Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 210 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 260 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 1990 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 IDP
↳11 IDPNonInfProof (⇒, 670 ms)
↳12 IDP
↳13 IDependencyGraphProof (⇔, 0 ms)
↳14 IDP
↳15 IDPNonInfProof (⇒, 120 ms)
↳16 IDP
↳17 IDependencyGraphProof (⇔, 0 ms)
↳18 TRUE

(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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Test2.main([Ljava/lang/String;)V: Graph of 104 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Test2.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 57 rules for P and 0 rules for R.


P rules:
1140_0_iter_Load(EOS(STATIC_1140), i328, i329, i330, i328) → 1142_0_iter_IntArithmetic(EOS(STATIC_1142), i328, i329, i330, i328, i329)
1142_0_iter_IntArithmetic(EOS(STATIC_1142), i328, i329, i330, i328, i329) → 1144_0_iter_ConstantStackPush(EOS(STATIC_1144), i328, i329, i330, +(i328, i329))
1144_0_iter_ConstantStackPush(EOS(STATIC_1144), i328, i329, i330, i341) → 1147_0_iter_Load(EOS(STATIC_1147), i328, i329, i330, i341, 3)
1147_0_iter_Load(EOS(STATIC_1147), i328, i329, i330, i341, matching1) → 1149_0_iter_IntArithmetic(EOS(STATIC_1149), i328, i329, i330, i341, 3, i330) | =(matching1, 3)
1149_0_iter_IntArithmetic(EOS(STATIC_1149), i328, i329, i330, i341, matching1, i330) → 1151_0_iter_IntArithmetic(EOS(STATIC_1151), i328, i329, i330, i341, *(3, i330)) | =(matching1, 3)
1151_0_iter_IntArithmetic(EOS(STATIC_1151), i328, i329, i330, i341, i346) → 1154_0_iter_LT(EOS(STATIC_1154), i328, i329, i330, +(i341, i346))
1154_0_iter_LT(EOS(STATIC_1154), i328, i329, i330, i353) → 1158_0_iter_LT(EOS(STATIC_1158), i328, i329, i330, i353)
1158_0_iter_LT(EOS(STATIC_1158), i328, i329, i330, i353) → 1163_0_iter_Load(EOS(STATIC_1163), i328, i329, i330) | >=(i353, 0)
1163_0_iter_Load(EOS(STATIC_1163), i328, i329, i330) → 1168_0_iter_Load(EOS(STATIC_1168), i328, i329, i330, i328)
1168_0_iter_Load(EOS(STATIC_1168), i328, i329, i330, i328) → 1172_0_iter_LE(EOS(STATIC_1172), i328, i329, i330, i328, i329)
1172_0_iter_LE(EOS(STATIC_1172), i328, i329, i330, i328, i329) → 1174_0_iter_LE(EOS(STATIC_1174), i328, i329, i330, i328, i329)
1172_0_iter_LE(EOS(STATIC_1172), i328, i329, i330, i328, i329) → 1176_0_iter_LE(EOS(STATIC_1176), i328, i329, i330, i328, i329)
1174_0_iter_LE(EOS(STATIC_1174), i328, i329, i330, i328, i329) → 1178_0_iter_Load(EOS(STATIC_1178), i328, i329, i330) | <=(i328, i329)
1178_0_iter_Load(EOS(STATIC_1178), i328, i329, i330) → 1182_0_iter_Load(EOS(STATIC_1182), i328, i329, i330, i329)
1182_0_iter_Load(EOS(STATIC_1182), i328, i329, i330, i329) → 1187_0_iter_LE(EOS(STATIC_1187), i328, i329, i330, i329, i330)
