YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 969 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 42 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 104 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 3 ms]
(14) IntTRS
(15) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(16) IntTRS
(17) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(18) YES


----------------------------------------

(0)
Obligation:
Rules:
l0(iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && nHAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x2 = x6 && x1 = x5 && x = x4 && x7 = x7 && 1 + x <= x2
l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= x8
l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x21 = 0 && x20 = 1 + x16 && 1 + x16 <= x18
l6(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l1(x40, x41, x42, x43) -> l7(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && x49 <= 0
l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = -1 + x56 && 1 <= x57
l8(x64, x65, x66, x67) -> l5(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x69 = 1 + x65 && x68 = 1 + x64
l2(x72, x73, x74, x75) -> l1(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 0 <= x75 && x75 <= 0
l2(x80, x81, x82, x83) -> l8(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84 && 1 <= x83
l2(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 && 1 + x91 <= 0
l9(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 0
l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108
Start term: l10(iHAT0, jHAT0, nHAT0, tmpHAT0)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && nHAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x2 = x6 && x1 = x5 && x = x4 && x7 = x7 && 1 + x <= x2
l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= x8
l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x21 = 0 && x20 = 1 + x16 && 1 + x16 <= x18
l6(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l1(x40, x41, x42, x43) -> l7(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && x49 <= 0
l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = -1 + x56 && 1 <= x57
l8(x64, x65, x66, x67) -> l5(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x69 = 1 + x65 && x68 = 1 + x64
l2(x72, x73, x74, x75) -> l1(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 0 <= x75 && x75 <= 0
l2(x80, x81, x82, x83) -> l8(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84 && 1 <= x83
l2(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 && 1 + x91 <= 0
l9(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 0
l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108
Start term: l10(iHAT0, jHAT0, nHAT0, tmpHAT0)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && nHAT0 <= iHAT0
(2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x2 = x6 && x1 = x5 && x = x4 && x7 = x7 && 1 + x <= x2
(3) l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= x8
(4) l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x21 = 0 && x20 = 1 + x16 && 1 + x16 <= x18
(5) l6(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(6) l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
(7) l1(x40, x41, x42, x43) -> l7(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
(8) l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && x49 <= 0
(9) l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = -1 + x56 && 1 <= x57
(10) l8(x64, x65, x66, x67) -> l5(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x69 = 1 + x65 && x68 = 1 + x64
(11) l2(x72, x73, x74, x75) -> l1(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 0 <= x75 && x75 <= 0
(12) l2(x80, x81, x82, x83) -> l8(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84 && 1 <= x83
(13) l2(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 && 1 + x91 <= 0
(14) l9(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 0
(15) l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108

Arcs:
(1) -> (7)
(2) -> (11), (12), (13)
(4) -> (6)
(5) -> (3), (4)
(6) -> (1), (2)
(7) -> (8), (9)
(8) -> (5)
(9) -> (5)
(10) -> (6)
(11) -> (7)
(12) -> (10)
(13) -> (10)
(14) -> (5)
(15) -> (14)

This digraph is fully evaluated!
----------------------------------------

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && nHAT0 <= iHAT0
(2) l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
(3) l8(x64, x65, x66, x67) -> l5(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x69 = 1 + x65 && x68 = 1 + x64
(4) l2(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 && 1 + x91 <= 0
(5) l2(x80, x81, x82, x83) -> l8(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84 && 1 <= x83
(6) l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x21 = 0 && x20 = 1 + x16 && 1 + x16 <= x18
(7) l6(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(8) l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = -1 + x56 && 1 <= x57
(9) l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && x49 <= 0
(10) l1(x40, x41, x42, x43) -> l7(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
(11) l2(x72, x73, x74, x75) -> l1(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 0 <= x75 && x75 <= 0
(12) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x2 = x6 && x1 = x5 && x = x4 && x7 = x7 && 1 + x <= x2

Arcs:
(1) -> (10)
(2) -> (1), (12)
(3) -> (2)
(4) -> (3)
(5) -> (3)
(6) -> (2)
(7) -> (6)
(8) -> (7)
(9) -> (7)
(10) -> (8), (9)
(11) -> (10)
(12) -> (4), (5), (11)

This digraph is fully evaluated!

----------------------------------------

(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
l5(x32:0, x33:0, x34:0, x35:0) -> l5(1 + x32:0, 1 + x33:0, x34:0, x71:0) :|: x34:0 >= 1 + x32:0 && x71:0 < 0
l7(x56:0, x29:0, x22:0, x23:0) -> l5(1 + (-1 + x56:0), 0, x22:0, x23:0) :|: x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0)
l5(iHATpost:0, jHATpost:0, nHATpost:0, tmpHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0, tmpHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2, x3) -> l7(x, x1, x2, x4) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
l7(x5, x6, x7, x8) -> l5(1 + x5, 0, x7, x8) :|: x6 < 1 && x7 >= 1 + x5
l5(x9, x10, x11, x12) -> l5(1 + x9, 1 + x10, x11, x13) :|: x11 >= 1 + x9 && x13 > 0

----------------------------------------

(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l5(x1, x2, x3, x4) -> l5(x1, x2, x3)
   l7(x1, x2, x3, x4) -> l7(x1, x2, x3)

----------------------------------------

(8)
Obligation:
Rules:
l5(x32:0, x33:0, x34:0) -> l5(1 + x32:0, 1 + x33:0, x34:0) :|: x34:0 >= 1 + x32:0 && x71:0 < 0
l7(x56:0, x29:0, x22:0) -> l5(1 + (-1 + x56:0), 0, x22:0) :|: x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0)
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
l7(x5, x6, x7) -> l5(1 + x5, 0, x7) :|: x6 < 1 && x7 >= 1 + x5
l5(x9, x10, x11) -> l5(1 + x9, 1 + x10, x11) :|: x11 >= 1 + x9 && x13 > 0

----------------------------------------

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l5(INTEGER, VARIABLE, INTEGER)
l7(INTEGER, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l5(x32:0, x33:0, x34:0) -> l5(c, c1, x34:0) :|: c1 = 1 + x33:0 && c = 1 + x32:0 && (x34:0 >= 1 + x32:0 && x71:0 < 0)
l7(x56:0, x29:0, x22:0) -> l5(c2, c3, x22:0) :|: c3 = 0 && c2 = 1 + (-1 + x56:0) && (x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0))
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
l7(x5, x6, x7) -> l5(c4, c5, x7) :|: c5 = 0 && c4 = 1 + x5 && (x6 < 1 && x7 >= 1 + x5)
l5(x9, x10, x11) -> l5(c6, c7, x11) :|: c7 = 1 + x10 && c6 = 1 + x9 && (x11 >= 1 + x9 && x13 > 0)

----------------------------------------

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x, x1, x2)] = -x + x2
[l7(x3, x4, x5)] = -x3 + x5

The following rules are decreasing:
l5(x32:0, x33:0, x34:0) -> l5(c, c1, x34:0) :|: c1 = 1 + x33:0 && c = 1 + x32:0 && (x34:0 >= 1 + x32:0 && x71:0 < 0)
l7(x5, x6, x7) -> l5(c4, c5, x7) :|: c5 = 0 && c4 = 1 + x5 && (x6 < 1 && x7 >= 1 + x5)
l5(x9, x10, x11) -> l5(c6, c7, x11) :|: c7 = 1 + x10 && c6 = 1 + x9 && (x11 >= 1 + x9 && x13 > 0)
The following rules are bounded:
l5(x32:0, x33:0, x34:0) -> l5(c, c1, x34:0) :|: c1 = 1 + x33:0 && c = 1 + x32:0 && (x34:0 >= 1 + x32:0 && x71:0 < 0)
l7(x56:0, x29:0, x22:0) -> l5(c2, c3, x22:0) :|: c3 = 0 && c2 = 1 + (-1 + x56:0) && (x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0))
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
l7(x5, x6, x7) -> l5(c4, c5, x7) :|: c5 = 0 && c4 = 1 + x5 && (x6 < 1 && x7 >= 1 + x5)
l5(x9, x10, x11) -> l5(c6, c7, x11) :|: c7 = 1 + x10 && c6 = 1 + x9 && (x11 >= 1 + x9 && x13 > 0)

----------------------------------------

(12)
Obligation:
Rules:
l7(x56:0, x29:0, x22:0) -> l5(c2, c3, x22:0) :|: c3 = 0 && c2 = 1 + (-1 + x56:0) && (x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0))
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x

----------------------------------------

(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l7(x, x1, x2)] = -2 - x + 2*x1 + x2
[l5(x3, x4, x5)] = -2 - x3 + 2*x4 + x5

The following rules are decreasing:
l7(x56:0, x29:0, x22:0) -> l5(c2, c3, x22:0) :|: c3 = 0 && c2 = 1 + (-1 + x56:0) && (x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0))
The following rules are bounded:
l7(x56:0, x29:0, x22:0) -> l5(c2, c3, x22:0) :|: c3 = 0 && c2 = 1 + (-1 + x56:0) && (x29:0 > 0 && x22:0 >= 1 + (-1 + x56:0))

----------------------------------------

(14)
Obligation:
Rules:
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x

----------------------------------------

(15) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x, x1, x2)] = x - x2
[l7(x3, x4, x5)] = -1 + x3 - x5

The following rules are decreasing:
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
The following rules are bounded:
l5(iHATpost:0, jHATpost:0, nHATpost:0) -> l7(iHATpost:0, jHATpost:0, nHATpost:0) :|: nHATpost:0 <= iHATpost:0

----------------------------------------

(16)
Obligation:
Rules:
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x

----------------------------------------

(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x, x1, x2)] = 1
[l7(x3, x4, x5)] = 0

The following rules are decreasing:
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x
The following rules are bounded:
l5(x, x1, x2) -> l7(x, x1, x2) :|: x4 < 1 && x4 > -1 && x2 >= 1 + x

----------------------------------------

(18)
YES