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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 2012 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 31 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 5 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x && x <= x2
l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && x25 = x16 && 1 + x16 <= x14
l4(x28, x29, x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x30 = x37 && x29 = x36 && x28 = x35 && x38 = 1 + x31
l5(x42, x43, x44, x45, x46, x47, x48) -> l2(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
l6(x56, x57, x58, x59, x60, x61, x62) -> l4(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
l7(x70, x71, x72, x73, x74, x75, x76) -> l8(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x71 <= x73
l7(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x97 = x87 && 1 + x87 <= x85
l3(x98, x99, x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
l9(x112, x113, x114, x115, x116, x117, x118) -> l5(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x113 = x120 && x112 = x119 && x121 = 1 + x114
l10(x126, x127, x128, x129, x130, x131, x132) -> l9(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133
l1(x140, x141, x142, x143, x144, x145, x146) -> l10(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && x152 = x142
l11(x154, x155, x156, x157, x158, x159, x160) -> l5(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x155 = x162 && x154 = x161 && x163 = 0
l12(x168, x169, x170, x171, x172, x173, x174) -> l11(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175
Start term: l12(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x && x <= x2
l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && x25 = x16 && 1 + x16 <= x14
l4(x28, x29, x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x30 = x37 && x29 = x36 && x28 = x35 && x38 = 1 + x31
l5(x42, x43, x44, x45, x46, x47, x48) -> l2(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
l6(x56, x57, x58, x59, x60, x61, x62) -> l4(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
l7(x70, x71, x72, x73, x74, x75, x76) -> l8(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x71 <= x73
l7(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x97 = x87 && 1 + x87 <= x85
l3(x98, x99, x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
l9(x112, x113, x114, x115, x116, x117, x118) -> l5(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x113 = x120 && x112 = x119 && x121 = 1 + x114
l10(x126, x127, x128, x129, x130, x131, x132) -> l9(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133
l1(x140, x141, x142, x143, x144, x145, x146) -> l10(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && x152 = x142
l11(x154, x155, x156, x157, x158, x159, x160) -> l5(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x155 = x162 && x154 = x161 && x163 = 0
l12(x168, x169, x170, x171, x172, x173, x174) -> l11(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175
Start term: l12(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost
(2) l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x && x <= x2
(3) l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && x25 = x16 && 1 + x16 <= x14
(4) l4(x28, x29, x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x30 = x37 && x29 = x36 && x28 = x35 && x38 = 1 + x31
(5) l5(x42, x43, x44, x45, x46, x47, x48) -> l2(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
(6) l6(x56, x57, x58, x59, x60, x61, x62) -> l4(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
(7) l7(x70, x71, x72, x73, x74, x75, x76) -> l8(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x71 <= x73
(8) l7(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x97 = x87 && 1 + x87 <= x85
(9) l3(x98, x99, x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
(10) l9(x112, x113, x114, x115, x116, x117, x118) -> l5(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x113 = x120 && x112 = x119 && x121 = 1 + x114
(11) l10(x126, x127, x128, x129, x130, x131, x132) -> l9(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133
(12) l1(x140, x141, x142, x143, x144, x145, x146) -> l10(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && x152 = x142
(13) l11(x154, x155, x156, x157, x158, x159, x160) -> l5(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x155 = x162 && x154 = x161 && x163 = 0
(14) l12(x168, x169, x170, x171, x172, x173, x174) -> l11(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost
(2) l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && x25 = x16 && 1 + x16 <= x14
(3) l5(x42, x43, x44, x45, x46, x47, x48) -> l2(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
(4) l9(x112, x113, x114, x115, x116, x117, x118) -> l5(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x113 = x120 && x112 = x119 && x121 = 1 + x114
(5) l10(x126, x127, x128, x129, x130, x131, x132) -> l9(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133
(6) l1(x140, x141, x142, x143, x144, x145, x146) -> l10(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && x152 = x142

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l10(__const_100HATpost:0, __const_200HATpost:0, x128:0, jHATpost:0, x123:0, x124:0, x125:0) -> l10(__const_100HATpost:0, __const_200HATpost:0, 1 + x128:0, jHATpost:0, 1 + x128:0, 1 + x128:0, x125:0) :|: __const_100HATpost:0 >= 1 + (1 + x128:0)

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

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

   l10(x1, x2, x3, x4, x5, x6, x7) -> l10(x1, x3)

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

(9)
Obligation:
Rules:
l10(__const_100HATpost:0, x128:0) -> l10(__const_100HATpost:0, 1 + x128:0) :|: __const_100HATpost:0 >= 1 + (1 + x128:0)

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l10(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l10(__const_100HATpost:0, x128:0) -> l10(__const_100HATpost:0, c) :|: c = 1 + x128:0 && __const_100HATpost:0 >= 1 + (1 + x128:0)

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l10(x, x1)] = x - x1

The following rules are decreasing:
l10(__const_100HATpost:0, x128:0) -> l10(__const_100HATpost:0, c) :|: c = 1 + x128:0 && __const_100HATpost:0 >= 1 + (1 + x128:0)
The following rules are bounded:
l10(__const_100HATpost:0, x128:0) -> l10(__const_100HATpost:0, c) :|: c = 1 + x128:0 && __const_100HATpost:0 >= 1 + (1 + x128:0)

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l3(x98, x99, x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
(2) l4(x28, x29, x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x30 = x37 && x29 = x36 && x28 = x35 && x38 = 1 + x31
(3) l6(x56, x57, x58, x59, x60, x61, x62) -> l4(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
(4) l7(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x97 = x87 && 1 + x87 <= x85

Arcs:
(1) -> (4)
(2) -> (1)
(3) -> (2)
(4) -> (3)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l6(x105:0, x106:0, x107:0, x59:0, x109:0, x110:0, x111:0) -> l6(x105:0, x106:0, x107:0, 1 + x59:0, x109:0, x110:0, 1 + x59:0) :|: x106:0 >= 1 + (1 + x59:0)

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

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

   l6(x1, x2, x3, x4, x5, x6, x7) -> l6(x2, x4)

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

(18)
Obligation:
Rules:
l6(x106:0, x59:0) -> l6(x106:0, 1 + x59:0) :|: x106:0 >= 1 + (1 + x59:0)

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

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

(20)
Obligation:
Rules:
l6(x106:0, x59:0) -> l6(x106:0, c) :|: c = 1 + x59:0 && x106:0 >= 1 + (1 + x59:0)

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

(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1)] = x - x1

The following rules are decreasing:
l6(x106:0, x59:0) -> l6(x106:0, c) :|: c = 1 + x59:0 && x106:0 >= 1 + (1 + x59:0)
The following rules are bounded:
l6(x106:0, x59:0) -> l6(x106:0, c) :|: c = 1 + x59:0 && x106:0 >= 1 + (1 + x59:0)

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

(22)
YES