1187_0_iter_LE(EOS(STATIC_1187), i328, i329, i330, i329, i330) → 1193_0_iter_LE(EOS(STATIC_1193), i328, i329, i330, i329, i330)
1187_0_iter_LE(EOS(STATIC_1187), i328, i329, i330, i329, i330) → 1195_0_iter_LE(EOS(STATIC_1195), i328, i329, i330, i329, i330)
1193_0_iter_LE(EOS(STATIC_1193), i328, i329, i330, i329, i330) → 1196_0_iter_Load(EOS(STATIC_1196), i328, i329, i330) | <=(i329, i330)
1196_0_iter_Load(EOS(STATIC_1196), i328, i329, i330) → 1199_0_iter_Load(EOS(STATIC_1199), i328, i329, i330, i329)
1199_0_iter_Load(EOS(STATIC_1199), i328, i329, i330, i329) → 1202_0_iter_GT(EOS(STATIC_1202), i328, i329, i330, i329, i330)
1202_0_iter_GT(EOS(STATIC_1202), i328, i329, i330, i329, i330) → 1205_0_iter_GT(EOS(STATIC_1205), i328, i329, i330, i329, i330)
1202_0_iter_GT(EOS(STATIC_1202), i328, i329, i330, i329, i330) → 1206_0_iter_GT(EOS(STATIC_1206), i328, i329, i330, i329, i330)
1205_0_iter_GT(EOS(STATIC_1205), i328, i329, i330, i329, i330) → 1211_0_iter_Load(EOS(STATIC_1211), i328, i329, i330) | >(i329, i330)
1211_0_iter_Load(EOS(STATIC_1211), i328, i329, i330) → 1135_0_iter_Load(EOS(STATIC_1135), i328, i329, i330)
1135_0_iter_Load(EOS(STATIC_1135), i328, i329, i330) → 1140_0_iter_Load(EOS(STATIC_1140), i328, i329, i330, i328)
1206_0_iter_GT(EOS(STATIC_1206), i328, i329, i330, i329, i330) → 1212_0_iter_Load(EOS(STATIC_1212), i328, i329, i330) | <=(i329, i330)
1212_0_iter_Load(EOS(STATIC_1212), i328, i329, i330) → 1213_0_iter_ConstantStackPush(EOS(STATIC_1213), i329, i330, i328)
1213_0_iter_ConstantStackPush(EOS(STATIC_1213), i329, i330, i328) → 1216_0_iter_InvokeMethod(EOS(STATIC_1216), i329, i330, i328, 1)
1216_0_iter_InvokeMethod(EOS(STATIC_1216), i329, i330, i328, matching1) → 1217_0_add_Load(EOS(STATIC_1217), i329, i330, i328, 1, i328, 1) | =(matching1, 1)
1217_0_add_Load(EOS(STATIC_1217), i329, i330, i328, matching1, i328, matching2) → 1219_0_add_Load(EOS(STATIC_1219), i329, i330, i328, 1, 1, i328) | &&(=(matching1, 1), =(matching2, 1))
1219_0_add_Load(EOS(STATIC_1219), i329, i330, i328, matching1, matching2, i328) → 1220_0_add_IntArithmetic(EOS(STATIC_1220), i329, i330, i328, 1, i328, 1) | &&(=(matching1, 1), =(matching2, 1))
1220_0_add_IntArithmetic(EOS(STATIC_1220), i329, i330, i328, matching1, i328, matching2) → 1221_0_add_Return(EOS(STATIC_1221), i329, i330, i328, 1, +(i328, 1)) | &&(=(matching1, 1), =(matching2, 1))
1221_0_add_Return(EOS(STATIC_1221), i329, i330, i328, matching1, i370) → 1223_0_iter_Store(EOS(STATIC_1223), i329, i330, i370) | =(matching1, 1)
1223_0_iter_Store(EOS(STATIC_1223), i329, i330, i370) → 1225_0_iter_Load(EOS(STATIC_1225), i370, i329, i330)
1225_0_iter_Load(EOS(STATIC_1225), i370, i329, i330) → 1226_0_iter_ConstantStackPush(EOS(STATIC_1226), i370, i330, i329)
1226_0_iter_ConstantStackPush(EOS(STATIC_1226), i370, i330, i329) → 1227_0_iter_InvokeMethod(EOS(STATIC_1227), i370, i330, i329, 1)
