### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: BubbleSort
`class BubbleSort {    public static void main(String[] args) {	sort(new int[100]);    }    public static void sort(int[] x) {	int n = x.length;	for (int pass=1; pass < n; pass++)  // count how many times	    // This next loop becomes shorter and shorter	    for (int i=0; i < n - pass; i++)		if (x[i] > x[i+1]) {		    // exchange elements		    int temp = x[i]; x[i] = x[i+1];  x[i+1] = temp;		}    }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
BubbleSort.main([Ljava/lang/String;)V: Graph of 193 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: BubbleSort.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 59 rules for P and 0 rules for R.

P rules:
1618_0_sort_Load(EOS(STATIC_1618), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i108, i108) → 1620_0_sort_GE(EOS(STATIC_1620), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i108, i108, 100) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1620_0_sort_GE(EOS(STATIC_1620), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i111, matching4) → 1622_0_sort_GE(EOS(STATIC_1622), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i111, 100) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1622_0_sort_GE(EOS(STATIC_1622), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i111, matching4) → 1624_0_sort_ConstantStackPush(EOS(STATIC_1624), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111) | &&(&&(&&(&&(<(i111, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1624_0_sort_ConstantStackPush(EOS(STATIC_1624), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111) → 1627_0_sort_Store(EOS(STATIC_1627), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, 0) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1627_0_sort_Store(EOS(STATIC_1627), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, matching4) → 1630_0_sort_Load(EOS(STATIC_1630), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, 0) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 0))
1630_0_sort_Load(EOS(STATIC_1630), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, matching4) → 1668_0_sort_Load(EOS(STATIC_1668), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, 0) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 0))
1668_0_sort_Load(EOS(STATIC_1668), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i113) → 1724_0_sort_Load(EOS(STATIC_1724), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i113) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1724_0_sort_Load(EOS(STATIC_1724), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i122) → 1779_0_sort_Load(EOS(STATIC_1779), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i122) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1779_0_sort_Load(EOS(STATIC_1779), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i131) → 1837_0_sort_Load(EOS(STATIC_1837), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i131) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1837_0_sort_Load(EOS(STATIC_1837), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140) → 1840_0_sort_Load(EOS(STATIC_1840), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1840_0_sort_Load(EOS(STATIC_1840), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140) → 1841_0_sort_Load(EOS(STATIC_1841), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140, 100) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1841_0_sort_Load(EOS(STATIC_1841), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, matching4) → 1843_0_sort_IntArithmetic(EOS(STATIC_1843), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140, 100, i111) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1843_0_sort_IntArithmetic(EOS(STATIC_1843), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, matching4, i111) → 1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140, -(100, i111)) | &&(&&(&&(&&(>(i111, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, i142) → 1846_0_sort_GE(EOS(STATIC_1846), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140, i142) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, i142) → 1847_0_sort_GE(EOS(STATIC_1847), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, i140, i142) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1846_0_sort_GE(EOS(STATIC_1846), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, i142) → 1848_0_sort_Inc(EOS(STATIC_1848), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111) | &&(&&(&&(>=(i140, i142), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
1848_0_sort_Inc(EOS(STATIC_1848), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111) → 1851_0_sort_JMP(EOS(STATIC_1851), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, +(i111, 1)) | &&(&&(&&(>(i111, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
1851_0_sort_JMP(EOS(STATIC_1851), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i143) → 1860_0_sort_Load(EOS(STATIC_1860), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i143) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1860_0_sort_Load(EOS(STATIC_1860), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i143) → 1616_0_sort_Load(EOS(STATIC_1616), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i143) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1616_0_sort_Load(EOS(STATIC_1616), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i108) → 1618_0_sort_Load(EOS(STATIC_1618), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i108, i108) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1847_0_sort_GE(EOS(STATIC_1847), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, i140, i142) → 1850_0_sort_Load(EOS(STATIC_1850), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140) | &&(&&(&&(<(i140, i142), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
