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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 768 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) FilterProof [EQUIVALENT, 0 ms]
        (11) IntTRS
        (12) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
        (13) NO
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 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(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, ___rho_2_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_2_HAT0 = ___rho_2_HATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6
l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l6(x24, x25, x26, x27, x28, x29) -> l7(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
l7(x36, x37, x38, x39, x40, x41) -> l6(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42
l3(x48, x49, x50, x51, x52, x53) -> l2(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53
l3(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 <= 0 && x72 = 1 && x67 = 0 && x68 = x68 && x70 = x68 && x60 = x66 && x63 = x69 && x65 = x71
l1(x73, x74, x75, x76, x77, x78) -> l2(x79, x80, x81, x82, x83, x84) :|: x77 <= 0 && x85 = 1 && x79 = 0 && x82 = x82 && x84 = x82 && x74 = x80 && x75 = x81 && x77 = x83
l1(x86, x87, x88, x89, x90, x91) -> l6(x92, x93, x94, x95, x96, x97) :|: x91 = x97 && x90 = x96 && x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && 1 <= x90
l8(x98, x99, x100, x101, x102, x103) -> l0(x104, x105, x106, x107, x108, x109) :|: x103 = x109 && x101 = x107 && x108 = x106 && x106 = x106 && x105 = 0 && x104 = 0
l9(x110, x111, x112, x113, x114, x115) -> l8(x116, x117, x118, x119, x120, x121) :|: x115 = x121 && x114 = x120 && x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116
Start term: l9(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0)

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

(2)
Obligation:
Rules:
l0(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, ___rho_2_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_2_HAT0 = ___rho_2_HATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6
l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l6(x24, x25, x26, x27, x28, x29) -> l7(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
l7(x36, x37, x38, x39, x40, x41) -> l6(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42
l3(x48, x49, x50, x51, x52, x53) -> l2(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53
l3(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 <= 0 && x72 = 1 && x67 = 0 && x68 = x68 && x70 = x68 && x60 = x66 && x63 = x69 && x65 = x71
l1(x73, x74, x75, x76, x77, x78) -> l2(x79, x80, x81, x82, x83, x84) :|: x77 <= 0 && x85 = 1 && x79 = 0 && x82 = x82 && x84 = x82 && x74 = x80 && x75 = x81 && x77 = x83
l1(x86, x87, x88, x89, x90, x91) -> l6(x92, x93, x94, x95, x96, x97) :|: x91 = x97 && x90 = x96 && x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && 1 <= x90
l8(x98, x99, x100, x101, x102, x103) -> l0(x104, x105, x106, x107, x108, x109) :|: x103 = x109 && x101 = x107 && x108 = x106 && x106 = x106 && x105 = 0 && x104 = 0
l9(x110, x111, x112, x113, x114, x115) -> l8(x116, x117, x118, x119, x120, x121) :|: x115 = x121 && x114 = x120 && x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116
Start term: l9(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, ___rho_2_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_2_HAT0 = ___rho_2_HATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
(2) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6
(3) l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
(4) l6(x24, x25, x26, x27, x28, x29) -> l7(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
(5) l7(x36, x37, x38, x39, x40, x41) -> l6(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42
(6) l3(x48, x49, x50, x51, x52, x53) -> l2(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53
(7) l3(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 <= 0 && x72 = 1 && x67 = 0 && x68 = x68 && x70 = x68 && x60 = x66 && x63 = x69 && x65 = x71
(8) l1(x73, x74, x75, x76, x77, x78) -> l2(x79, x80, x81, x82, x83, x84) :|: x77 <= 0 && x85 = 1 && x79 = 0 && x82 = x82 && x84 = x82 && x74 = x80 && x75 = x81 && x77 = x83
(9) l1(x86, x87, x88, x89, x90, x91) -> l6(x92, x93, x94, x95, x96, x97) :|: x91 = x97 && x90 = x96 && x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && 1 <= x90
(10) l8(x98, x99, x100, x101, x102, x103) -> l0(x104, x105, x106, x107, x108, x109) :|: x103 = x109 && x101 = x107 && x108 = x106 && x106 = x106 && x105 = 0 && x104 = 0
(11) l9(x110, x111, x112, x113, x114, x115) -> l8(x116, x117, x118, x119, x120, x121) :|: x115 = x121 && x114 = x120 && x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(AHAT0, RHAT0, ___rho_1_HAT0, ___rho_2_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, ___rho_2_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_2_HAT0 = ___rho_2_HATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
(2) l3(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 <= 0 && x72 = 1 && x67 = 0 && x68 = x68 && x70 = x68 && x60 = x66 && x63 = x69 && x65 = x71
(3) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6
(4) l1(x73, x74, x75, x76, x77, x78) -> l2(x79, x80, x81, x82, x83, x84) :|: x77 <= 0 && x85 = 1 && x79 = 0 && x82 = x82 && x84 = x82 && x74 = x80 && x75 = x81 && x77 = x83
(5) l3(x48, x49, x50, x51, x52, x53) -> l2(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
l2(x54:0, x1:0, x2:0, x3:0, x10:0, x11:0) -> l2(x54:0, x1:0, x2:0, x3:0, x10:0, x11:0) :|: x11:0 > 0
l2(x, x1, x2, x3, x4, x5) -> l2(0, 0, x6, x7, x6, x7) :|: x5 < 1 && x6 < 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:

   l2(x1, x2, x3, x4, x5, x6) -> l2(x6)

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(9)
Obligation:
Rules:
l2(x11:0) -> l2(x11:0) :|: x11:0 > 0
l2(x5) -> l2(x7) :|: x5 < 1 && x6 < 1

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(10) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l2(x11:0) -> l2(x11:0) :|: x11:0 > 0
l2(x5) -> l2(x7) :|: x5 < 1 && x6 < 1

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(12) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x11:0) -> f(1, x11:0) :|: pc = 1 && x11:0 > 0
f(pc, x5) -> f(1, x7) :|: pc = 1 && (x5 < 1 && x6 < 1)
Witness term starting non-terminating reduction: f(1, 4)
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(13)
NO

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l6(x24, x25, x26, x27, x28, x29) -> l7(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
(2) l7(x36, x37, x38, x39, x40, x41) -> l6(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
l6(x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) -> l6(x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) :|: TRUE

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l6(VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
l6(x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) -> l6(x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) :|: TRUE

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(19) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) -> f(1, x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) :|: 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 run2_6 = ((run1_6 * 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 run2_6 = ((run1_6 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T))

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(20)
NO