1227_0_iter_InvokeMethod(EOS(STATIC_1227), i370, i330, i329, matching1) → 1228_0_add_Load(EOS(STATIC_1228), i370, i330, i329, 1, i329, 1) | =(matching1, 1)
1228_0_add_Load(EOS(STATIC_1228), i370, i330, i329, matching1, i329, matching2) → 1230_0_add_Load(EOS(STATIC_1230), i370, i330, i329, 1, 1, i329) | &&(=(matching1, 1), =(matching2, 1))
1230_0_add_Load(EOS(STATIC_1230), i370, i330, i329, matching1, matching2, i329) → 1232_0_add_IntArithmetic(EOS(STATIC_1232), i370, i330, i329, 1, i329, 1) | &&(=(matching1, 1), =(matching2, 1))
1232_0_add_IntArithmetic(EOS(STATIC_1232), i370, i330, i329, matching1, i329, matching2) → 1233_0_add_Return(EOS(STATIC_1233), i370, i330, i329, 1, +(i329, 1)) | &&(=(matching1, 1), =(matching2, 1))
1233_0_add_Return(EOS(STATIC_1233), i370, i330, i329, matching1, i373) → 1235_0_iter_Store(EOS(STATIC_1235), i370, i330, i373) | =(matching1, 1)
1235_0_iter_Store(EOS(STATIC_1235), i370, i330, i373) → 1236_0_iter_Load(EOS(STATIC_1236), i370, i373, i330)
1236_0_iter_Load(EOS(STATIC_1236), i370, i373, i330) → 1238_0_iter_ConstantStackPush(EOS(STATIC_1238), i370, i373, i330)
1238_0_iter_ConstantStackPush(EOS(STATIC_1238), i370, i373, i330) → 1240_0_iter_IntArithmetic(EOS(STATIC_1240), i370, i373, i330, 1)
1240_0_iter_IntArithmetic(EOS(STATIC_1240), i370, i373, i330, matching1) → 1242_0_iter_Store(EOS(STATIC_1242), i370, i373, -(i330, 1)) | =(matching1, 1)
1242_0_iter_Store(EOS(STATIC_1242), i370, i373, i377) → 1243_0_iter_JMP(EOS(STATIC_1243), i370, i373, i377)
1243_0_iter_JMP(EOS(STATIC_1243), i370, i373, i377) → 1245_0_iter_Load(EOS(STATIC_1245), i370, i373, i377)
1245_0_iter_Load(EOS(STATIC_1245), i370, i373, i377) → 1135_0_iter_Load(EOS(STATIC_1135), i370, i373, i377)
1195_0_iter_LE(EOS(STATIC_1195), i328, i329, i330, i329, i330) → 1198_0_iter_Inc(EOS(STATIC_1198), i328, i329, i330) | >(i329, i330)
1198_0_iter_Inc(EOS(STATIC_1198), i328, i329, i330) → 1201_0_iter_Inc(EOS(STATIC_1201), +(i328, 1), i329, i330)
1201_0_iter_Inc(EOS(STATIC_1201), i362, i329, i330) → 1204_0_iter_JMP(EOS(STATIC_1204), i362, +(i329, -2), i330)
1204_0_iter_JMP(EOS(STATIC_1204), i362, i363, i330) → 1209_0_iter_Load(EOS(STATIC_1209), i362, i363, i330)
1209_0_iter_Load(EOS(STATIC_1209), i362, i363, i330) → 1135_0_iter_Load(EOS(STATIC_1135), i362, i363, i330)
1176_0_iter_LE(EOS(STATIC_1176), i328, i329, i330, i328, i329) → 1180_0_iter_Inc(EOS(STATIC_1180), i328, i329, i330) | >(i328, i329)
1180_0_iter_Inc(EOS(STATIC_1180), i328, i329, i330) → 1184_0_iter_JMP(EOS(STATIC_1184), +(i328, -1), i329, i330)
1184_0_iter_JMP(EOS(STATIC_1184), i357, i329, i330) → 1191_0_iter_Load(EOS(STATIC_1191), i357, i329, i330)
1191_0_iter_Load(EOS(STATIC_1191), i357, i329, i330) → 1135_0_iter_Load(EOS(STATIC_1135), i357, i329, i330)
R rules:

