Termination proof of /tmp/tmpJ-jEbW/complete1.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoIDPProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

eval(i, j) → Cond_eval(&&(&&(>=@z(-@z(i, j), 1@z), >=@z(nat, 0@z)), >@z(pos, 0@z)), i, j, nat, pos)
Cond_eval(TRUE, i, j, nat, pos) → eval(-@z(i, nat), +@z(j, pos))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval(TRUE, x0, x1, x2, x3)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

eval(i, j) → Cond_eval(&&(&&(>=@z(-@z(i, j), 1@z), >=@z(nat, 0@z)), >@z(pos, 0@z)), i, j, nat, pos)
Cond_eval(TRUE, i, j, nat, pos) → eval(-@z(i, nat), +@z(j, pos))

The integer pair graph contains the following rules and edges:

(0): EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])
(1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) → EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))

(0) -> (1), if ((nat[0] →^* nat[1])∧(pos[0] →^* pos[1])∧(i[0] →^* i[1])∧(j[0] →^* j[1])∧(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)) →^* TRUE))

(1) -> (0), if ((+@z(j[1], pos[1]) →^* j[0])∧(-@z(i[1], nat[1]) →^* i[0]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval(TRUE, x0, x1, x2, x3)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(0): EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])
(1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) → EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))

(0) -> (1), if ((nat[0] →^* nat[1])∧(pos[0] →^* pos[1])∧(i[0] →^* i[1])∧(j[0] →^* j[1])∧(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)) →^* TRUE))

(1) -> (0), if ((+@z(j[1], pos[1]) →^* j[0])∧(-@z(i[1], nat[1]) →^* i[0]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval(TRUE, x0, x1, x2, x3)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair EVAL(i, j) → COND_EVAL(&&(&&(>=@z(-@z(i, j), 1@z), >=@z(nat, 0@z)), >@z(pos, 0@z)), i, j, nat, pos) the following chains were created:

We consider the chain EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0]) which results in the following constraint:
(1)    (EVAL(i[0], j[0])≥NonInfC∧EVAL(i[0], j[0])≥COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])∧(U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)

For Pair COND_EVAL(TRUE, i, j, nat, pos) → EVAL(-@z(i, nat), +@z(j, pos)) the following chains were created:

We consider the chain EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0]), COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) → EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1])), EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0]) which results in the following constraint:
(6)    (&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z))=TRUE∧+@z(j[1], pos[1])=j[0]1∧i[0]=i[1]∧pos[0]=pos[1]∧j[0]=j[1]∧nat[0]=nat[1]∧-@z(i[1], nat[1])=i[0]1 ⇒ COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1])≥NonInfC∧COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1])≥EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(pos[0], 0@z)=TRUE∧>=@z(-@z(i[0], j[0]), 1@z)=TRUE∧>=@z(nat[0], 0@z)=TRUE ⇒ COND_EVAL(TRUE, i[0], j[0], nat[0], pos[0])≥NonInfC∧COND_EVAL(TRUE, i[0], j[0], nat[0], pos[0])≥EVAL(-@z(i[0], nat[0]), +@z(j[0], pos[0]))∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (pos[0] + -1 ≥ 0∧-1 + i[0] + (-1)j[0] ≥ 0∧nat[0] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥)∧-1 + (-1)Bound + (-1)j[0] + i[0] ≥ 0∧-1 + pos[0] + nat[0] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (pos[0] + -1 ≥ 0∧-1 + i[0] + (-1)j[0] ≥ 0∧nat[0] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥)∧-1 + (-1)Bound + (-1)j[0] + i[0] ≥ 0∧-1 + pos[0] + nat[0] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + i[0] + (-1)j[0] ≥ 0∧pos[0] + -1 ≥ 0∧nat[0] ≥ 0 ⇒ -1 + (-1)Bound + (-1)j[0] + i[0] ≥ 0∧-1 + pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (-1 + i[0] + (-1)j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0 ⇒ -1 + (-1)Bound + (-1)j[0] + i[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0 ⇒ (-1)Bound + j[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(13)    (j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0∧i[0] ≥ 0 ⇒ (-1)Bound + j[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

(14)    (j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0∧i[0] ≥ 0 ⇒ (-1)Bound + j[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

To summarize, we get the following constraints P_≥ for the following pairs.

EVAL(i, j) → COND_EVAL(&&(&&(>=@z(-@z(i, j), 1@z), >=@z(nat, 0@z)), >@z(pos, 0@z)), i, j, nat, pos)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL(TRUE, i, j, nat, pos) → EVAL(-@z(i, nat), +@z(j, pos))
- (j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0∧i[0] ≥ 0 ⇒ (-1)Bound + j[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))
- (j[0] ≥ 0∧pos[0] ≥ 0∧nat[0] ≥ 0∧i[0] ≥ 0 ⇒ (-1)Bound + j[0] ≥ 0∧pos[0] + nat[0] ≥ 0∧(U^Increasing(EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))), ≥))

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(0@z) = 0
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(EVAL(x₁, x₂)) = -1 + (-1)x₂ + x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(COND_EVAL(x₁, x₂, x₃, x₄, x₅)) = -1 + (-1)x₃ + x₂ + (-1)x₁
POL(FALSE) = 1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) → EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))

The following pairs are in P_bound:

COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) → EVAL(-@z(i[1], nat[1]), +@z(j[1], pos[1]))

The following pairs are in P_≥:

EVAL(i[0], j[0]) → COND_EVAL(&&(&&(>=@z(-@z(i[0], j[0]), 1@z), >=@z(nat[0], 0@z)), >@z(pos[0], 0@z)), i[0], j[0], nat[0], pos[0])

At least the following rules have been oriented under context sensitive arithmetic replacement:

FALSE¹ → &&(FALSE, FALSE)¹
-@z¹ ↔
&&(TRUE, TRUE)¹ ↔ TRUE¹
+@z¹ ↔
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

eval(x0, x1)
Cond_eval(TRUE, x0, x1, x2, x3)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.