Term Rewriting System R:
[x, y, z, l, l1, l2, l3]
0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(0(x), 0(y)) -> 0'(+(x, y))
+'(0(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(1(x), 1(y)) -> 0'(+(+(x, y), 1(#)))
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 1(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
-'(0(x), 0(y)) -> 0'(-(x, y))
-'(0(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(0(x), 1(y)) -> -'(x, y)
-'(1(x), 0(y)) -> -'(x, y)
-'(1(x), 1(y)) -> 0'(-(x, y))
-'(1(x), 1(y)) -> -'(x, y)
EQ(#, 0(y)) -> EQ(#, y)
EQ(0(x), #) -> EQ(x, #)
EQ(1(x), 1(y)) -> EQ(x, y)
EQ(0(x), 0(y)) -> EQ(x, y)
GE(0(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> NOT(ge(y, x))
GE(0(x), 1(y)) -> GE(y, x)
GE(1(x), 0(y)) -> GE(x, y)
GE(1(x), 1(y)) -> GE(x, y)
GE(#, 0(x)) -> GE(#, x)
LOG(x) -> -'(log'(x), 1(#))
LOG(x) -> LOG'(x)
LOG'(1(x)) -> +'(log'(x), 1(#))
LOG'(1(x)) -> LOG'(x)
LOG'(0(x)) -> IF(ge(x, 1(#)), +(log'(x), 1(#)), #)
LOG'(0(x)) -> GE(x, 1(#))
LOG'(0(x)) -> +'(log'(x), 1(#))
LOG'(0(x)) -> LOG'(x)
*'(0(x), y) -> 0'(*(x, y))
*'(0(x), y) -> *'(x, y)
*'(1(x), y) -> +'(0(*(x, y)), y)
*'(1(x), y) -> 0'(*(x, y))
*'(1(x), y) -> *'(x, y)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(*(x, y), z) -> *'(y, z)
*'(x, +(y, z)) -> +'(*(x, y), *(x, z))
*'(x, +(y, z)) -> *'(x, y)
*'(x, +(y, z)) -> *'(x, z)
APP(cons(x, l1), l2) -> APP(l1, l2)
SUM(nil) -> 0'(#)
SUM(cons(x, l)) -> +'(x, sum(l))
SUM(cons(x, l)) -> SUM(l)
SUM(app(l1, l2)) -> +'(sum(l1), sum(l2))
SUM(app(l1, l2)) -> SUM(l1)
SUM(app(l1, l2)) -> SUM(l2)
PROD(cons(x, l)) -> *'(x, prod(l))
PROD(cons(x, l)) -> PROD(l)
PROD(app(l1, l2)) -> *'(prod(l1), prod(l2))
PROD(app(l1, l2)) -> PROD(l1)
PROD(app(l1, l2)) -> PROD(l2)
MEM(x, cons(y, l)) -> IF(eq(x, y), true, mem(x, l))
MEM(x, cons(y, l)) -> EQ(x, y)
MEM(x, cons(y, l)) -> MEM(x, l)
INTER(app(l1, l2), l3) -> APP(inter(l1, l3), inter(l2, l3))
INTER(app(l1, l2), l3) -> INTER(l1, l3)
INTER(app(l1, l2), l3) -> INTER(l2, l3)
INTER(l1, app(l2, l3)) -> APP(inter(l1, l2), inter(l1, l3))
INTER(l1, app(l2, l3)) -> INTER(l1, l2)
INTER(l1, app(l2, l3)) -> INTER(l1, l3)
INTER(cons(x, l1), l2) -> IFINTER(mem(x, l2), x, l1, l2)
INTER(cons(x, l1), l2) -> MEM(x, l2)
INTER(l1, cons(x, l2)) -> IFINTER(mem(x, l1), x, l2, l1)
INTER(l1, cons(x, l2)) -> MEM(x, l1)
IFINTER(true, x, l1, l2) -> INTER(l1, l2)
IFINTER(false, x, l1, l2) -> INTER(l1, l2)

Furthermore, R contains 14 SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
0(#) -> #

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  x1 POL(1(x1)) =  1 + x1 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 15`
`             ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(0(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