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


P rules:
1140_0_iter_Load(EOS(STATIC_1140), x0, x1, x2, x0) → 1140_0_iter_Load(EOS(STATIC_1140), x0, x1, x2, x0) | FALSE
1140_0_iter_Load(EOS(STATIC_1140), x0, x1, x2, x0) → 1140_0_iter_Load(EOS(STATIC_1140), +(x0, 1), +(x1, 1), -(x2, 1), +(x0, 1)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1140_0_iter_Load(EOS(STATIC_1140), x0, x1, x2, x0) → 1140_0_iter_Load(EOS(STATIC_1140), +(x0, 1), +(x1, -2), x2, +(x0, 1)) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1140_0_iter_Load(EOS(STATIC_1140), x0, x1, x2, x0) → 1140_0_iter_Load(EOS(STATIC_1140), +(x0, -1), x1, x2, +(x0, -1)) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:

Filtered ground terms:



1140_0_iter_Load(x1, x2, x3, x4, x5) → 1140_0_iter_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_1140_0_iter_Load2(x1, x2, x3, x4, x5, x6) → Cond_1140_0_iter_Load2(x1, x3, x4, x5, x6)
Cond_1140_0_iter_Load1(x1, x2, x3, x4, x5, x6) → Cond_1140_0_iter_Load1(x1, x3, x4, x5, x6)
Cond_1140_0_iter_Load(x1, x2, x3, x4, x5, x6) → Cond_1140_0_iter_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



1140_0_iter_Load(x1, x2, x3, x4) → 1140_0_iter_Load(x2, x3, x4)
Cond_1140_0_iter_Load(x1, x2, x3, x4, x5) → Cond_1140_0_iter_Load(x1, x3, x4, x5)
Cond_1140_0_iter_Load1(x1, x2, x3, x4, x5) → Cond_1140_0_iter_Load1(x1, x3, x4, x5)
Cond_1140_0_iter_Load2(x1, x2, x3, x4, x5) → Cond_1140_0_iter_Load2(x1, x3, x4, x5)

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


P rules:
1140_0_iter_Load(x1, x2, x0) → 1140_0_iter_Load(+(x1, 1), -(x2, 1), +(x0, 1)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1140_0_iter_Load(x1, x2, x0) → 1140_0_iter_Load(+(x1, -2), x2, +(x0, 1)) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1140_0_iter_Load(x1, x2, x0) → 1140_0_iter_Load(x1, x2, +(x0, -1)) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.


P rules:
1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1140_0_ITER_LOAD(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, 1), -(x2, 1), +(x0, 1))
1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1140_0_ITER_LOAD1(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, -2), x2, +(x0, 1))
1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1140_0_ITER_LOAD2(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(x1, x2, +(x0, -1))
R rules:

(6) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(1): COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(x1[1] + 1, x2[1] - 1, x0[1] + 1)
(2): 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(3): COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(x1[3] + -2, x2[3], x0[3] + 1)
(4): 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_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] ∧x1[0] →* x1[1]∧x2[0] →* x2[1]∧x0[0] →* x0[1])


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


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


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


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


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


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


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


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


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


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


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



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@c024b1f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_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]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 1140_0_ITER_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1140_0_ITER_LOAD(x1[0], x2[0], x0[0])≥COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(UIncreasing(COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥))



    We simplified constraint (1) using rules (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])=TRUE ⇒ 1140_0_ITER_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1140_0_ITER_LOAD(x1[0], x2[0], x0[0])≥COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(UIncreasing(COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), 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_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 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_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 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_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 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_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 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_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 0)