1850_0_sort_Load(EOS(STATIC_1850), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140) → 1853_0_sort_Load(EOS(STATIC_1853), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, java.lang.Object(ARRAY(100))) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1853_0_sort_Load(EOS(STATIC_1853), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i140, java.lang.Object(ARRAY(matching4))) → 1862_0_sort_ArrayAccess(EOS(STATIC_1862), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i140, java.lang.Object(ARRAY(100)), i140) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1862_0_sort_ArrayAccess(EOS(STATIC_1862), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 1864_0_sort_ArrayAccess(EOS(STATIC_1864), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1864_0_sort_ArrayAccess(EOS(STATIC_1864), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 1867_0_sort_Load(EOS(STATIC_1867), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149) | &&(&&(&&(&&(<(i147, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1867_0_sort_Load(EOS(STATIC_1867), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149) → 1870_0_sort_Load(EOS(STATIC_1870), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, java.lang.Object(ARRAY(100))) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1870_0_sort_Load(EOS(STATIC_1870), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, java.lang.Object(ARRAY(matching4))) → 1872_0_sort_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1872_0_sort_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, java.lang.Object(ARRAY(matching4)), i147) → 1875_0_sort_IntArithmetic(EOS(STATIC_1875), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, java.lang.Object(ARRAY(100)), i147, 1) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1875_0_sort_IntArithmetic(EOS(STATIC_1875), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, java.lang.Object(ARRAY(matching4)), i147, matching5) → 1879_0_sort_ArrayAccess(EOS(STATIC_1879), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, java.lang.Object(ARRAY(100)), +(i147, 1)) | &&(&&(&&(&&(&&(>=(i147, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 1))
1879_0_sort_ArrayAccess(EOS(STATIC_1879), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, java.lang.Object(ARRAY(matching4)), i152) → 1880_0_sort_ArrayAccess(EOS(STATIC_1880), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, java.lang.Object(ARRAY(100)), i152) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1880_0_sort_ArrayAccess(EOS(STATIC_1880), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, java.lang.Object(ARRAY(matching4)), i152) → 1885_0_sort_LE(EOS(STATIC_1885), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, i153) | &&(&&(&&(&&(<(i152, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1885_0_sort_LE(EOS(STATIC_1885), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, i153) → 1889_0_sort_LE(EOS(STATIC_1889), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, i153) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1885_0_sort_LE(EOS(STATIC_1885), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, i153) → 1890_0_sort_LE(EOS(STATIC_1890), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, i149, i153) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1889_0_sort_LE(EOS(STATIC_1889), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, i153) → 1892_0_sort_Inc(EOS(STATIC_1892), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(&&(<=(i149, i153), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
1892_0_sort_Inc(EOS(STATIC_1892), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 2191_0_sort_Inc(EOS(STATIC_2191), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
2191_0_sort_Inc(EOS(STATIC_2191), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 2202_0_sort_JMP(EOS(STATIC_2202), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, +(i147, 1)) | &&(&&(&&(>=(i147, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
2202_0_sort_JMP(EOS(STATIC_2202), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i164) → 2215_0_sort_Load(EOS(STATIC_2215), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i164) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
2215_0_sort_Load(EOS(STATIC_2215), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i164) → 1837_0_sort_Load(EOS(STATIC_1837), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i164) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1890_0_sort_LE(EOS(STATIC_1890), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, i149, i153) → 1894_0_sort_Load(EOS(STATIC_1894), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(&&(>(i149, i153), =(matching1, 100)), =(matching2, 100)), =(matching3, 100))
1894_0_sort_Load(EOS(STATIC_1894), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 1901_0_sort_Load(EOS(STATIC_1901), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100))) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1901_0_sort_Load(EOS(STATIC_1901), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4))) → 1911_0_sort_ArrayAccess(EOS(STATIC_1911), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1911_0_sort_ArrayAccess(EOS(STATIC_1911), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 1912_0_sort_Store(EOS(STATIC_1912), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(&&(&&(<(i147, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1912_0_sort_Store(EOS(STATIC_1912), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 1918_0_sort_Load(EOS(STATIC_1918), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1918_0_sort_Load(EOS(STATIC_1918), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 1922_0_sort_Load(EOS(STATIC_1922), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100))) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
1922_0_sort_Load(EOS(STATIC_1922), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4))) → 1924_0_sort_Load(EOS(STATIC_1924), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1924_0_sort_Load(EOS(STATIC_1924), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 1929_0_sort_Load(EOS(STATIC_1929), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, java.lang.Object(ARRAY(100))) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
1929_0_sort_Load(EOS(STATIC_1929), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, java.lang.Object(ARRAY(matching5))) → 1934_0_sort_ConstantStackPush(EOS(STATIC_1934), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 100))
1934_0_sort_ConstantStackPush(EOS(STATIC_1934), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, java.lang.Object(ARRAY(matching5)), i147) → 1937_0_sort_IntArithmetic(EOS(STATIC_1937), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, java.lang.Object(ARRAY(100)), i147, 1) | &&(&&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 100))
1937_0_sort_IntArithmetic(EOS(STATIC_1937), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, java.lang.Object(ARRAY(matching5)), i147, matching6) → 1943_0_sort_ArrayAccess(EOS(STATIC_1943), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, java.lang.Object(ARRAY(100)), +(i147, 1)) | &&(&&(&&(&&(&&(&&(>=(i147, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 100)), =(matching6, 1))
1943_0_sort_ArrayAccess(EOS(STATIC_1943), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, java.lang.Object(ARRAY(matching5)), i159) → 1956_0_sort_ArrayAccess(EOS(STATIC_1956), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, java.lang.Object(ARRAY(100)), i159) | &&(&&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 100))
1956_0_sort_ArrayAccess(EOS(STATIC_1956), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, java.lang.Object(ARRAY(matching5)), i159) → 1961_0_sort_ArrayAccess(EOS(STATIC_1961), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(&&(&&(<(i159, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 100))
1961_0_sort_ArrayAccess(EOS(STATIC_1961), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 2125_0_sort_Load(EOS(STATIC_2125), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(&&(&&(<(i147, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
2125_0_sort_Load(EOS(STATIC_2125), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147) → 2131_0_sort_Load(EOS(STATIC_2131), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100))) | &&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100))
2131_0_sort_Load(EOS(STATIC_2131), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4))) → 2138_0_sort_ConstantStackPush(EOS(STATIC_2138), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
2138_0_sort_ConstantStackPush(EOS(STATIC_2138), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147) → 2145_0_sort_IntArithmetic(EOS(STATIC_2145), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i147, 1) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
2145_0_sort_IntArithmetic(EOS(STATIC_2145), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i147, matching5) → 2161_0_sort_Load(EOS(STATIC_2161), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), +(i147, 1)) | &&(&&(&&(&&(&&(>=(i147, 0), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100)), =(matching5, 1))
2161_0_sort_Load(EOS(STATIC_2161), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i161) → 2169_0_sort_ArrayAccess(EOS(STATIC_2169), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i161) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
2169_0_sort_ArrayAccess(EOS(STATIC_2169), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i163) → 2176_0_sort_ArrayAccess(EOS(STATIC_2176), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147, java.lang.Object(ARRAY(100)), i163) | &&(&&(&&(=(matching1, 100), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
2176_0_sort_ArrayAccess(EOS(STATIC_2176), java.lang.Object(ARRAY(matching1)), java.lang.Object(ARRAY(matching2)), matching3, i111, i147, java.lang.Object(ARRAY(matching4)), i163) → 2191_0_sort_Inc(EOS(STATIC_2191), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, i111, i147) | &&(&&(&&(&&(<(i163, 100), =(matching1, 100)), =(matching2, 100)), =(matching3, 100)), =(matching4, 100))
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, x3, x4, x4, x5) → 1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, +(x3, 1), 0, 0, -(100, +(x3, 1))) | &&(&&(<=(x5, x4), >(x3, 0)), <(x3, 99))
1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, x3, x4, x4, x5) → 1844_0_sort_GE(EOS(STATIC_1844), java.lang.Object(ARRAY(100)), java.lang.Object(ARRAY(100)), 100, x3, +(x4, 1), +(x4, 1), -(100, x3)) | &&(&&(&&(&&(>(x5, x4), >(+(x4, 1), 0)), <(x4, 99)), <(x4, 100)), >(x3, 0))
R rules:

