Termination Proof using AProVE

Term Rewriting System R:
[N, M, NzN, NzM]
p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

+'(s(N), s(M)) -> +'(N, M)
*'(s(N), s(M)) -> +'(N, +(M, *(N, M)))
*'(s(N), s(M)) -> +'(M, *(N, M))
*'(s(N), s(M)) -> *'(N, M)
GT(NzN, 0) -> U₄(is_NzNat(NzN))
GT(NzN, 0) -> IS_NZNAT(NzN)
GT(s(N), s(M)) -> GT(N, M)
LT(N, M) -> GT(M, N)
D(s(N), s(M)) -> D(N, M)
QUOT(N, NzM) -> U₁₁(is_NzNat(NzM), N, NzM)
QUOT(N, NzM) -> IS_NZNAT(NzM)
QUOT(NzM, NzM) -> U₀₁(is_NzNat(NzM))
QUOT(NzM, NzM) -> IS_NZNAT(NzM)
QUOT(N, NzM) -> U₂₁(is_NzNat(NzM), NzM, N)
U₁₁(True, N, NzM) -> U₁(gt(N, NzM), N, NzM)
U₁₁(True, N, NzM) -> GT(N, NzM)
U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)
U₁(True, N, NzM) -> D(N, NzM)
U₂₁(True, NzM, N) -> U₂(gt(NzM, N))
U₂₁(True, NzM, N) -> GT(NzM, N)
GCD(NzM, NzM) -> U₀₂(is_NzNat(NzM), NzM)
GCD(NzM, NzM) -> IS_NZNAT(NzM)
GCD(NzN, NzM) -> U₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
GCD(NzN, NzM) -> IS_NZNAT(NzN)
GCD(NzN, NzM) -> IS_NZNAT(NzM)
U₃₁(True, True, NzN, NzM) -> U₃(gt(NzN, NzM), NzN, NzM)
U₃₁(True, True, NzN, NzM) -> GT(NzN, NzM)
U₃(True, NzN, NzM) -> GCD(d(NzN, NzM), NzM)
U₃(True, NzN, NzM) -> D(NzN, NzM)

Furthermore, R contains six SCCs.

     ↳DPs

       →DP Problem 1

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

+'(s(N), s(M)) -> +'(N, M)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:

innermost

As we are in the innermost case, we can delete all 28 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 7

             ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

+'(s(N), s(M)) -> +'(N, M)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

+'(s(N), s(M)) -> +'(N, M)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

GT(s(N), s(M)) -> GT(N, M)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:

innermost

As we are in the innermost case, we can delete all 28 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 8

             ↳Size-Change Principle

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

GT(s(N), s(M)) -> GT(N, M)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

GT(s(N), s(M)) -> GT(N, M)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Usable Rules (Innermost)

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

D(s(N), s(M)) -> D(N, M)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:

innermost

As we are in the innermost case, we can delete all 28 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 9

             ↳Size-Change Principle

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

D(s(N), s(M)) -> D(N, M)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

D(s(N), s(M)) -> D(N, M)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳Usable Rules (Innermost)

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

*'(s(N), s(M)) -> *'(N, M)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:

innermost

As we are in the innermost case, we can delete all 28 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

           →DP Problem 10

             ↳Size-Change Principle

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining

Dependency Pair:

*'(s(N), s(M)) -> *'(N, M)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

*'(s(N), s(M)) -> *'(N, M)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳Usable Rules (Innermost)

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)
U₁₁(True, N, NzM) -> U₁(gt(N, NzM), N, NzM)
QUOT(N, NzM) -> U₁₁(is_NzNat(NzM), N, NzM)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:

innermost

As we are in the innermost case, we can delete all 20 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 11

             ↳Narrowing Transformation

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)
U₁₁(True, N, NzM) -> U₁(gt(N, NzM), N, NzM)
QUOT(N, NzM) -> U₁₁(is_NzNat(NzM), N, NzM)

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:

innermost

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

U₁₁(True, N, NzM) -> U₁(gt(N, NzM), N, NzM)

three new Dependency Pairs are created:

U₁₁(True, N', 0) -> U₁(u₄(is_NzNat(N')), N', 0)
U₁₁(True, 0, NzM') -> U₁(False, 0, NzM')
U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 11

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))
U₁₁(True, N', 0) -> U₁(u₄(is_NzNat(N')), N', 0)
QUOT(N, NzM) -> U₁₁(is_NzNat(NzM), N, NzM)
U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:

innermost

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

QUOT(N, NzM) -> U₁₁(is_NzNat(NzM), N, NzM)

two new Dependency Pairs are created:

QUOT(N, 0) -> U₁₁(False, N, 0)
QUOT(N, s(N'')) -> U₁₁(True, N, s(N''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 11

             ↳Nar

...

