Termination Proof using AProVE

Term Rewriting System R:
[Y, X]
minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS(n_₀, Y) -> 0'
MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(X)
MINUS(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(Y)
GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))
GEQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(X)
GEQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(Y)
DIV(s(X), n_{_s}(Y)) -> IF(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
DIV(s(X), n_{_s}(Y)) -> GEQ(X, activate(Y))
DIV(s(X), n_{_s}(Y)) -> ACTIVATE(Y)
DIV(s(X), n_{_s}(Y)) -> DIV(minus(X, activate(Y)), n_{_s}(activate(Y)))
DIV(s(X), n_{_s}(Y)) -> MINUS(X, activate(Y))
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n_₀) -> 0'
ACTIVATE(n_{_s}(X)) -> S(X)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

Dependency Pair:

MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))

The following usable rules for innermost can be oriented:

activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X
0 -> n_₀
s(X) -> n_{_s}(X)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{n_{_s}, s} > {activate, 0} > n_₀

resulting in one new DP problem.
Used Argument Filtering System:

MINUS(x₁, x₂) -> MINUS(x₁, x₂)
n_{_s}(x₁) -> n_{_s}(x₁)
activate(x₁) -> activate(x₁)
0 -> 0
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

Dependency Pair:

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

Dependency Pair:

GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))

The following usable rules for innermost can be oriented:

activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X
0 -> n_₀
s(X) -> n_{_s}(X)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{n_{_s}, s} > {activate, 0} > n_₀

resulting in one new DP problem.
Used Argument Filtering System:

GEQ(x₁, x₂) -> GEQ(x₁, x₂)
n_{_s}(x₁) -> n_{_s}(x₁)
activate(x₁) -> activate(x₁)
0 -> 0
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

Dependency Pair:

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes