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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 393 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 2 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 48 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 26 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms]
        (20) NO


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(0)
Obligation:
Rules:
l0(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(___rho_1_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && ___rho_1_HAT0 = ___rho_1_HATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
l4(x10, x11, x12, x13, x14) -> l5(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l5(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l6(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = -1 + x32 && 1 + x31 <= x32 && x30 <= 0
l6(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x40 = x45 && x48 = 1 + x43 && x47 = -1 + x42 && x46 = -1 + x41 && 1 <= x40
l1(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x52
l1(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x62 <= 0
l7(x70, x71, x72, x73, x74) -> l0(x75, x76, x77, x78, x79) :|: x70 = x75 && x77 = x79 && x78 = 0 && x79 = 8 && 1 <= x76 && x76 = x76
l8(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85
Start term: l8(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
l0(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(___rho_1_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && ___rho_1_HAT0 = ___rho_1_HATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
l4(x10, x11, x12, x13, x14) -> l5(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l5(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l6(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = -1 + x32 && 1 + x31 <= x32 && x30 <= 0
l6(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x40 = x45 && x48 = 1 + x43 && x47 = -1 + x42 && x46 = -1 + x41 && 1 <= x40
l1(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x52
l1(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x62 <= 0
l7(x70, x71, x72, x73, x74) -> l0(x75, x76, x77, x78, x79) :|: x70 = x75 && x77 = x79 && x78 = 0 && x79 = 8 && 1 <= x76 && x76 = x76
l8(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85
Start term: l8(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(___rho_1_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && ___rho_1_HAT0 = ___rho_1_HATpost
(2) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
(3) l4(x10, x11, x12, x13, x14) -> l5(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
(4) l5(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(5) l6(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = -1 + x32 && 1 + x31 <= x32 && x30 <= 0
(6) l6(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x40 = x45 && x48 = 1 + x43 && x47 = -1 + x42 && x46 = -1 + x41 && 1 <= x40
(7) l1(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x52
(8) l1(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x62 <= 0
(9) l7(x70, x71, x72, x73, x74) -> l0(x75, x76, x77, x78, x79) :|: x70 = x75 && x77 = x79 && x78 = 0 && x79 = 8 && 1 <= x76 && x76 = x76
(10) l8(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85

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

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(___rho_1_HAT0, cHAT0, curr_servHAT0, respHAT0, serversHAT0) -> l1(___rho_1_HATpost, cHATpost, curr_servHATpost, respHATpost, serversHATpost) :|: serversHAT0 = serversHATpost && respHAT0 = respHATpost && curr_servHAT0 = curr_servHATpost && cHAT0 = cHATpost && ___rho_1_HAT0 = ___rho_1_HATpost
(2) l6(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x40 = x45 && x48 = 1 + x43 && x47 = -1 + x42 && x46 = -1 + x41 && 1 <= x40
(3) l6(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = -1 + x32 && 1 + x31 <= x32 && x30 <= 0
(4) l1(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x52

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l6(___rho_1_HATpost:0, x41:0, x42:0, x43:0, serversHATpost:0) -> l6(x55:0, -1 + x41:0, -1 + x42:0, 1 + x43:0, serversHATpost:0) :|: ___rho_1_HATpost:0 > 0 && x42:0 > 1
l6(x, x1, x2, x3, x4) -> l6(x5, x1, -1 + x2, x3, x4) :|: x2 >= 1 + x1 && x2 > 1 && x < 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) -> l6(x1, x2, x3)

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(9)
Obligation:
Rules:
l6(___rho_1_HATpost:0, x41:0, x42:0) -> l6(x55:0, -1 + x41:0, -1 + x42:0) :|: ___rho_1_HATpost:0 > 0 && x42:0 > 1
l6(x, x1, x2) -> l6(x5, x1, -1 + x2) :|: x2 >= 1 + x1 && x2 > 1 && x < 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.
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(11)
Obligation:
Rules:
l6(___rho_1_HATpost:0, x41:0, x42:0) -> l6(x55:0, c, c1) :|: c1 = -1 + x42:0 && c = -1 + x41:0 && (___rho_1_HATpost:0 > 0 && x42:0 > 1)
l6(x, x1, x2) -> l6(x5, x1, c2) :|: c2 = -1 + x2 && (x2 >= 1 + x1 && x2 > 1 && x < 1)

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

The following rules are decreasing:
l6(___rho_1_HATpost:0, x41:0, x42:0) -> l6(x55:0, c, c1) :|: c1 = -1 + x42:0 && c = -1 + x41:0 && (___rho_1_HATpost:0 > 0 && x42:0 > 1)
l6(x, x1, x2) -> l6(x5, x1, c2) :|: c2 = -1 + x2 && (x2 >= 1 + x1 && x2 > 1 && x < 1)

The following rules are bounded:
l6(___rho_1_HATpost:0, x41:0, x42:0) -> l6(x55:0, c, c1) :|: c1 = -1 + x42:0 && c = -1 + x41:0 && (___rho_1_HATpost:0 > 0 && x42:0 > 1)
l6(x, x1, x2) -> l6(x5, x1, c2) :|: c2 = -1 + x2 && (x2 >= 1 + x1 && x2 > 1 && x < 1)


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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l4(x10, x11, x12, x13, x14) -> l5(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
(2) l5(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25

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

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l4(x10:0, x11:0, x12:0, x13:0, x14:0) -> l4(x10:0, x11:0, x12:0, x13:0, x14:0) :|: TRUE

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l4(VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
l4(x10:0, x11:0, x12:0, x13:0, x14:0) -> l4(x10:0, x11:0, x12:0, x13:0, x14:0) :|: TRUE

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(19) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x10:0, x11:0, x12:0, x13:0, x14:0) -> f(1, x10:0, x11:0, x12:0, x13:0, x14:0) :|: pc = 1 && TRUE
Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
(((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1)) and run2_2 = ((run1_2 * 1)) and run2_3 = ((run1_3 * 1)) and run2_4 = ((run1_4 * 1)) and run2_5 = ((run1_5 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
Proved satisfiability of the following formula, indicating that the system is entered at least once:
((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1)) and run2_2 = ((run1_2 * 1)) and run2_3 = ((run1_3 * 1)) and run2_4 = ((run1_4 * 1)) and run2_5 = ((run1_5 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T))

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(20)
NO