Filtered ground terms:

1844_0_sort_GE(x1, x2, x3, x4, x5, x6, x7, x8) → 1844_0_sort_GE(x5, x6, x7, x8)
ARRAY(x1) → ARRAY
java.lang.Object(x1) → java.lang.Object
EOS(x1) → EOS
Cond_1844_0_sort_GE1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1844_0_sort_GE1(x1, x6, x7, x8, x9)
Cond_1844_0_sort_GE(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1844_0_sort_GE(x1, x6, x7, x8, x9)

Filtered duplicate args:

1844_0_sort_GE(x1, x2, x3, x4) → 1844_0_sort_GE(x1, x3, x4)
Cond_1844_0_sort_GE(x1, x2, x3, x4, x5) → Cond_1844_0_sort_GE(x1, x2, x4, x5)
Cond_1844_0_sort_GE1(x1, x2, x3, x4, x5) → Cond_1844_0_sort_GE1(x1, x2, x4, x5)

Filtered unneeded arguments:

Cond_1844_0_sort_GE(x1, x2, x3, x4) → Cond_1844_0_sort_GE(x1, x2)
Cond_1844_0_sort_GE1(x1, x2, x3, x4) → Cond_1844_0_sort_GE1(x1, x2, x3)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1844_0_sort_GE(x3, x4, x5) → 1844_0_sort_GE(+(x3, 1), 0, -(100, +(x3, 1))) | &&(&&(<=(x5, x4), >(x3, 0)), <(x3, 99))
1844_0_sort_GE(x3, x4, x5) → 1844_0_sort_GE(x3, +(x4, 1), -(100, x3)) | &&(&&(&&(&&(>(x5, x4), >(x4, -1)), <(x4, 99)), <(x4, 100)), >(x3, 0))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE(&&(&&(<=(x5, x4), >(x3, 0)), <(x3, 99)), x3, x4, x5)
COND_1844_0_SORT_GE(TRUE, x3, x4, x5) → 1844_0_SORT_GE(+(x3, 1), 0, -(100, +(x3, 1)))
1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5, x4), >(x4, -1)), <(x4, 99)), <(x4, 100)), >(x3, 0)), x3, x4, x5)
COND_1844_0_SORT_GE1(TRUE, x3, x4, x5) → 1844_0_SORT_GE(x3, +(x4, 1), -(100, x3))
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): 1844_0_SORT_GE(x3[0], x4[0], x5[0]) → COND_1844_0_SORT_GE(x5[0] <= x4[0] && x3[0] > 0 && x3[0] < 99, x3[0], x4[0], x5[0])
(1): COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1]) → 1844_0_SORT_GE(x3[1] + 1, 0, 100 - x3[1] + 1)
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])

(0) -> (1), if (x5[0] <= x4[0] && x3[0] > 0 && x3[0] < 99x3[0]* x3[1]x4[0]* x4[1]x5[0]* x5[1])

(1) -> (0), if (x3[1] + 1* x3[0]0* x4[0]100 - x3[1] + 1* x5[0])

(1) -> (2), if (x3[1] + 1* x3[2]0* x4[2]100 - x3[1] + 1* x5[2])

(2) -> (3), if (x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0x3[2]* x3[3]x4[2]* x4[3]x5[2]* x5[3])

