### (0) Obligation:

JBC Problem based on JBC Program:
`No human-readable program information known.`

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PlusSwap

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 175 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (4) Obligation:

ITRS problem:

The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The TRS R consists of the following rules:
Load668(i78, i84, i80) → Cond_Load668(i84 > 0 && i80 + 1 > 0, i78, i84, i80)
Cond_Load668(TRUE, i78, i84, i80) → Load668(i84 - 1, i78, i80 + 1)
The set Q consists of the following terms:

### (6) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load668(i78, i84, i80) → Cond_Load668(i84 > 0 && i80 + 1 > 0, i78, i84, i80)
Cond_Load668(TRUE, i78, i84, i80) → Load668(i84 - 1, i78, i80 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD668(i78[0], i84[0], i80[0]) → COND_LOAD668(i84[0] > 0 && i80[0] + 1 > 0, i78[0], i84[0], i80[0])
(1): COND_LOAD668(TRUE, i78[1], i84[1], i80[1]) → LOAD668(i84[1] - 1, i78[1], i80[1] + 1)

(0) -> (1), if ((i84[0] > 0 && i80[0] + 1 > 0* TRUE)∧(i84[0]* i84[1])∧(i80[0]* i80[1])∧(i78[0]* i78[1]))

(1) -> (0), if ((i80[1] + 1* i80[0])∧(i78[1]* i84[0])∧(i84[1] - 1* i78[0]))

The set Q consists of the following terms:

### (7) UsableRulesProof (EQUIVALENT transformation)

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.

### (8) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD668(i78[0], i84[0], i80[0]) → COND_LOAD668(i84[0] > 0 && i80[0] + 1 > 0, i78[0], i84[0], i80[0])
(1): COND_LOAD668(TRUE, i78[1], i84[1], i80[1]) → LOAD668(i84[1] - 1, i78[1], i80[1] + 1)

(0) -> (1), if ((i84[0] > 0 && i80[0] + 1 > 0* TRUE)∧(i84[0]* i84[1])∧(i80[0]* i80[1])∧(i78[0]* i78[1]))

(1) -> (0), if ((i80[1] + 1* i80[0])∧(i78[1]* i84[0])∧(i84[1] - 1* i78[0]))

The set Q consists of the following terms:

### (9) IDPNonInfProof (SOUND transformation)

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 LOAD668(i78, i84, i80) → COND_LOAD668(&&(>(i84, 0), >(+(i80, 1), 0)), i78, i84, i80) the following chains were created:
• We consider the chain LOAD668(i78[0], i84[0], i80[0]) → COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0]), COND_LOAD668(TRUE, i78[1], i84[1], i80[1]) → LOAD668(-(i84[1], 1), i78[1], +(i80[1], 1)) which results in the following constraint:

(1)    (&&(>(i84[0], 0), >(+(i80[0], 1), 0))=TRUEi84[0]=i84[1]i80[0]=i80[1]i78[0]=i78[1]LOAD668(i78[0], i84[0], i80[0])≥NonInfC∧LOAD668(i78[0], i84[0], i80[0])≥COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])∧(UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(2)    (>(i84[0], 0)=TRUE>(+(i80[0], 1), 0)=TRUELOAD668(i78[0], i84[0], i80[0])≥NonInfC∧LOAD668(i78[0], i84[0], i80[0])≥COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])∧(UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(3)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] + [bni_14]i78[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(4)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] + [bni_14]i78[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(5)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] + [bni_14]i78[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(6)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14] = 0∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] ≥ 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(7)    (i84[0] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14] = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] ≥ 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_LOAD668(TRUE, i78, i84, i80) → LOAD668(-(i84, 1), i78, +(i80, 1)) the following chains were created:

