↳38 BEST
↳39 typed CpxTrs
↳40 RewriteLemmaProof (LOWER BOUND(ID), 101 ms)
↳41 BEST
↳42 typed CpxTrs
↳43 typed CpxTrs
↳44 typed CpxTrs
↳45 typed CpxTrs
↳46 typed CpxTrs
↳47 typed CpxTrs
↳48 typed CpxTrs
↳49 typed CpxTrs
↳50 typed CpxTrs
↳51 typed CpxTrs
↳52 typed CpxTrs
(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0) → ok(0)
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
U11(mark(X1), X2, X3) →+ mark(U11(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, topThey will be analysed ascendingly in the following order:
U12 < active
isNat < active
U13 < active
U22 < active
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
U12 < proper
isNat < proper
U13 < proper
U22 < proper
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U12, active, isNat, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
U12 < active
isNat < active
U13 < active
U22 < active
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
U12 < proper
isNat < proper
U13 < proper
U22 < proper
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Induction Base:
U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
isNat, active, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
isNat < active
U13 < active
U22 < active
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
isNat < proper
U13 < proper
U22 < proper
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNat.
(13) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U13, active, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
U13 < active
U22 < active
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
U13 < proper
U22 < proper
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)Induction Base:
U13(gen_tt:mark:0':ok3_0(+(1, 0)))
Induction Step:
U13(gen_tt:mark:0':ok3_0(+(1, +(n1343_0, 1)))) →RΩ(1)
mark(U13(gen_tt:mark:0':ok3_0(+(1, n1343_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(15) Complex Obligation (BEST)
(16) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U22, active, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
U22 < active
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
U22 < proper
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)Induction Base:
U22(gen_tt:mark:0':ok3_0(+(1, 0)))
Induction Step:
U22(gen_tt:mark:0':ok3_0(+(1, +(n1958_0, 1)))) →RΩ(1)
mark(U22(gen_tt:mark:0':ok3_0(+(1, n1958_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(18) Complex Obligation (BEST)
(19) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
s, active, plus, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
s < active
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
s < proper
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)Induction Base:
s(gen_tt:mark:0':ok3_0(+(1, 0)))
Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n2674_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n2674_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(21) Complex Obligation (BEST)
(22) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
plus, active, U11, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
plus < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
plus < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)Induction Base:
plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
plus(gen_tt:mark:0':ok3_0(+(1, +(n3491_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(24) Complex Obligation (BEST)
(25) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U11, active, and, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)Induction Base:
U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
Induction Step:
U11(gen_tt:mark:0':ok3_0(+(1, +(n5941_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(27) Complex Obligation (BEST)
(28) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
and, active, isNatKind, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)Induction Base:
and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
and(gen_tt:mark:0':ok3_0(+(1, +(n10087_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(30) Complex Obligation (BEST)
(31) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
isNatKind, active, U21, U31, U41, proper, top
They will be analysed ascendingly in the following order:
isNatKind < active
U21 < active
U31 < active
U41 < active
active < top
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
proper < top
(32) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNatKind.
(33) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U21, active, U31, U41, proper, top
They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
active < top
U21 < proper
U31 < proper
U41 < proper
proper < top
(34) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)Induction Base:
U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
U21(gen_tt:mark:0':ok3_0(+(1, +(n13079_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(35) Complex Obligation (BEST)
(36) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U31, active, U41, proper, top
They will be analysed ascendingly in the following order:
U31 < active
U41 < active
active < top
U31 < proper
U41 < proper
proper < top
(37) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n163400)Induction Base:
U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
U31(gen_tt:mark:0':ok3_0(+(1, +(n16340_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(38) Complex Obligation (BEST)
(39) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n163400)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
U41, active, proper, top
They will be analysed ascendingly in the following order:
U41 < active
active < top
U41 < proper
proper < top
(40) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U41(gen_tt:mark:0':ok3_0(+(1, n19905_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n199050)Induction Base:
U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
Induction Step:
U41(gen_tt:mark:0':ok3_0(+(1, +(n19905_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U41(gen_tt:mark:0':ok3_0(+(1, n19905_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(41) Complex Obligation (BEST)
(42) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n163400)
U41(gen_tt:mark:0':ok3_0(+(1, n19905_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n199050)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':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
(43) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n163400)
U41(gen_tt:mark:0':ok3_0(+(1, n19905_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n199050)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(44) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
U31(gen_tt:mark:0':ok3_0(+(1, n16340_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n163400)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(45) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
U21(gen_tt:mark:0':ok3_0(+(1, n13079_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n130790)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(46) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
and(gen_tt:mark:0':ok3_0(+(1, n10087_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n100870)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(47) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
U11(gen_tt:mark:0':ok3_0(+(1, n5941_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n59410)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(48) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
plus(gen_tt:mark:0':ok3_0(+(1, n3491_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n34910)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(49) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
s(gen_tt:mark:0':ok3_0(+(1, n2674_0))) → *4_0, rt ∈ Ω(n26740)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(50) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
U22(gen_tt:mark:0':ok3_0(+(1, n1958_0))) → *4_0, rt ∈ Ω(n19580)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(51) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1343_0))) → *4_0, rt ∈ Ω(n13430)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(52) Obligation:
TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0')) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0') → ok(0')
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
U11 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
U12 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: tt:mark:0':ok → tt:mark:0':ok
U21 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U22 :: tt:mark:0':ok → tt:mark:0':ok
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
isNatKind :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.