NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 1336 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 54 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) FilterProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
        (22) NO


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

(0)
Obligation:
Rules:
l0(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(__const_5HATpost, __rho_1_HATpost, __rho_2_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && __rho_2_HAT0 = __rho_2_HATpost && __rho_1_HAT0 = __rho_1_HATpost && __const_5HAT0 = __const_5HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7
l4(x14, x15, x16, x17, x18, x19, x20) -> l5(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
l5(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l6(x42, x43, x44, x45, x46, x47, x48) -> l0(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 && x53 = -1 + x46 && 1 + x45 <= x46 && x43 <= 0
l6(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x58 = x65 && x57 = x64 && x56 = x63 && x68 = 1 + x61 && x67 = -1 + x60 && x66 = -1 + x59 && 1 <= x57
l1(x70, x71, x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x70 = x77 && x78 = x78 && 1 <= x74
l1(x84, x85, x86, x87, x88, x89, x90) -> l4(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x88 <= 0
l7(x98, x99, x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x98 <= x101
l7(x112, x113, x114, x115, x116, x117, x118) -> l0(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 && x115 <= x112
l8(x126, x127, x128, x129, x130, x131, x132) -> l7(x133, x134, x135, x136, x137, x138, x139) :|: x127 = x134 && x126 = x133 && x137 = x139 && x138 = 0 && x139 = 4 && 1 <= x136 && x136 = x135 && x135 = x135
l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147
Start term: l9(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(__const_5HATpost, __rho_1_HATpost, __rho_2_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && __rho_2_HAT0 = __rho_2_HATpost && __rho_1_HAT0 = __rho_1_HATpost && __const_5HAT0 = __const_5HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7
l4(x14, x15, x16, x17, x18, x19, x20) -> l5(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
l5(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l6(x42, x43, x44, x45, x46, x47, x48) -> l0(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 && x53 = -1 + x46 && 1 + x45 <= x46 && x43 <= 0
l6(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x58 = x65 && x57 = x64 && x56 = x63 && x68 = 1 + x61 && x67 = -1 + x60 && x66 = -1 + x59 && 1 <= x57
l1(x70, x71, x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x70 = x77 && x78 = x78 && 1 <= x74
l1(x84, x85, x86, x87, x88, x89, x90) -> l4(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x88 <= 0
l7(x98, x99, x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x98 <= x101
l7(x112, x113, x114, x115, x116, x117, x118) -> l0(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 && x115 <= x112
l8(x126, x127, x128, x129, x130, x131, x132) -> l7(x133, x134, x135, x136, x137, x138, x139) :|: x127 = x134 && x126 = x133 && x137 = x139 && x138 = 0 && x139 = 4 && 1 <= x136 && x136 = x135 && x135 = x135
l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147
Start term: l9(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(__const_5HATpost, __rho_1_HATpost, __rho_2_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && __rho_2_HAT0 = __rho_2_HATpost && __rho_1_HAT0 = __rho_1_HATpost && __const_5HAT0 = __const_5HATpost
(2) l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7
(3) l4(x14, x15, x16, x17, x18, x19, x20) -> l5(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
(4) l5(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
(5) l6(x42, x43, x44, x45, x46, x47, x48) -> l0(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 && x53 = -1 + x46 && 1 + x45 <= x46 && x43 <= 0
(6) l6(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x58 = x65 && x57 = x64 && x56 = x63 && x68 = 1 + x61 && x67 = -1 + x60 && x66 = -1 + x59 && 1 <= x57
(7) l1(x70, x71, x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x70 = x77 && x78 = x78 && 1 <= x74
(8) l1(x84, x85, x86, x87, x88, x89, x90) -> l4(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && x88 <= 0
(9) l7(x98, x99, x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x98 <= x101
(10) l7(x112, x113, x114, x115, x116, x117, x118) -> l0(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 && x115 <= x112
(11) l8(x126, x127, x128, x129, x130, x131, x132) -> l7(x133, x134, x135, x136, x137, x138, x139) :|: x127 = x134 && x126 = x133 && x137 = x139 && x138 = 0 && x139 = 4 && 1 <= x136 && x136 = x135 && x135 = x135
(12) l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(__const_5HAT0, __rho_1_HAT0, __rho_2_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(__const_5HATpost, __rho_1_HATpost, __rho_2_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && __rho_2_HAT0 = __rho_2_HATpost && __rho_1_HAT0 = __rho_1_HATpost && __const_5HAT0 = __const_5HATpost
(2) l6(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x58 = x65 && x57 = x64 && x56 = x63 && x68 = 1 + x61 && x67 = -1 + x60 && x66 = -1 + x59 && 1 <= x57
(3) l6(x42, x43, x44, x45, x46, x47, x48) -> l0(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 && x53 = -1 + x46 && 1 + x45 <= x46 && x43 <= 0
(4) l1(x70, x71, x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x70 = x77 && x78 = x78 && 1 <= x74

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
l6(__const_5HATpost:0, __rho_1_HATpost:0, __rho_2_HATpost:0, x59:0, x60:0, x61:0, serversHATpost:0) -> l6(__const_5HATpost:0, x78:0, __rho_2_HATpost:0, -1 + x59:0, -1 + x60:0, 1 + x61:0, serversHATpost:0) :|: __rho_1_HATpost:0 > 0 && x60:0 > 1
l6(x, x1, x2, x3, x4, x5, x6) -> l6(x, x7, x2, x3, -1 + x4, x5, x6) :|: x4 >= 1 + x3 && x4 > 1 && x1 < 1

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(8) 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, x5)

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

(9)
Obligation:
Rules:
l6(__rho_1_HATpost:0, x59:0, x60:0) -> l6(x78:0, -1 + x59:0, -1 + x60:0) :|: __rho_1_HATpost:0 > 0 && x60:0 > 1
l6(x1, x3, x4) -> l6(x7, x3, -1 + x4) :|: x4 >= 1 + x3 && x4 > 1 && x1 < 1

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

(11)
Obligation:
Rules:
l6(__rho_1_HATpost:0, x59:0, x60:0) -> l6(x78:0, c, c1) :|: c1 = -1 + x60:0 && c = -1 + x59:0 && (__rho_1_HATpost:0 > 0 && x60:0 > 1)
l6(x1, x3, x4) -> l6(x7, x3, c2) :|: c2 = -1 + x4 && (x4 >= 1 + x3 && x4 > 1 && x1 < 1)

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1, x2)] = -x1 + x2

The following rules are decreasing:
l6(x1, x3, x4) -> l6(x7, x3, c2) :|: c2 = -1 + x4 && (x4 >= 1 + x3 && x4 > 1 && x1 < 1)
The following rules are bounded:
l6(x1, x3, x4) -> l6(x7, x3, c2) :|: c2 = -1 + x4 && (x4 >= 1 + x3 && x4 > 1 && x1 < 1)

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

(13)
Obligation:
Rules:
l6(__rho_1_HATpost:0, x59:0, x60:0) -> l6(x78:0, c, c1) :|: c1 = -1 + x60:0 && c = -1 + x59:0 && (__rho_1_HATpost:0 > 0 && x60:0 > 1)

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(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l6 ] = l6_3

The following rules are decreasing:
l6(__rho_1_HATpost:0, x59:0, x60:0) -> l6(x78:0, c, c1) :|: c1 = -1 + x60:0 && c = -1 + x59:0 && (__rho_1_HATpost:0 > 0 && x60:0 > 1)

The following rules are bounded:
l6(__rho_1_HATpost:0, x59:0, x60:0) -> l6(x78:0, c, c1) :|: c1 = -1 + x60:0 && c = -1 + x59:0 && (__rho_1_HATpost:0 > 0 && x60:0 > 1)


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(15)
YES

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

(16)
Obligation:

Termination digraph:
Nodes:
(1) l4(x14, x15, x16, x17, x18, x19, x20) -> l5(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
(2) l5(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35

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

This digraph is fully evaluated!

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

(18)
Obligation:
Rules:
l4(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) -> l4(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) :|: TRUE

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(19) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l4(VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l4(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) -> l4(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) :|: TRUE

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(21) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) -> f(1, x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1, -8, -8, -8, -8, -8, -8, -8)
----------------------------------------

(22)
NO