(3) -> (0), if (x3[3]* x3[0]x4[3] + 1* x4[0]100 - x3[3]* x5[0])

(3) -> (2), if (x3[3]* x3[2]x4[3] + 1* x4[2]100 - x3[3]* x5[2])

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.IdpCand1ShapeHeuristic@2710e05e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE(&&(&&(<=(x5, x4), >(x3, 0)), <(x3, 99)), x3, x4, x5) the following chains were created:
• We consider the chain 1844_0_SORT_GE(x3[0], x4[0], x5[0]) → COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0]), COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1]) → 1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1))) which results in the following constraint:

(1)    (&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99))=TRUEx3[0]=x3[1]x4[0]=x4[1]x5[0]=x5[1]1844_0_SORT_GE(x3[0], x4[0], x5[0])≥NonInfC∧1844_0_SORT_GE(x3[0], x4[0], x5[0])≥COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])∧(UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥))

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

(2)    (<(x3[0], 99)=TRUE<=(x5[0], x4[0])=TRUE>(x3[0], 0)=TRUE1844_0_SORT_GE(x3[0], x4[0], x5[0])≥NonInfC∧1844_0_SORT_GE(x3[0], x4[0], x5[0])≥COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])∧(UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥))

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

(3)    ([98] + [-1]x3[0] ≥ 0∧x4[0] + [-1]x5[0] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(4)    ([98] + [-1]x3[0] ≥ 0∧x4[0] + [-1]x5[0] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(5)    ([98] + [-1]x3[0] ≥ 0∧x4[0] + [-1]x5[0] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(6)    ([97] + [-1]x3[0] ≥ 0∧x4[0] + [-1]x5[0] ≥ 0∧x3[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(7)    ([97] + [-1]x3[0] ≥ 0∧x4[0] ≥ 0∧x3[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(8)    ([97] + [-1]x3[0] ≥ 0∧x4[0] ≥ 0∧x3[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

(9)    ([97] + [-1]x3[0] ≥ 0∧x4[0] ≥ 0∧x3[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_1844_0_SORT_GE(TRUE, x3, x4, x5) → 1844_0_SORT_GE(+(x3, 1), 0, -(100, +(x3, 1))) the following chains were created:
• We consider the chain COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1]) → 1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1))) which results in the following constraint:

(10)    (COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1])≥NonInfC∧COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1])≥1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))∧(UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥))

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

(11)    ((UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥)∧[bni_15] = 0∧[1 + (-1)bso_16] ≥ 0)

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

(12)    ((UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥)∧[bni_15] = 0∧[1 + (-1)bso_16] ≥ 0)

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

(13)    ((UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥)∧[bni_15] = 0∧[1 + (-1)bso_16] ≥ 0)

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

(14)    ((UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥)∧[bni_15] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5, x4), >(x4, -1)), <(x4, 99)), <(x4, 100)), >(x3, 0)), x3, x4, x5) the following chains were created:
• We consider the chain 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2]), COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(15)    (&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0))=TRUEx3[2]=x3[3]x4[2]=x4[3]x5[2]=x5[3]1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(16)    (>(x3[2], 0)=TRUE<(x4[2], 100)=TRUE<(x4[2], 99)=TRUE>(x5[2], x4[2])=TRUE>(x4[2], -1)=TRUE1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(17)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(18)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(19)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(20)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(21)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_1844_0_SORT_GE1(TRUE, x3, x4, x5) → 1844_0_SORT_GE(x3, +(x4, 1), -(100, x3)) the following chains were created:
• We consider the chain COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(22)    (COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥NonInfC∧COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))∧(UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥))

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

(23)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_19] = 0∧[(-1)bso_20] ≥ 0)

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

(24)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_19] = 0∧[(-1)bso_20] ≥ 0)

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

(25)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_19] = 0∧[(-1)bso_20] ≥ 0)

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

(26)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE(&&(&&(<=(x5, x4), >(x3, 0)), <(x3, 99)), x3, x4, x5)
• ([97] + [-1]x3[0] ≥ 0∧x4[0] ≥ 0∧x3[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)
• ([97] + [-1]x3[0] ≥ 0∧x4[0] ≥ 0∧x3[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x3[0] ≥ 0∧[(-1)bso_14] ≥ 0)

• COND_1844_0_SORT_GE(TRUE, x3, x4, x5) → 1844_0_SORT_GE(+(x3, 1), 0, -(100, +(x3, 1)))
• ((UIncreasing(1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))), ≥)∧[bni_15] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• 1844_0_SORT_GE(x3, x4, x5) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5, x4), >(x4, -1)), <(x4, 99)), <(x4, 100)), >(x3, 0)), x3, x4, x5)
• (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x3[2] ≥ 0∧[(-1)bso_18] ≥ 0)

• COND_1844_0_SORT_GE1(TRUE, x3, x4, x5) → 1844_0_SORT_GE(x3, +(x4, 1), -(100, x3))
• ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_20] ≥ 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(1844_0_SORT_GE(x1, x2, x3)) = [-1] + [-1]x1
POL(COND_1844_0_SORT_GE(x1, x2, x3, x4)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(99) = [99]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(100) = [100]
POL(COND_1844_0_SORT_GE1(x1, x2, x3, x4)) = [-1] + [-1]x2
POL(-1) = [-1]

The following pairs are in P>:

COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1]) → 1844_0_SORT_GE(+(x3[1], 1), 0, -(100, +(x3[1], 1)))

The following pairs are in Pbound:

1844_0_SORT_GE(x3[0], x4[0], x5[0]) → COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])

The following pairs are in P:

1844_0_SORT_GE(x3[0], x4[0], x5[0]) → COND_1844_0_SORT_GE(&&(&&(<=(x5[0], x4[0]), >(x3[0], 0)), <(x3[0], 99)), x3[0], x4[0], x5[0])
1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])
COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))

There are no usable rules.

### (9) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1844_0_SORT_GE(x3[0], x4[0], x5[0]) → COND_1844_0_SORT_GE(x5[0] <= x4[0] && x3[0] > 0 && x3[0] < 99, x3[0], x4[0], x5[0])
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])

(3) -> (0), if (x3[3]* x3[0]x4[3] + 1* x4[0]100 - x3[3]* x5[0])

(3) -> (2), if (x3[3]* x3[2]x4[3] + 1* x4[2]100 - x3[3]* x5[2])

(2) -> (3), if (x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0x3[2]* x3[3]x4[2]* x4[3]x5[2]* x5[3])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (11) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])

(3) -> (2), if (x3[3]* x3[2]x4[3] + 1* x4[2]100 - x3[3]* x5[2])

(2) -> (3), if (x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0x3[2]* x3[3]x4[2]* x4[3]x5[2]* x5[3])

The set Q is empty.

### (12) 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.IdpCand1ShapeHeuristic@2710e05e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) the following chains were created:
• We consider the chain COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(1)    (COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥NonInfC∧COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))∧(UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥))

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

(2)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2]) the following chains were created:
• We consider the chain 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2]), COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(6)    (&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0))=TRUEx3[2]=x3[3]x4[2]=x4[3]x5[2]=x5[3]1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(7)    (>(x3[2], 0)=TRUE<(x4[2], 100)=TRUE<(x4[2], 99)=TRUE>(x5[2], x4[2])=TRUE>(x4[2], -1)=TRUE1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(8)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(12)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))
• ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])
• (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)Bound*bni_10] + [(-1)bni_10]x4[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1844_0_SORT_GE1(x1, x2, x3, x4)) = [-1]x3
POL(1844_0_SORT_GE(x1, x2, x3)) = [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(100) = [100]
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = 0
POL(-1) = [-1]
POL(<(x1, x2)) = [-1]
POL(99) = [99]
POL(0) = 0

The following pairs are in P>:

COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))

The following pairs are in Pbound:

1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])

The following pairs are in P:

1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])

There are no usable rules.

### (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:
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])

The set Q is empty.

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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

### (17) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])

The set Q is empty.

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1844_0_SORT_GE(TRUE, x3[1], x4[1], x5[1]) → 1844_0_SORT_GE(x3[1] + 1, 0, 100 - x3[1] + 1)
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])

(1) -> (2), if (x3[1] + 1* x3[2]0* x4[2]100 - x3[1] + 1* x5[2])

(3) -> (2), if (x3[3]* x3[2]x4[3] + 1* x4[2]100 - x3[3]* x5[2])

(2) -> (3), if (x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0x3[2]* x3[3]x4[2]* x4[3]x5[2]* x5[3])

The set Q is empty.

### (21) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) 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:
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])

(3) -> (2), if (x3[3]* x3[2]x4[3] + 1* x4[2]100 - x3[3]* x5[2])

(2) -> (3), if (x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0x3[2]* x3[3]x4[2]* x4[3]x5[2]* x5[3])

The set Q is empty.

### (23) 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.IdpCand1ShapeHeuristic@2710e05e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) the following chains were created:
• We consider the chain COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(1)    (COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥NonInfC∧COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3])≥1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))∧(UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥))

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

(2)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(3)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(4)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(5)    ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

For Pair 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2]) the following chains were created:
• We consider the chain 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2]), COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3])) which results in the following constraint:

(6)    (&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0))=TRUEx3[2]=x3[3]x4[2]=x4[3]x5[2]=x5[3]1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(7)    (>(x3[2], 0)=TRUE<(x4[2], 100)=TRUE<(x4[2], 99)=TRUE>(x5[2], x4[2])=TRUE>(x4[2], -1)=TRUE1844_0_SORT_GE(x3[2], x4[2], x5[2])≥NonInfC∧1844_0_SORT_GE(x3[2], x4[2], x5[2])≥COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])∧(UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥))

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

(8)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(9)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(10)    (x3[2] + [-1] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(11)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] + [-1] + [-1]x4[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(12)    (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))
• ((UIncreasing(1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

• 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])
• (x3[2] ≥ 0∧[99] + [-1]x4[2] ≥ 0∧[98] + [-1]x4[2] ≥ 0∧x5[2] ≥ 0∧x4[2] ≥ 0 ⇒ (UIncreasing(COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x4[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1844_0_SORT_GE1(x1, x2, x3, x4)) = [-1] + [-1]x3
POL(1844_0_SORT_GE(x1, x2, x3)) = [-1] + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(100) = [100]
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(<(x1, x2)) = [-1]
POL(99) = [99]
POL(0) = 0

The following pairs are in P>:

COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], +(x4[3], 1), -(100, x3[3]))

The following pairs are in Pbound:

1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])

The following pairs are in P:

1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(&&(&&(&&(&&(>(x5[2], x4[2]), >(x4[2], -1)), <(x4[2], 99)), <(x4[2], 100)), >(x3[2], 0)), x3[2], x4[2], x5[2])

There are no usable rules.

### (25) 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:
(2): 1844_0_SORT_GE(x3[2], x4[2], x5[2]) → COND_1844_0_SORT_GE1(x5[2] > x4[2] && x4[2] > -1 && x4[2] < 99 && x4[2] < 100 && x3[2] > 0, x3[2], x4[2], x5[2])

The set Q is empty.

### (26) IDependencyGraphProof (EQUIVALENT transformation)

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

### (28) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_1844_0_SORT_GE1(TRUE, x3[3], x4[3], x5[3]) → 1844_0_SORT_GE(x3[3], x4[3] + 1, 100 - x3[3])

The set Q is empty.

### (29) IDependencyGraphProof (EQUIVALENT transformation)

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