               →DP Problem 13

                 ↳Instantiation Transformation

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

QUOT(N, s(N'')) -> U₁₁(True, N, s(N''))
U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)
U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:

innermost

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

U₁(True, N, NzM) -> QUOT(d(N, NzM), NzM)

one new Dependency Pair is created:

U₁(True, s(N''''), s(M''')) -> QUOT(d(s(N''''), s(M''')), s(M'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 11

             ↳Nar

...

               →DP Problem 14

                 ↳Rewriting Transformation

       →DP Problem 6

         ↳Remaining

Dependency Pairs:

U₁(True, s(N''''), s(M''')) -> QUOT(d(s(N''''), s(M''')), s(M'''))
U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))
QUOT(N, s(N'')) -> U₁₁(True, N, s(N''))

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:

innermost

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

U₁(True, s(N''''), s(M''')) -> QUOT(d(s(N''''), s(M''')), s(M'''))

one new Dependency Pair is created:

U₁(True, s(N''''), s(M''')) -> QUOT(d(N'''', M'''), s(M'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
QUOT(N, s(N'')) -> U₁₁(True, N, s(N''))
U₁(True, s(N''''), s(M''')) -> QUOT(d(N'''', M'''), s(M'''))
U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:
innermost
Dependency Pairs:
U₃(True, NzN, NzM) -> GCD(d(NzN, NzM), NzM)
U₃₁(True, True, NzN, NzM) -> U₃(gt(NzN, NzM), NzN, NzM)
GCD(NzN, NzM) -> U₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
QUOT(N, s(N'')) -> U₁₁(True, N, s(N''))
U₁(True, s(N''''), s(M''')) -> QUOT(d(N'''', M'''), s(M'''))
U₁₁(True, s(N''), s(M')) -> U₁(gt(N'', M'), s(N''), s(M'))

Rules:

d(s(N), s(M)) -> d(N, M)
d(0, N) -> N
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
u₄(True) -> True
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(0, M) -> False
gt(s(N), s(M)) -> gt(N, M)

Strategy:
innermost
Dependency Pairs:
U₃(True, NzN, NzM) -> GCD(d(NzN, NzM), NzM)
U₃₁(True, True, NzN, NzM) -> U₃(gt(NzN, NzM), NzN, NzM)
GCD(NzN, NzM) -> U₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)

Rules:

p(s(N)) -> N
+(N, 0) -> N
+(s(N), s(M)) -> s(s(+(N, M)))
*(N, 0) -> 0
*(s(N), s(M)) -> s(+(N, +(M, *(N, M))))
gt(0, M) -> False
gt(NzN, 0) -> u₄(is_NzNat(NzN))
gt(s(N), s(M)) -> gt(N, M)
u₄(True) -> True
is_NzNat(0) -> False
is_NzNat(s(N)) -> True
lt(N, M) -> gt(M, N)
d(0, N) -> N
d(s(N), s(M)) -> d(N, M)
quot(N, NzM) -> u₁₁(is_NzNat(NzM), N, NzM)
quot(NzM, NzM) -> u₀₁(is_NzNat(NzM))
quot(N, NzM) -> u₂₁(is_NzNat(NzM), NzM, N)
u₁₁(True, N, NzM) -> u₁(gt(N, NzM), N, NzM)
u₁(True, N, NzM) -> s(quot(d(N, NzM), NzM))
u₀₁(True) -> s(0)
u₂₁(True, NzM, N) -> u₂(gt(NzM, N))
u₂(True) -> 0
gcd(0, N) -> 0
gcd(NzM, NzM) -> u₀₂(is_NzNat(NzM), NzM)
gcd(NzN, NzM) -> u₃₁(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u₀₂(True, NzM) -> NzM
u₃₁(True, True, NzN, NzM) -> u₃(gt(NzN, NzM), NzN, NzM)
u₃(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM)

Strategy:
innermost

The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:00 minutes