Termination Proof using AProVE

Term Rewriting System R:
[x, y, n, m]
eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

EQ(s(x), s(y)) -> EQ(x, y)
LE(s(x), s(y)) -> LE(x, y)
APP(add(n, x), y) -> APP(x, y)
MIN(add(n, add(m, x))) -> IF_MIN(le(n, m), add(n, add(m, x)))
MIN(add(n, add(m, x))) -> LE(n, m)
IF_MIN(true, add(n, add(m, x))) -> MIN(add(n, x))
IF_MIN(false, add(n, add(m, x))) -> MIN(add(m, x))
RM(n, add(m, x)) -> IF_RM(eq(n, m), n, add(m, x))
RM(n, add(m, x)) -> EQ(n, m)
IF_RM(true, n, add(m, x)) -> RM(n, x)
IF_RM(false, n, add(m, x)) -> RM(n, x)
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)
MINSORT(add(n, x), y) -> EQ(n, min(add(n, x)))
MINSORT(add(n, x), y) -> MIN(add(n, x))
IF_MINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
IF_MINSORT(true, add(n, x), y) -> APP(rm(n, x), y)
IF_MINSORT(true, add(n, x), y) -> RM(n, x)
IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))

Furthermore, R contains six SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

EQ(s(x), s(y)) -> EQ(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

EQ(s(x), s(y)) -> EQ(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

EQ(x₁, x₂) -> EQ(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

LE(s(x), s(y)) -> LE(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

LE(s(x), s(y)) -> LE(x, y)

LE(x₁, x₂) -> LE(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

APP(add(n, x), y) -> APP(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(add(n, x), y) -> APP(x, y)

APP(x₁, x₂) -> APP(x₁, x₂)
add(x₁, x₂) -> add(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

IF_RM(false, n, add(m, x)) -> RM(n, x)
IF_RM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IF_RM(eq(n, m), n, add(m, x))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IF_RM(false, n, add(m, x)) -> RM(n, x)
IF_RM(true, n, add(m, x)) -> RM(n, x)

IF_RM(x₁, x₂, x₃) -> x₃
add(x₁, x₂) -> add(x₁, x₂)
RM(x₁, x₂) -> x₂

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 10

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining

Dependency Pair:

RM(n, add(m, x)) -> IF_RM(eq(n, m), n, add(m, x))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

IF_MIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IF_MIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IF_MIN(le(n, m), add(n, add(m, x)))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IF_MIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IF_MIN(true, add(n, add(m, x))) -> MIN(add(n, x))

MIN(x₁) -> x₁
add(x₁, x₂) -> add(x₁, x₂)
IF_MIN(x₁, x₂) -> x₂

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 11

             ↳Dependency Graph

       →DP Problem 6

         ↳Remaining

Dependency Pair:

MIN(add(n, add(m, x))) -> IF_MIN(le(n, m), add(n, add(m, x)))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
IF_MINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes