Termination proof of AG_#3.56

Termination of the given RelTRS could be proven:

0 RelTRS

↳1 RelTRStoRelADPProof (⇔, 0 ms)

↳2 RelADPP

↳3 RelADPDepGraphProof (⇔, 0 ms)

↳4 AND

    ↳5 RelADPP

        ↳6 RelADPDerelatifyingProof (⇔, 0 ms)

        ↳7 QDP

        ↳8 MRRProof (⇔, 0 ms)

        ↳9 QDP

        ↳10 MRRProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPOrderProof (⇔, 4 ms)

        ↳13 QDP

        ↳14 PisEmptyProof (⇔, 0 ms)

        ↳15 YES

    ↳16 RelADPP

        ↳17 RelADPCleverAfsProof (⇒, 42 ms)

        ↳18 QDP

        ↳19 MRRProof (⇔, 12 ms)

        ↳20 QDP

        ↳21 MRRProof (⇔, 0 ms)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔, 0 ms)

        ↳24 QDP

        ↳25 PisEmptyProof (⇔, 0 ms)

        ↳26 YES

    ↳27 RelADPP

        ↳28 RelADPCleverAfsProof (⇒, 54 ms)

        ↳29 QDP

        ↳30 MRRProof (⇔, 8 ms)

        ↳31 QDP

        ↳32 PisEmptyProof (⇔, 0 ms)

        ↳33 YES

(0) Obligation:

Relative term rewrite system:
The relative TRS consists of the following R rules:

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

The relative TRS consists of the following S rules:

rand(x) → rand(s(x))
rand(x) → x

(1) RelTRStoRelADPProof (EQUIVALENT transformation)

We upgrade the RelTRS problem to an equivalent Relative ADP Problem [IJCAR24].

(2) Obligation:

Relative ADP Problem with
absolute ADPs:

g(c(x, s(y))) → G(c(s(x), y))
f(c(s(x), y)) → F(c(x, s(y)))
f(f(x)) → F(d(f(x)))
f(f(x)) → f(d(F(x)))
f(x) → x

and relative ADPs:

rand(x) → RAND(s(x))
rand(x) → x

(3) RelADPDepGraphProof (EQUIVALENT transformation)

We use the relative dependency graph processor [IJCAR24].
The approximation of the Relative Dependency Graph contains:
3 SCCs with nodes from P_abs,
0 Lassos,
Result: This relative DT problem is equivalent to 3 subproblems.

(4) Complex Obligation (AND)

(5) Obligation:

Relative ADP Problem with
absolute ADPs:

f(x) → x
g(c(x, s(y))) → G(c(s(x), y))
f(f(x)) → f(d(f(x)))
f(c(s(x), y)) → f(c(x, s(y)))

and relative ADPs:

rand(x) → rand(s(x))
rand(x) → x

(6) RelADPDerelatifyingProof (EQUIVALENT transformation)

We use the first derelatifying processor [IJCAR24].
There are no annotations in relative ADPs, so the relative ADP problem can be transformed into a non-relative DP problem.

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(c(x, s(y))) → G(c(s(x), y))

The TRS R consists of the following rules:

g(c(x, s(y))) → g(c(s(x), y))
f(x) → x
f(f(x)) → f(d(f(x)))
rand(x) → rand(s(x))
f(c(s(x), y)) → f(c(x, s(y)))
rand(x) → x

Q is empty.
We have to consider all (P,Q,R)-chains.

(8) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

rand(x) → x

Used ordering: Polynomial interpretation [POLO]:

POL(G(x₁)) = x₁
POL(c(x₁, x₂)) = 2·x₁ + x₂
POL(d(x₁)) = x₁
POL(f(x₁)) = 2·x₁
POL(g(x₁)) = x₁
POL(rand(x₁)) = 2 + x₁
POL(s(x₁)) = x₁

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(c(x, s(y))) → G(c(s(x), y))

The TRS R consists of the following rules:

g(c(x, s(y))) → g(c(s(x), y))
f(x) → x
f(f(x)) → f(d(f(x)))
rand(x) → rand(s(x))
f(c(s(x), y)) → f(c(x, s(y)))

Q is empty.
We have to consider all (P,Q,R)-chains.