For Pair COND_1140_0_ITER_LOAD(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, 1), -(x2, 1), +(x0, 1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)), 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]) which results in the following constraint:

    (8)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧+(x1[1], 1)=x1[0]1∧-(x2[1], 1)=x2[0]1∧+(x0[1], 1)=x0[0]1 ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



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

    (9)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE∧>=(x2[0], x1[0])=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥1140_0_ITER_LOAD(+(x1[0], 1), -(x2[0], 1), +(x0[0], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



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

    (10)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (11)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (12)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (13)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)), 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) which results in the following constraint:

    (15)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧+(x1[1], 1)=x1[2]∧-(x2[1], 1)=x2[2]∧+(x0[1], 1)=x0[2] ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



    We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (16)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE∧>=(x2[0], x1[0])=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥1140_0_ITER_LOAD(+(x1[0], 1), -(x2[0], 1), +(x0[0], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



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

    (17)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (18)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (19)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (20)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (21)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (22)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧+(x1[1], 1)=x1[4]∧-(x2[1], 1)=x2[4]∧+(x0[1], 1)=x0[4] ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1])≥1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



    We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (23)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE∧>=(x2[0], x1[0])=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_1140_0_ITER_LOAD(TRUE, x1[0], x2[0], x0[0])≥1140_0_ITER_LOAD(+(x1[0], 1), -(x2[0], 1), +(x0[0], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥))



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

    (24)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (25)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (26)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



    We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)







For Pair 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)) which results in the following constraint:

    (29)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))



    We simplified constraint (29) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (30)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))



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

    (31)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (32)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (33)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (34)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (35)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (36)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)


    (37)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)







For Pair COND_1140_0_ITER_LOAD1(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, -2), x2, +(x0, 1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)), 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]) which results in the following constraint:

    (38)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧+(x1[3], -2)=x1[0]∧x2[3]=x2[0]∧+(x0[3], 1)=x0[0] ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (38) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (39)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥1140_0_ITER_LOAD(+(x1[2], -2), x2[2], +(x0[2], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (40)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (41)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (42)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (43)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



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

    (44)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



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

    (45)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)


    (46)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)), 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) which results in the following constraint:

    (47)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧+(x1[3], -2)=x1[2]1∧x2[3]=x2[2]1∧+(x0[3], 1)=x0[2]1 ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (47) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (48)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥1140_0_ITER_LOAD(+(x1[2], -2), x2[2], +(x0[2], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (49)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (50)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (51)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (52)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (53)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (53) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (54)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)


    (55)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (56)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧+(x1[3], -2)=x1[4]∧x2[3]=x2[4]∧+(x0[3], 1)=x0[4] ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (56) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (57)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥1140_0_ITER_LOAD(+(x1[2], -2), x2[2], +(x0[2], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (57) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (58)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (58) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (59)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (59) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (60)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (60) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (61)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (61) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (62)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (63)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)


    (64)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)







For Pair 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)) which results in the following constraint:

    (65)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (65) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (66)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (67)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (67) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (68)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (68) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (69)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (70)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (71)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)


    (72)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (73)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)


    (74)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (72) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (75)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)


    (76)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)







For Pair COND_1140_0_ITER_LOAD2(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(x1, x2, +(x0, -1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]) which results in the following constraint:

    (77)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[0]∧x2[5]=x2[0]∧+(x0[5], -1)=x0[0] ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (77) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (78)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (78) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (79)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (79) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (80)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (80) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (81)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (81) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (82)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (82) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (83)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (84)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (83) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (85)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (86)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (84) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (87)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (88)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) which results in the following constraint:

    (89)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[2]∧x2[5]=x2[2]∧+(x0[5], -1)=x0[2] ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (89) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (90)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (90) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (91)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (91) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (92)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (92) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (93)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (93) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (94)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (94) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (95)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (96)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (95) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (97)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (98)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (96) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (99)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (100)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (101)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[4]1∧x2[5]=x2[4]1∧+(x0[5], -1)=x0[4]1 ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (101) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (102)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    We simplified constraint (102) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (103)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (103) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (104)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (104) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (105)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (105) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (106)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (106) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (107)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (108)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (107) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (109)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (110)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (108) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (111)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)


    (112)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[(-1)bso_27] ≥ 0)

  • COND_1140_0_ITER_LOAD(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, 1), -(x2, 1), +(x0, 1))
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[0] ≥ 0∧[1 + (-1)bso_29] ≥ 0)

  • 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), 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_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x2[2] ≥ 0∧[(-1)bso_31] ≥ 0)

  • COND_1140_0_ITER_LOAD1(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(+(x1, -2), x2, +(x0, 1))
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]x2[2] ≥ 0∧[(-1)bso_33] ≥ 0)

  • 1140_0_ITER_LOAD(x1, x2, x0) → COND_1140_0_ITER_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x2[4] ≥ 0∧[(-1)bso_35] ≥ 0)

  • COND_1140_0_ITER_LOAD2(TRUE, x1, x2, x0) → 1140_0_ITER_LOAD(x1, x2, +(x0, -1))
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x2[4] ≥ 0∧[(-1)bso_37] ≥ 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) = [1]   
POL(FALSE) = [3]   
POL(1140_0_ITER_LOAD(x1, x2, x3)) = [-1] + x2   
POL(COND_1140_0_ITER_LOAD(x1, x2, x3, x4)) = [-1] + x3   
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_1140_0_ITER_LOAD1(x1, x2, x3, x4)) = [-1] + x3   
POL(<(x1, x2)) = [-1]   
POL(-2) = [-2]   
POL(COND_1140_0_ITER_LOAD2(x1, x2, x3, x4)) = [-1] + x3   
POL(-1) = [-1]   

The following pairs are in P>:

COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))

The following pairs are in Pbound:

1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])
COND_1140_0_ITER_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1140_0_ITER_LOAD(+(x1[1], 1), -(x2[1], 1), +(x0[1], 1))

The following pairs are in P≥:

1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])
1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])
COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))
1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))

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

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

(8) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1140_0_ITER_LOAD(x1[0], x2[0], x0[0]) → COND_1140_0_ITER_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(2): 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(3): COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(x1[3] + -2, x2[3], x0[3] + 1)
(4): 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], x0[5] + -1)

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


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


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


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


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


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


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


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



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) 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_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], x0[5] + -1)
(4): 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(x1[3] + -2, x2[3], x0[3] + 1)
(2): 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])

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


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


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


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


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


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



The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@c024b1f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) which results in the following constraint:

    (1)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[2]∧x2[5]=x2[2]∧+(x0[5], -1)=x0[2] ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



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

    (2)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



    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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 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(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (13)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[4]1∧x2[5]=x2[4]1∧+(x0[5], -1)=x0[4]1 ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



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

    (14)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



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

    (15)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (16)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (17)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[4] + [bni_21]x1[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (18)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (19)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (20)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (21)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (22)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (23)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)


    (24)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)







For Pair 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)) which results in the following constraint:

    (25)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (25) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (26)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (27)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[4] + [bni_23]x1[4] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (28)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[4] + [bni_23]x1[4] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[4] + [bni_23]x1[4] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (30)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (31)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)


    (32)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (31) 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∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)


    (34)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (32) 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)


    (36)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x1[4] + [bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)







For Pair COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)), 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) which results in the following constraint:

    (37)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧+(x1[3], -2)=x1[2]1∧x2[3]=x2[2]1∧+(x0[3], 1)=x0[2]1 ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (37) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (38)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥1140_0_ITER_LOAD(+(x1[2], -2), x2[2], +(x0[2], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



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

    (39)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (40)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (41)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (42)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x1[2] + [(-1)bni_25]x2[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (43)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



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

    (44)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)