+'(0(x), 0(y)) -> +'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  1 + x1 POL(1(x1)) =  0 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 15`
`             ↳Polo`
`             ...`
`               →DP Problem 16`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  0 POL(1(x1)) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 15`
`             ↳Polo`
`             ...`
`               →DP Problem 17`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Narrowing Transformation`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

-'(1(x), 1(y)) -> -'(x, y)
-'(1(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(0(x), 0(y)) -> -'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
six new Dependency Pairs are created:

-'(0(#), 1(y')) -> -'(#, 1(#))
-'(0(x''), 1(#)) -> -'(x'', 1(#))
-'(0(0(x'')), 1(0(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(0(0(x'')), 1(1(y''))) -> -'(1(-(-(x'', y''), 1(#))), 1(#))
-'(0(1(x'')), 1(0(y''))) -> -'(1(-(x'', y'')), 1(#))
-'(0(1(x'')), 1(1(y''))) -> -'(0(-(x'', y'')), 1(#))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 18`
`             ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

-'(0(1(x'')), 1(1(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(0(1(x'')), 1(0(y''))) -> -'(1(-(x'', y'')), 1(#))
-'(0(0(x'')), 1(1(y''))) -> -'(1(-(-(x'', y''), 1(#))), 1(#))
-'(0(0(x'')), 1(0(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(0(x''), 1(#)) -> -'(x'', 1(#))
-'(1(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(x, y)
-'(0(x), 0(y)) -> -'(x, y)
-'(1(x), 1(y)) -> -'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

-'(0(1(x'')), 1(0(y''))) -> -'(1(-(x'', y'')), 1(#))
-'(0(0(x'')), 1(0(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(1(x), 0(y)) -> -'(x, y)
-'(0(x), 0(y)) -> -'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  1 + x1 POL(-'(x1, x2)) =  x2 POL(1(x1)) =  x1 POL(-(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 18`
`             ↳Polo`
`             ...`
`               →DP Problem 19`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

-'(0(1(x'')), 1(1(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(0(0(x'')), 1(1(y''))) -> -'(1(-(-(x'', y''), 1(#))), 1(#))
-'(0(x''), 1(#)) -> -'(x'', 1(#))
-'(0(x), 1(y)) -> -'(x, y)
-'(1(x), 1(y)) -> -'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

-'(0(1(x'')), 1(1(y''))) -> -'(0(-(x'', y'')), 1(#))
-'(0(0(x'')), 1(1(y''))) -> -'(1(-(-(x'', y''), 1(#))), 1(#))
-'(0(x), 1(y)) -> -'(x, y)
-'(1(x), 1(y)) -> -'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  0 POL(-'(x1, x2)) =  x2 POL(1(x1)) =  1 + x1 POL(-(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 18`
`             ↳Polo`
`             ...`
`               →DP Problem 20`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

-'(0(x''), 1(#)) -> -'(x'', 1(#))

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

-'(0(x''), 1(#)) -> -'(x'', 1(#))

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(-'(x1, x2)) =  x1 POL(0(x1)) =  1 + x1 POL(1(x1)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 18`
`             ↳Polo`
`             ...`
`               →DP Problem 21`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

EQ(#, 0(y)) -> EQ(#, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

EQ(#, 0(y)) -> EQ(#, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(EQ(x1, x2)) =  x2 POL(0(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 22`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

EQ(0(x), #) -> EQ(x, #)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

EQ(0(x), #) -> EQ(x, #)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(EQ(x1, x2)) =  x1 POL(0(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 23`
`             ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polynomial Ordering`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

GE(#, 0(x)) -> GE(#, x)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

GE(#, 0(x)) -> GE(#, x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  1 + x1 POL(GE(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Dependency Graph`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polynomial Ordering`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

APP(cons(x, l1), l2) -> APP(l1, l2)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

APP(cons(x, l1), l2) -> APP(l1, l2)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(cons(x1, x2)) =  1 + x2 POL(APP(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`           →DP Problem 25`
`             ↳Dependency Graph`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(1(x), y) -> *'(x, y)
*'(0(x), y) -> *'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

*'(0(x), y) -> *'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  1 + x1 POL(*'(x1, x2)) =  x1 POL(1(x1)) =  x1 POL(*(x1, x2)) =  x1 + x2 POL(+(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 26`
`             ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(1(x), y) -> *'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(0(x1)) =  0 POL(*'(x1, x2)) =  x1 POL(1(x1)) =  x1 POL(*(x1, x2)) =  1 + x1 + x2 POL(+(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 26`
`             ↳Polo`
`             ...`
`               →DP Problem 27`
`                 ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
*'(1(x), y) -> *'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