(10) MRRProof (EQUIVALENT transformation)

f(x) → x

Used ordering: Polynomial interpretation [POLO]:

POL(G(x₁)) = x₁
POL(c(x₁, x₂)) = 2·x₁ + x₂
POL(d(x₁)) = x₁
POL(f(x₁)) = 1 + x₁
POL(g(x₁)) = x₁
POL(rand(x₁)) = x₁
POL(s(x₁)) = x₁

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(c(x, s(y))) → G(c(s(x), y))

The TRS R consists of the following rules:

g(c(x, s(y))) → g(c(s(x), y))
f(f(x)) → f(d(f(x)))
rand(x) → rand(s(x))
f(c(s(x), y)) → f(c(x, s(y)))

Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

G(c(x, s(y))) → G(c(s(x), y))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(x1) = G(x1)
c(x1, x2) = c(x2)
s(x1) = s(x1)
g(x1) = g(x1)
f(x1) = f
d(x1) = d
rand(x1) = rand

Recursive path order with status [RPO].
Quasi-Precedence:

g₁ > [G₁, c₁, f] > s₁
rand > s₁

Status:

G₁: [1]
c₁: multiset
s₁: multiset
g₁: [1]
f: multiset
d: multiset
rand: multiset

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

g(c(x, s(y))) → g(c(s(x), y))
f(f(x)) → f(d(f(x)))
rand(x) → rand(s(x))
f(c(s(x), y)) → f(c(x, s(y)))

(13) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

g(c(x, s(y))) → g(c(s(x), y))
f(f(x)) → f(d(f(x)))
rand(x) → rand(s(x))
f(c(s(x), y)) → f(c(x, s(y)))

Q is empty.
We have to consider all (P,Q,R)-chains.

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) YES

(16) Obligation:

Relative ADP Problem with
absolute ADPs:

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → F(c(x, s(y)))
f(x) → x
f(f(x)) → f(d(f(x)))

and relative ADPs:

rand(x) → rand(s(x))
rand(x) → x

(17) RelADPCleverAfsProof (SOUND transformation)

We use the first derelatifying processor [IJCAR24].
There are no annotations in relative ADPs, so the relative ADP problem can be transformed into a non-relative DP problem.

Furthermore, We use an argument filter [LPAR04].
Filtering:s_1 = c_2 = 1 d_1 = 0 f_1 = F_1 = rand_1 = g_1 = 0
Found this filtering by looking at the following order that orders at least one DP strictly:Combined order from the following AFS and order.
F(x1) = x1
c(x1, x2) = c(x1)
s(x1) = s(x1)
g(x1) = g
f(x1) = f(x1)
d(x1) = d

Recursive path order with status [RPO].
Quasi-Precedence:

f₁ > [c₁, s₁, g, d]

Status:

c₁: multiset
s₁: multiset
g: multiset
f₁: multiset
d: multiset

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F0(c(s0(x))) → F0(c(x))

The TRS R consists of the following rules:

g → g
f0(x) → x
f0(f0(x)) → f0(d)
rand0(x) → rand0(s0(x))
f0(c(s0(x))) → f0(c(x))
rand0(x) → x

Q is empty.
We have to consider all (P,Q,R)-chains.

(19) MRRProof (EQUIVALENT transformation)

rand0(x) → x

Used ordering: Polynomial interpretation [POLO]:

POL(F0(x₁)) = x₁
POL(c(x₁)) = x₁
POL(d) = 0
POL(f0(x₁)) = x₁
POL(g) = 0
POL(rand0(x₁)) = 2 + x₁
POL(s0(x₁)) = x₁

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F0(c(s0(x))) → F0(c(x))

The TRS R consists of the following rules:

g → g
f0(x) → x
f0(f0(x)) → f0(d)
rand0(x) → rand0(s0(x))
f0(c(s0(x))) → f0(c(x))

Q is empty.
We have to consider all (P,Q,R)-chains.

(21) MRRProof (EQUIVALENT transformation)

f0(x) → x
f0(f0(x)) → f0(d)

Used ordering: Polynomial interpretation [POLO]:

POL(F0(x₁)) = x₁
POL(c(x₁)) = x₁
POL(d) = 0
POL(f0(x₁)) = 1 + x₁
POL(g) = 0
POL(rand0(x₁)) = x₁
POL(s0(x₁)) = x₁

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F0(c(s0(x))) → F0(c(x))

The TRS R consists of the following rules:

g → g
rand0(x) → rand0(s0(x))
f0(c(s0(x))) → f0(c(x))

Q is empty.
We have to consider all (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

F0(c(s0(x))) → F0(c(x))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F0(x1) = x1
c(x1) = x1
s0(x1) = s0(x1)
g = g
rand0(x1) = rand0
f0(x1) = f0

Knuth-Bendix order [KBO] with precedence:

trivial

and weight map:

s0_1=1
f0=1
rand0=2
g=2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

g → g
rand0(x) → rand0(s0(x))
f0(c(s0(x))) → f0(c(x))

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

g → g
rand0(x) → rand0(s0(x))
f0(c(s0(x))) → f0(c(x))

Q is empty.
We have to consider all (P,Q,R)-chains.

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) YES

(27) Obligation:

Relative ADP Problem with
absolute ADPs:

g(c(x, s(y))) → g(c(s(x), y))
f(x) → x
f(f(x)) → f(d(f(x)))
f(f(x)) → f(d(F(x)))
f(c(s(x), y)) → f(c(x, s(y)))

and relative ADPs:

rand(x) → rand(s(x))
rand(x) → x

(28) RelADPCleverAfsProof (SOUND transformation)

We use the first derelatifying processor [IJCAR24].
There are no annotations in relative ADPs, so the relative ADP problem can be transformed into a non-relative DP problem.

Furthermore, We use an argument filter [LPAR04].
Filtering:s_1 = 0 c_2 = 0 d_1 = 0 f_1 = F_1 = rand_1 = g_1 = 0
Found this filtering by looking at the following order that orders at least one DP strictly:Combined order from the following AFS and order.
F(x1) = F(x1)
f(x1) = f(x1)
g(x1) = g
c(x1, x2) = x2
s(x1) = s
d(x1) = d

Recursive path order with status [RPO].
Quasi-Precedence:

f₁ > F₁ > s
f₁ > d > s
g > s

Status:

F₁: multiset
f₁: [1]
g: multiset
s: multiset
d: multiset

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F0(f0(x)) → F0(x)

The TRS R consists of the following rules:

g → g
f0(x) → x
f0(f0(x)) → f0(d)
rand0(x) → rand0(s)
f0(c(y)) → f0(c(s))
rand0(x) → x

Q is empty.
We have to consider all (P,Q,R)-chains.

(30) MRRProof (EQUIVALENT transformation)

F0(f0(x)) → F0(x)

Strictly oriented rules of the TRS R:

f0(x) → x
f0(f0(x)) → f0(d)
rand0(x) → x

Used ordering: Knuth-Bendix order [KBO] with precedence:

F0₁ > c₁ > f0₁ > d > g > s > rand0₁

and weight map:

g=1
d=2
s=1
f0_1=1
rand0_1=1
c_1=1
F0_1=1

The variable weight is 1

(31) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

g → g
rand0(x) → rand0(s)
f0(c(y)) → f0(c(s))

Q is empty.
We have to consider all (P,Q,R)-chains.

(32) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) RelTRStoRelADPProof (EQUIVALENT transformation)

(2) Obligation:

(3) RelADPDepGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) RelADPDerelatifyingProof (EQUIVALENT transformation)

(7) Obligation:

(8) MRRProof (EQUIVALENT transformation)

(9) Obligation:

(10) MRRProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPOrderProof (EQUIVALENT transformation)

(13) Obligation:

(14) PisEmptyProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) RelADPCleverAfsProof (SOUND transformation)

(18) Obligation:

(19) MRRProof (EQUIVALENT transformation)

(20) Obligation:

(21) MRRProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) PisEmptyProof (EQUIVALENT transformation)

(26) YES

(27) Obligation:

(28) RelADPCleverAfsProof (SOUND transformation)

(29) Obligation:

(30) MRRProof (EQUIVALENT transformation)

(31) Obligation:

(32) PisEmptyProof (EQUIVALENT transformation)

(33) YES