    (45)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (46)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧+(x1[3], -2)=x1[4]∧x2[3]=x2[4]∧+(x0[3], 1)=x0[4] ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (46) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (47)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_1140_0_ITER_LOAD1(TRUE, x1[2], x2[2], x0[2])≥1140_0_ITER_LOAD(+(x1[2], -2), x2[2], +(x0[2], 1))∧(UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (48)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (49)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (50)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (51)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x1[2] + [(-1)bni_25]x2[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (52)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)



    We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (53)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)


    (54)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)







For Pair 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1)) which results in the following constraint:

    (55)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))



    We simplified constraint (55) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (56)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1140_0_ITER_LOAD(x1[2], x2[2], x0[2])≥COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))



    We simplified constraint (56) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (57)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (57) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (58)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (58) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (59)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (60)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[2] + [(-1)bni_27]x2[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (60) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (61)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (61) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (62)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)


    (63)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [(-1)bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x1[4] + [bni_21]x2[4] ≥ 0∧[(-1)bso_22] ≥ 0)

  • 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[4] + [bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x1[4] + [(-1)bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x1[4] + [bni_23]x2[4] ≥ 0∧[(-1)bso_24] ≥ 0)

  • COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_25] + [bni_25]x1[2] ≥ 0∧[2 + (-1)bso_26] ≥ 0)

  • 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 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) = [3]   
POL(COND_1140_0_ITER_LOAD2(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2   
POL(1140_0_ITER_LOAD(x1, x2, x3)) = [-1] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(&&(x1, x2)) = [-1]   
POL(<(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(0) = 0   
POL(*(x1, x2)) = x1·x2   
POL(3) = [3]   
POL(COND_1140_0_ITER_LOAD1(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2   
POL(-2) = [-2]   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))

The following pairs are in Pbound:

COND_1140_0_ITER_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1140_0_ITER_LOAD(+(x1[3], -2), x2[3], +(x0[3], 1))
1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])

The following pairs are in P≥:

COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))
1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])

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

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

(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_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], x0[5] + -1)
(4): 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(2): 1140_0_ITER_LOAD(x1[2], x2[2], x0[2]) → COND_1140_0_ITER_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])

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


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


(4) -> (5), if (x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4] ∧x1[4] →* x1[5]∧x2[4] →* x2[5]∧x0[4] →* 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): 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], x0[5] + -1)

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


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



The set Q is empty.

(15) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@c024b1f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_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]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



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

    (2)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1140_0_ITER_LOAD(x1[4], x2[4], x0[4])≥COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] + [(-1)bni_15]x1[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] + [(-1)bni_15]x1[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] + [(-1)bni_15]x1[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 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_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)







For Pair COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)) the following chains were created:
  • We consider the chain 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1)), 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) which results in the following constraint:

    (13)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5]∧x1[5]=x1[4]1∧x2[5]=x2[4]1∧+(x0[5], -1)=x0[4]1 ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



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

    (14)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥NonInfC∧COND_1140_0_ITER_LOAD2(TRUE, x1[4], x2[4], x0[4])≥1140_0_ITER_LOAD(x1[4], x2[4], +(x0[4], -1))∧(UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥))



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

    (15)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[4] + [(-1)bni_17]x1[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (16)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[4] + [(-1)bni_17]x1[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (17)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[4] + [(-1)bni_17]x1[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (18)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (19)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)


    (20)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (21)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)


    (22)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (23)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)


    (24)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[4] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

  • COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))), ≥)∧[(-1)Bound*bni_17 + bni_17] + [bni_17]x0[4] ≥ 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(1140_0_ITER_LOAD(x1, x2, x3)) = [1] + x3 + [-1]x1   
POL(COND_1140_0_ITER_LOAD2(x1, x2, x3, x4)) = x4 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
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]   

The following pairs are in P>:

1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

The following pairs are in Pbound:

1140_0_ITER_LOAD(x1[4], x2[4], x0[4]) → COND_1140_0_ITER_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))

The following pairs are in P≥:

COND_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], +(x0[5], -1))

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

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

(16) 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_1140_0_ITER_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1140_0_ITER_LOAD(x1[5], x2[5], x0[5] + -1)


The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE