### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: Test2
`public class Test2 {    public static void main(String[] args) {	iter(args.length, args.length % 5, args.length % 4);    }    private static void iter(int x, int y, int z) {	while (x + y + 3 * z >= 0) {	    if (x > y)		x--;	    else if (y > z) {		x++;		y -= 2;	    }	    else if (y <= z) {		x = add(x, 1);		y = add(y, 1);		z = z - 1;	    }	}    }    private static int add(int v, int w) {	return v + w;    }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Test2.main([Ljava/lang/String;)V: Graph of 80 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 57 rules for P and 3 rules for R.

Combined rules. Obtained 4 rules for P and 0 rules for R.

Filtered ground terms:

830_0_iter_Load(x1, x2, x3, x4, x5) → 830_0_iter_Load(x2, x3, x4, x5)

Filtered duplicate args:

830_0_iter_Load(x1, x2, x3, x4) → 830_0_iter_Load(x2, x3, x4)

Combined rules. Obtained 3 rules for P and 0 rules for R.

Finished conversion. Obtained 3 rules for P and 0 rules for R. System has predefined symbols.

### (4) 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): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))
(2): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2])) → COND_830_1_MAIN_INVOKEMETHOD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], 830_0_iter_Load(x1[2], x2[2], x0[2]))
(3): COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0]* TRUE)∧(830_0_iter_Load(x1[0], x2[0], x0[0]) →* 830_0_iter_Load(x1[1], x2[1], x0[1])))

(1) -> (0), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(1) -> (2), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[2], x2[2], x0[2])))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(2) -> (3), if ((x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2]* TRUE)∧(830_0_iter_Load(x1[2], x2[2], x0[2]) →* 830_0_iter_Load(x1[3], x2[3], x0[3])))

(3) -> (0), if ((830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(3) -> (2), if ((830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1) →* 830_0_iter_Load(x1[2], x2[2], x0[2])))

(3) -> (4), if ((830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (2), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[2], x2[2], x0[2])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

The set Q is empty.

### (5) 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 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0)) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0])), COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(1)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE830_0_iter_Load(x1[0], x2[0], x0[0])=830_0_iter_Load(x1[1], x2[1], x0[1]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

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

(2)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE>=(x2[0], x1[0])=TRUE>=(x1[0], x0[0])=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

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

(3)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x2[0] + [bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(4)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x2[0] + [bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(5)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x2[0] + [bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x1[0] + [bni_19]x2[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [(2)bni_19]x2[0] + [(-1)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1, 1), -(x2, 1), +(x0, 1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(8)    (COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(9)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_22] ≥ 0)

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

(10)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_22] ≥ 0)

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

(11)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_22] ≥ 0)

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

(12)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0)) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2])) → COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2])), COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1))) which results in the following constraint:

(13)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE830_0_iter_Load(x1[2], x2[2], x0[2])=830_0_iter_Load(x1[3], x2[3], x0[3]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2]))≥COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥))

We simplified constraint (13) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(14)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE<(x2[2], x1[2])=TRUE>=(x1[2], x0[2])=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2]))≥COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥))

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

(15)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(16)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(17)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(18)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(19)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23 + bni_23] + [(2)bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(20)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23 + bni_23] + [(2)bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

(21)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23 + bni_23] + [(-2)bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1, -2), x2, +(x0, 1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1))) which results in the following constraint:

(22)    (COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(23)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥)∧[2 + (-1)bso_26] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(24)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥)∧[2 + (-1)bso_26] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(25)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥)∧[2 + (-1)bso_26] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(26)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_26] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0)) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])), COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(27)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE830_0_iter_Load(x1[4], x2[4], x0[4])=830_0_iter_Load(x1[5], x2[5], x0[5]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (27) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(28)    (<(x1[4], x0[4])=TRUE<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(29)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x2[4] + [bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(30)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x2[4] + [bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(31)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x2[4] + [bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(32)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(33)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(34)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(35)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(36)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(37)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(38)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (35) using rule (IDP_POLY_GCD) which results in the following new constraint:

(39)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_GCD) which results in the following new constraint:

(40)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_GCD) which results in the following new constraint:

(41)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_GCD) which results in the following new constraint:

(42)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, +(x0, -1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(43)    (COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥))

We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(44)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(45)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(46)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (46) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(47)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_19] + [(2)bni_19]x2[0] + [(-1)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1, 1), -(x2, 1), +(x0, 1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0))
• (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23 + bni_23] + [(2)bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)
• (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))), ≥)∧[(-1)Bound*bni_23 + bni_23] + [(-2)bni_23]x2[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1, -2), x2, +(x0, 1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_26] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, x0)) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), 830_0_iter_Load(x1, x2, x0))
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]x1[4] + [(-1)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1, x2, x0)) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1, x2, +(x0, -1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 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) = 0
POL(FALSE) = 0
POL(830_1_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(830_0_iter_Load(x1, x2, x3)) = [-1] + [-1]x2 + [-1]x1
POL(COND_830_1_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(*(x1, x2)) = x1·x2
POL(3) = [3]
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_830_1_MAIN_INVOKEMETHOD1(x1, x2)) = [-1] + [-1]x2
POL(<(x1, x2)) = [-1]
POL(-2) = [-2]
POL(COND_830_1_MAIN_INVOKEMETHOD2(x1, x2)) = [-1] + [-1]x2
POL(-1) = [-1]

The following pairs are in P>:

COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[3], -2), x2[3], +(x0[3], 1)))

The following pairs are in Pbound:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2])) → COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))

The following pairs are in P:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))
COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2])) → COND_830_1_MAIN_INVOKEMETHOD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), 830_0_iter_Load(x1[2], x2[2], x0[2]))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))

There are no usable rules.

### (7) 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): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))
(2): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[2], x2[2], x0[2])) → COND_830_1_MAIN_INVOKEMETHOD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], 830_0_iter_Load(x1[2], x2[2], x0[2]))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

(1) -> (0), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0]* TRUE)∧(830_0_iter_Load(x1[0], x2[0], x0[0]) →* 830_0_iter_Load(x1[1], x2[1], x0[1])))

(1) -> (2), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[2], x2[2], x0[2])))

(5) -> (2), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[2], x2[2], x0[2])))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (9) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))
(0): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))

(1) -> (0), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0]* TRUE)∧(830_0_iter_Load(x1[0], x2[0], x0[0]) →* 830_0_iter_Load(x1[1], x2[1], x0[1])))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (10) 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 COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(1)    (COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(3)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(4)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(5)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])), COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(6)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE830_0_iter_Load(x1[4], x2[4], x0[4])=830_0_iter_Load(x1[5], x2[5], x0[5]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (6) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (<(x1[4], x0[4])=TRUE<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(9)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(10)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(13)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(15)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(16)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(17)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(18)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(19)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_GCD) which results in the following new constraint:

(20)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_GCD) which results in the following new constraint:

(21)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(22)    (COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(23)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(24)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(25)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(26)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0])), COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(27)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE830_0_iter_Load(x1[0], x2[0], x0[0])=830_0_iter_Load(x1[1], x2[1], x0[1]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

We simplified constraint (27) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(28)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE>=(x2[0], x1[0])=TRUE>=(x1[0], x0[0])=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(29)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(30)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(31)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(32)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(33)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 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) = 0
POL(FALSE) = 0
POL(COND_830_1_MAIN_INVOKEMETHOD2(x1, x2)) = [-1] + [-1]x2
POL(830_0_iter_Load(x1, x2, x3)) = [-1] + [-1]x2
POL(830_1_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = 0
POL(<(x1, x2)) = [-1]
POL(<=(x1, x2)) = 0
POL(0) = 0
POL(*(x1, x2)) = x1·x2
POL(3) = [3]
POL(COND_830_1_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(>=(x1, x2)) = [-1]

The following pairs are in P>:

COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))

The following pairs are in Pbound:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))

The following pairs are in P:

COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))

There are no usable rules.

### (12) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(0): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC 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:
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (15) 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 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])), COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(1)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE830_0_iter_Load(x1[4], x2[4], x0[4])=830_0_iter_Load(x1[5], x2[5], x0[5]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

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

(2)    (<(x1[4], x0[4])=TRUE<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

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

(3)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(4)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(5)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(6)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(7)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(9)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(11)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(13)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(14)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(15)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(16)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(17)    (COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥))

We simplified constraint (17) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(18)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (18) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(19)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(20)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (20) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(21)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 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) = 0
POL(FALSE) = 0
POL(830_1_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(830_0_iter_Load(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_830_1_MAIN_INVOKEMETHOD2(x1, x2)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(<(x1, x2)) = 0
POL(<=(x1, x2)) = [1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(*(x1, x2)) = x1·x2
POL(3) = [3]
POL(-1) = [-1]

The following pairs are in P>:

COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))

The following pairs are in Pbound:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))

The following pairs are in P:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))

There are no usable rules.

### (17) 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:
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))

The set Q is empty.

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) 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:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

The set Q is empty.

### (21) IDependencyGraphProof (EQUIVALENT transformation)

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

### (23) 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:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))

The set Q is empty.

### (24) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (26) 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): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))
(3): COND_830_1_MAIN_INVOKEMETHOD1(TRUE, 830_0_iter_Load(x1[3], x2[3], x0[3])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

(1) -> (0), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(3) -> (0), if ((830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0]* TRUE)∧(830_0_iter_Load(x1[0], x2[0], x0[0]) →* 830_0_iter_Load(x1[1], x2[1], x0[1])))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(3) -> (4), if ((830_0_iter_Load(x1[3] + -2, x2[3], x0[3] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (27) IDependencyGraphProof (EQUIVALENT transformation)

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

### (28) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))
(0): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], 830_0_iter_Load(x1[0], x2[0], x0[0]))

(1) -> (0), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(5) -> (0), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[0], x2[0], x0[0])))

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0]* TRUE)∧(830_0_iter_Load(x1[0], x2[0], x0[0]) →* 830_0_iter_Load(x1[1], x2[1], x0[1])))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (29) 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 COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(1)    (COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(3)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(4)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[(-1)bso_12] ≥ 0)