(8)    (&&(>(i84[0], 0), >(+(i80[0], 1), 0))=TRUEi84[0]=i84[1]i80[0]=i80[1]i78[0]=i78[1]+(i80[1], 1)=i80[0]1i78[1]=i84[0]1-(i84[1], 1)=i78[0]1&&(>(i84[0]1, 0), >(+(i80[0]1, 1), 0))=TRUEi84[0]1=i84[1]1i80[0]1=i80[1]1i78[0]1=i78[1]1+(i80[1]1, 1)=i80[0]2i78[1]1=i84[0]2-(i84[1]1, 1)=i78[0]2&&(>(i84[0]2, 0), >(+(i80[0]2, 1), 0))=TRUEi84[0]2=i84[1]2i80[0]2=i80[1]2i78[0]2=i78[1]2COND_LOAD668(TRUE, i78[1]1, i84[1]1, i80[1]1)≥NonInfC∧COND_LOAD668(TRUE, i78[1]1, i84[1]1, i80[1]1)≥LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))∧(UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(9)    (>(i84[0], 0)=TRUE>(+(i80[0], 1), 0)=TRUE>(i84[0]1, 0)=TRUE>(+(+(i80[0], 1), 1), 0)=TRUE>(-(i84[0], 1), 0)=TRUE>(+(+(+(i80[0], 1), 1), 1), 0)=TRUECOND_LOAD668(TRUE, -(i84[0], 1), i84[0]1, +(i80[0], 1))≥NonInfC∧COND_LOAD668(TRUE, -(i84[0], 1), i84[0]1, +(i80[0], 1))≥LOAD668(-(i84[0]1, 1), -(i84[0], 1), +(+(i80[0], 1), 1))∧(UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(10)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0∧i84[0]1 + [-1] ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] + [-2] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(11)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0∧i84[0]1 + [-1] ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] + [-2] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(12)    (i84[0] + [-1] ≥ 0∧i80[0] ≥ 0∧i84[0]1 + [-1] ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] + [-2] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(13)    (i84[0] ≥ 0∧i80[0] ≥ 0∧i84[0]1 + [-1] ≥ 0∧i80[0] + [1] ≥ 0∧[-1] + i84[0] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(14)    ([1] + i84[0] ≥ 0∧i80[0] ≥ 0∧i84[0]1 + [-1] ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(15)    ([1] + i84[0] ≥ 0∧i80[0] ≥ 0∧i84[0]1 ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] ≥ 0∧i80[0] + [2] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_GCD) which results in the following new constraint:

(16)    ([1] + i84[0] ≥ 0∧i80[0] ≥ 0∧i84[0]1 ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD668(i78, i84, i80) → COND_LOAD668(&&(>(i84, 0), >(+(i80, 1), 0)), i78, i84, i80)
• (i84[0] ≥ 0∧i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])), ≥)∧[bni_14] = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]i84[0] ≥ 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

• ([1] + i84[0] ≥ 0∧i80[0] ≥ 0∧i84[0]1 ≥ 0∧i80[0] + [1] ≥ 0∧i84[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD668(-(i84[1]1, 1), i78[1]1, +(i80[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]i84[0]1 + [bni_16]i84[0] ≥ 0∧[(-1)bso_17] ≥ 0)

The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [2]
POL(FALSE) = [2]
POL(LOAD668(x1, x2, x3)) = [1] + x2 + x1
POL(COND_LOAD668(x1, x2, x3, x4)) = [2] + x3 + x2 + [-1]x1
POL(&&(x1, x2)) = [2]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2

The following pairs are in P>:

LOAD668(i78[0], i84[0], i80[0]) → COND_LOAD668(&&(>(i84[0], 0), >(+(i80[0], 1), 0)), i78[0], i84[0], i80[0])

The following pairs are in Pbound:

The following pairs are in P:

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

&&(TRUE, TRUE)1TRUE1
&&(TRUE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(FALSE, FALSE)1FALSE1

### (11) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD668(TRUE, i78[1], i84[1], i80[1]) → LOAD668(i84[1] - 1, i78[1], i80[1] + 1)

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD668(i78[0], i84[0], i80[0]) → COND_LOAD668(i84[0] > 0 && i80[0] + 1 > 0, i78[0], i84[0], i80[0])

The set Q consists of the following terms: