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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1122 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 103 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) AND
    (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (15) IntTRS
        (16) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (20) YES


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

(0)
Obligation:
Rules:
l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + yHAT0 <= 0
l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x2 <= x4
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 + x13 <= 0
l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x21 <= x23
l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = x33
l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= 0
l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x55 = 1 && x57 = x54 && x56 = x53
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x60 <= 0
l5(x70, x71, x72, x73, x74) -> l2(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x70
l6(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x89 = -1 + x84
l6(x90, x91, x92, x93, x94) -> l5(x95, x96, x97, x98, x99) :|: x94 = x99 && x92 = x97 && x91 = x96 && x90 = x95 && x98 = -1 + x93
l3(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 <= x104 && 1 <= x103
l7(x110, x111, x112, x113, x114) -> l3(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && x110 <= 0
l8(x120, x121, x122, x123, x124) -> l7(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125
Start term: l8(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0)

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

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

(2)
Obligation:
Rules:
l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + yHAT0 <= 0
l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x2 <= x4
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 + x13 <= 0
l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x21 <= x23
l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = x33
l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= 0
l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x55 = 1 && x57 = x54 && x56 = x53
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x60 <= 0
l5(x70, x71, x72, x73, x74) -> l2(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x70
l6(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x89 = -1 + x84
l6(x90, x91, x92, x93, x94) -> l5(x95, x96, x97, x98, x99) :|: x94 = x99 && x92 = x97 && x91 = x96 && x90 = x95 && x98 = -1 + x93
l3(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 <= x104 && 1 <= x103
l7(x110, x111, x112, x113, x114) -> l3(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && x110 <= 0
l8(x120, x121, x122, x123, x124) -> l7(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125
Start term: l8(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + yHAT0 <= 0
(2) l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x2 <= x4
(3) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 + x13 <= 0
(4) l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x21 <= x23
(5) l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = x33
(6) l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= 0
(7) l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x55 = 1 && x57 = x54 && x56 = x53
(8) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x60 <= 0
(9) l5(x70, x71, x72, x73, x74) -> l2(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x70
(10) l6(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x89 = -1 + x84
(11) l6(x90, x91, x92, x93, x94) -> l5(x95, x96, x97, x98, x99) :|: x94 = x99 && x92 = x97 && x91 = x96 && x90 = x95 && x98 = -1 + x93
(12) l3(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 <= x104 && 1 <= x103
(13) l7(x110, x111, x112, x113, x114) -> l3(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && x110 <= 0
(14) l8(x120, x121, x122, x123, x124) -> l7(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = x33
(2) l5(x70, x71, x72, x73, x74) -> l2(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x70
(3) l6(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x89 = -1 + x84
(4) l3(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 <= x104 && 1 <= x103
(5) l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x55 = 1 && x57 = x54 && x56 = x53
(6) l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= 0
(7) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x60 <= 0
(8) l6(x90, x91, x92, x93, x94) -> l5(x95, x96, x97, x98, x99) :|: x94 = x99 && x92 = x97 && x91 = x96 && x90 = x95 && x98 = -1 + x93

Arcs:
(1) -> (4)
(2) -> (1)
(3) -> (2), (7)
(4) -> (3), (8)
(5) -> (4)
(6) -> (4)
(7) -> (5), (6)
(8) -> (2), (7)

This digraph is fully evaluated!

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

(6)
Obligation:
Rules:
l5(x105:0, x106:0, x107:0, x108:0, x109:0) -> l6(x105:0, x106:0, x107:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x1, x2, x3, x4) -> l6(1, x3, x4, x3, x4) :|: x3 > 0 && x4 > 0 && x < 1
l6(x80:0, x81:0, x82:0, x83:0, x84:0) -> l5(x80:0, x81:0, x82:0, x83:0, -1 + x84:0) :|: TRUE
l6(x90:0, x91:0, x92:0, x93:0, x94:0) -> l5(x90:0, x91:0, x92:0, -1 + x93:0, x94:0) :|: TRUE
l5(x5, x6, x7, x8, x9) -> l6(x5, x6, x7, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

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(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, x5) -> l5(x1, x4, x5)
   l6(x1, x2, x3, x4, x5) -> l6(x1, x4, x5)

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

(8)
Obligation:
Rules:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(1, x3, x4) :|: x3 > 0 && x4 > 0 && x < 1
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, -1 + x84:0) :|: TRUE
l6(x90:0, x93:0, x94:0) -> l5(x90:0, -1 + x93:0, x94:0) :|: TRUE
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l5(VARIABLE, VARIABLE, VARIABLE)
l6(VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(c, x3, x4) :|: c = 1 && (x3 > 0 && x4 > 0 && x < 1)
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, c1) :|: c1 = -1 + x84:0 && TRUE
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x, x1, x2)] = -1 + x2
[l6(x3, x4, x5)] = -1 + x5

The following rules are decreasing:
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, c1) :|: c1 = -1 + x84:0 && TRUE
The following rules are bounded:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(c, x3, x4) :|: c = 1 && (x3 > 0 && x4 > 0 && x < 1)
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

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(12)
Complex Obligation (AND)

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

(13)
Obligation:
Rules:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(c, x3, x4) :|: c = 1 && (x3 > 0 && x4 > 0 && x < 1)
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

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(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l5 ] = 2*l5_3 + 2*l5_2 + 1
[ l6 ] = 2*l6_2 + 2*l6_3

The following rules are decreasing:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(c, x3, x4) :|: c = 1 && (x3 > 0 && x4 > 0 && x < 1)
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0

The following rules are bounded:
l5(x105:0, x108:0, x109:0) -> l6(x105:0, x108:0, x109:0) :|: x105:0 < 1 && x109:0 > 0 && x108:0 > 0
l5(x, x3, x4) -> l6(c, x3, x4) :|: c = 1 && (x3 > 0 && x4 > 0 && x < 1)
l5(x5, x8, x9) -> l6(x5, x8, x9) :|: x8 > 0 && x9 > 0 && x5 > 0


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

(15)
Obligation:
Rules:
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE

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

(16) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l6 ] = 0
[ l5 ] = -1

The following rules are decreasing:
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE

The following rules are bounded:
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE


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(17)
YES

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

(18)
Obligation:
Rules:
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, c1) :|: c1 = -1 + x84:0 && TRUE
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE

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

(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1, x2)] = 0
[l5(x3, x4, x5)] = -1

The following rules are decreasing:
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, c1) :|: c1 = -1 + x84:0 && TRUE
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE
The following rules are bounded:
l6(x80:0, x83:0, x84:0) -> l5(x80:0, x83:0, c1) :|: c1 = -1 + x84:0 && TRUE
l6(x90:0, x93:0, x94:0) -> l5(x90:0, c2, x94:0) :|: c2 = -1 + x93:0 && TRUE

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(20)
YES