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

(5)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])), COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(6)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE830_0_iter_Load(x1[4], x2[4], x0[4])=830_0_iter_Load(x1[5], x2[5], x0[5]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (6) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (<(x1[4], x0[4])=TRUE<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(9)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(10)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(13)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(15)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(16)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

(17)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(18)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(19)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_GCD) which results in the following new constraint:

(20)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_GCD) which results in the following new constraint:

(21)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(22)    (COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(23)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_16] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(24)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_16] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(25)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧[(-1)bso_16] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(26)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

For Pair 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0])), COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))) which results in the following constraint:

(27)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE830_0_iter_Load(x1[0], x2[0], x0[0])=830_0_iter_Load(x1[1], x2[1], x0[1]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

We simplified constraint (27) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(28)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE>=(x2[0], x1[0])=TRUE>=(x1[0], x0[0])=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0]))≥COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(29)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(30)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(31)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(32)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(33)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[4] ≥ 0∧[(-1)bso_14] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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) = 0
POL(FALSE) = 0
POL(COND_830_1_MAIN_INVOKEMETHOD2(x1, x2)) = [2] + [-1]x2
POL(830_0_iter_Load(x1, x2, x3)) = [-1] + [-1]x2
POL(830_1_MAIN_INVOKEMETHOD(x1)) = [2] + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = 0
POL(<(x1, x2)) = [-1]
POL(<=(x1, x2)) = 0
POL(0) = 0
POL(*(x1, x2)) = x1·x2
POL(3) = [3]
POL(COND_830_1_MAIN_INVOKEMETHOD(x1, x2)) = [1] + [-1]x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(>=(x1, x2)) = [-1]

The following pairs are in P>:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))

The following pairs are in Pbound:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[0], x2[0], x0[0])) → COND_830_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 830_0_iter_Load(x1[0], x2[0], x0[0]))

The following pairs are in P:

COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)))

There are no usable rules.

### (30) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(1): COND_830_1_MAIN_INVOKEMETHOD(TRUE, 830_0_iter_Load(x1[1], x2[1], x0[1])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1))

(1) -> (4), if ((830_0_iter_Load(x1[1] + 1, x2[1] - 1, x0[1] + 1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (31) IDependencyGraphProof (EQUIVALENT transformation)

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

### (32) 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:
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

(5) -> (4), if ((830_0_iter_Load(x1[5], x2[5], x0[5] + -1) →* 830_0_iter_Load(x1[4], x2[4], x0[4])))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4]* TRUE)∧(830_0_iter_Load(x1[4], x2[4], x0[4]) →* 830_0_iter_Load(x1[5], x2[5], x0[5])))

The set Q is empty.

### (33) 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 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])) the following chains were created:
• We consider the chain 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4])), COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(1)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE830_0_iter_Load(x1[4], x2[4], x0[4])=830_0_iter_Load(x1[5], x2[5], x0[5]) ⇒ 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

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

(2)    (<(x1[4], x0[4])=TRUE<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥NonInfC∧830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4]))≥COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))∧(UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥))

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

(3)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(4)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(5)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[4] + [(-1)bni_9]x1[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(6)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(7)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(9)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(11)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

(12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(13)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(14)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(15)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(16)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) the following chains were created:
• We consider the chain COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1))) which results in the following constraint:

(17)    (COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥NonInfC∧COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5]))≥830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))∧(UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥))

We simplified constraint (17) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(18)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (18) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(19)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(20)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (20) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(21)    ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)
• (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))), ≥)∧[(-1)Bound*bni_9 + bni_9] + [bni_9]x0[4] ≥ 0∧[(-1)bso_10] ≥ 0)

• COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))
• ((UIncreasing(830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 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) = 0
POL(FALSE) = 0
POL(830_1_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(830_0_iter_Load(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_830_1_MAIN_INVOKEMETHOD2(x1, x2)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(<(x1, x2)) = 0
POL(<=(x1, x2)) = [1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(*(x1, x2)) = x1·x2
POL(3) = [3]
POL(-1) = [-1]

The following pairs are in P>:

COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], +(x0[5], -1)))

The following pairs are in Pbound:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))

The following pairs are in P:

830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 830_0_iter_Load(x1[4], x2[4], x0[4]))

There are no usable rules.

### (35) 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:
(4): 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[4], x2[4], x0[4])) → COND_830_1_MAIN_INVOKEMETHOD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], 830_0_iter_Load(x1[4], x2[4], x0[4]))

The set Q is empty.

### (36) IDependencyGraphProof (EQUIVALENT transformation)

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

### (38) 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:
(5): COND_830_1_MAIN_INVOKEMETHOD2(TRUE, 830_0_iter_Load(x1[5], x2[5], x0[5])) → 830_1_MAIN_INVOKEMETHOD(830_0_iter_Load(x1[5], x2[5], x0[5] + -1))

The set Q is empty.

### (39) IDependencyGraphProof (EQUIVALENT transformation)

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