*'(1(x), y) -> *'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x1 POL(1(x1)) =  1 + x1 POL(+(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 26`
`             ↳Polo`
`             ...`
`               →DP Problem 28`
`                 ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x2 POL(+(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 26`
`             ↳Polo`
`             ...`
`               →DP Problem 29`
`                 ↳Dependency Graph`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

SUM(app(l1, l2)) -> SUM(l2)
SUM(app(l1, l2)) -> SUM(l1)
SUM(cons(x, l)) -> SUM(l)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

SUM(app(l1, l2)) -> SUM(l2)
SUM(app(l1, l2)) -> SUM(l1)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(SUM(x1)) =  x1 POL(cons(x1, x2)) =  x2 POL(app(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 30`
`             ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

SUM(cons(x, l)) -> SUM(l)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

SUM(cons(x, l)) -> SUM(l)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(SUM(x1)) =  x1 POL(cons(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 30`
`             ↳Polo`
`             ...`
`               →DP Problem 31`
`                 ↳Dependency Graph`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polynomial Ordering`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

EQ(0(x), 0(y)) -> EQ(x, y)
EQ(1(x), 1(y)) -> EQ(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

EQ(0(x), 0(y)) -> EQ(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(EQ(x1, x2)) =  x1 POL(0(x1)) =  1 + x1 POL(1(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`           →DP Problem 32`
`             ↳Polynomial Ordering`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

EQ(1(x), 1(y)) -> EQ(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

EQ(1(x), 1(y)) -> EQ(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(EQ(x1, x2)) =  x1 POL(1(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`           →DP Problem 32`
`             ↳Polo`
`             ...`
`               →DP Problem 33`
`                 ↳Dependency Graph`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polynomial Ordering`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

GE(1(x), 1(y)) -> GE(x, y)
GE(1(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> GE(y, x)
GE(0(x), 0(y)) -> GE(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

GE(1(x), 1(y)) -> GE(x, y)
GE(1(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> GE(y, x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0(x1)) =  x1 POL(GE(x1, x2)) =  1 + x1 + x2 POL(1(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`           →DP Problem 34`
`             ↳Polynomial Ordering`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

GE(0(x), 0(y)) -> GE(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

GE(0(x), 0(y)) -> GE(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0(x1)) =  1 + x1 POL(GE(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`           →DP Problem 34`
`             ↳Polo`
`             ...`
`               →DP Problem 35`
`                 ↳Dependency Graph`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polynomial Ordering`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

PROD(app(l1, l2)) -> PROD(l2)
PROD(app(l1, l2)) -> PROD(l1)
PROD(cons(x, l)) -> PROD(l)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

PROD(app(l1, l2)) -> PROD(l2)
PROD(app(l1, l2)) -> PROD(l1)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(cons(x1, x2)) =  x2 POL(app(x1, x2)) =  1 + x1 + x2 POL(PROD(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`           →DP Problem 36`
`             ↳Polynomial Ordering`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

PROD(cons(x, l)) -> PROD(l)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

PROD(cons(x, l)) -> PROD(l)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(cons(x1, x2)) =  1 + x2 POL(PROD(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`           →DP Problem 36`
`             ↳Polo`
`             ...`
`               →DP Problem 37`
`                 ↳Dependency Graph`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polynomial Ordering`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

MEM(x, cons(y, l)) -> MEM(x, l)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

MEM(x, cons(y, l)) -> MEM(x, l)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MEM(x1, x2)) =  x2 POL(cons(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`           →DP Problem 38`
`             ↳Dependency Graph`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polynomial Ordering`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pairs:

LOG'(0(x)) -> LOG'(x)
LOG'(1(x)) -> LOG'(x)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

LOG'(0(x)) -> LOG'(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LOG'(x1)) =  x1 POL(0(x1)) =  1 + x1 POL(1(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`           →DP Problem 39`
`             ↳Polynomial Ordering`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

LOG'(1(x)) -> LOG'(x)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pair can be strictly oriented:

LOG'(1(x)) -> LOG'(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LOG'(x1)) =  x1 POL(1(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`           →DP Problem 39`
`             ↳Polo`
`             ...`
`               →DP Problem 40`
`                 ↳Dependency Graph`
`       →DP Problem 14`
`         ↳Polo`

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polynomial Ordering`

Dependency Pairs:

IFINTER(false, x, l1, l2) -> INTER(l1, l2)
INTER(l1, cons(x, l2)) -> IFINTER(mem(x, l1), x, l2, l1)
IFINTER(true, x, l1, l2) -> INTER(l1, l2)
INTER(cons(x, l1), l2) -> IFINTER(mem(x, l2), x, l1, l2)
INTER(l1, app(l2, l3)) -> INTER(l1, l3)
INTER(l1, app(l2, l3)) -> INTER(l1, l2)
INTER(app(l1, l2), l3) -> INTER(l2, l3)
INTER(app(l1, l2), l3) -> INTER(l1, l3)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

INTER(l1, app(l2, l3)) -> INTER(l1, l3)
INTER(l1, app(l2, l3)) -> INTER(l1, l2)
INTER(app(l1, l2), l3) -> INTER(l2, l3)
INTER(app(l1, l2), l3) -> INTER(l1, l3)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(false) =  0 POL(true) =  0 POL(IFINTER(x1, x2, x3, x4)) =  x3 + x4 POL(INTER(x1, x2)) =  x1 + x2 POL(#) =  0 POL(if(x1, x2, x3)) =  0 POL(mem(x1, x2)) =  0 POL(eq(x1, x2)) =  0 POL(0(x1)) =  0 POL(cons(x1, x2)) =  x2 POL(1(x1)) =  0 POL(nil) =  0 POL(app(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`
`           →DP Problem 41`
`             ↳Polynomial Ordering`

Dependency Pairs:

IFINTER(false, x, l1, l2) -> INTER(l1, l2)
INTER(l1, cons(x, l2)) -> IFINTER(mem(x, l1), x, l2, l1)
IFINTER(true, x, l1, l2) -> INTER(l1, l2)
INTER(cons(x, l1), l2) -> IFINTER(mem(x, l2), x, l1, l2)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

The following dependency pairs can be strictly oriented:

INTER(l1, cons(x, l2)) -> IFINTER(mem(x, l1), x, l2, l1)
INTER(cons(x, l1), l2) -> IFINTER(mem(x, l2), x, l1, l2)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(#) =  0 POL(if(x1, x2, x3)) =  0 POL(mem(x1, x2)) =  0 POL(eq(x1, x2)) =  0 POL(0(x1)) =  0 POL(false) =  0 POL(cons(x1, x2)) =  1 + x2 POL(1(x1)) =  0 POL(nil) =  0 POL(true) =  0 POL(IFINTER(x1, x2, x3, x4)) =  x3 + x4 POL(INTER(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Nar`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Polo`
`       →DP Problem 10`
`         ↳Polo`
`       →DP Problem 11`
`         ↳Polo`
`       →DP Problem 12`
`         ↳Polo`
`       →DP Problem 13`
`         ↳Polo`
`       →DP Problem 14`
`         ↳Polo`
`           →DP Problem 41`
`             ↳Polo`
`             ...`
`               →DP Problem 42`
`                 ↳Dependency Graph`

Dependency Pairs:

IFINTER(false, x, l1, l2) -> INTER(l1, l2)
IFINTER(true, x, l1, l2) -> INTER(l1, l2)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
-(#, x) -> #
-(x, #) -> x
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(true) -> false
not(false) -> true
if(true, x, y) -> x
if(false, x, y) -> y
eq(#, #) -> true
eq(#, 1(y)) -> false
eq(1(x), #) -> false
eq(#, 0(y)) -> eq(#, y)
eq(0(x), #) -> eq(x, #)
eq(1(x), 1(y)) -> eq(x, y)
eq(0(x), 1(y)) -> false
eq(1(x), 0(y)) -> false
eq(0(x), 0(y)) -> eq(x, y)
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 0(x)) -> ge(#, x)
ge(#, 1(x)) -> false
log(x) -> -(log'(x), 1(#))
log'(#) -> #
log'(1(x)) -> +(log'(x), 1(#))
log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))
mem(x, nil) -> false
mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes