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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 577 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) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (13) IntTRS
        (14) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (15) NO
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) FilterProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (22) NO


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(0)
Obligation:
Rules:
l0(AHAT0, RHAT0, ___rho_1_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
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
l6(x20, x21, x22, x23, x24) -> l7(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l7(x30, x31, x32, x33, x34) -> l6(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l3(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x44
l3(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 <= 0 && x60 = 1 && x56 = 0 && x57 = x57 && x58 = x57 && x50 = x55 && x54 = x59
l1(x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70) :|: x64 <= 0 && x71 = 1 && x66 = 0 && x70 = x70 && 1 <= x70 && x62 = x67 && x63 = x68 && x64 = x69
l1(x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 && 1 <= x75
l8(x82, x83, x84, x85, x86) -> l0(x87, x88, x89, x90, x91) :|: x86 = x91 && x90 = x89 && x89 = x89 && x88 = 0 && x87 = 0
l9(x92, x93, x94, x95, x96) -> l8(x97, x98, x99, x100, x101) :|: x96 = x101 && x95 = x100 && x94 = x99 && x93 = x98 && x92 = x97
Start term: l9(AHAT0, RHAT0, ___rho_1_HAT0, dobreakHAT0, nHAT0)

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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(AHAT0, RHAT0, ___rho_1_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
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
l6(x20, x21, x22, x23, x24) -> l7(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l7(x30, x31, x32, x33, x34) -> l6(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l3(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x44
l3(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 <= 0 && x60 = 1 && x56 = 0 && x57 = x57 && x58 = x57 && x50 = x55 && x54 = x59
l1(x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70) :|: x64 <= 0 && x71 = 1 && x66 = 0 && x70 = x70 && 1 <= x70 && x62 = x67 && x63 = x68 && x64 = x69
l1(x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 && 1 <= x75
l8(x82, x83, x84, x85, x86) -> l0(x87, x88, x89, x90, x91) :|: x86 = x91 && x90 = x89 && x89 = x89 && x88 = 0 && x87 = 0
l9(x92, x93, x94, x95, x96) -> l8(x97, x98, x99, x100, x101) :|: x96 = x101 && x95 = x100 && x94 = x99 && x93 = x98 && x92 = x97
Start term: l9(AHAT0, RHAT0, ___rho_1_HAT0, dobreakHAT0, nHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(AHAT0, RHAT0, ___rho_1_HAT0, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
(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) l6(x20, x21, x22, x23, x24) -> l7(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(5) l7(x30, x31, x32, x33, x34) -> l6(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
(6) l3(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x44
(7) l3(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 <= 0 && x60 = 1 && x56 = 0 && x57 = x57 && x58 = x57 && x50 = x55 && x54 = x59
(8) l1(x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70) :|: x64 <= 0 && x71 = 1 && x66 = 0 && x70 = x70 && 1 <= x70 && x62 = x67 && x63 = x68 && x64 = x69
(9) l1(x72, x73, x74, x75, x76) -> l6(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 && 1 <= x75
(10) l8(x82, x83, x84, x85, x86) -> l0(x87, x88, x89, x90, x91) :|: x86 = x91 && x90 = x89 && x89 = x89 && x88 = 0 && x87 = 0
(11) l9(x92, x93, x94, x95, x96) -> l8(x97, x98, x99, x100, x101) :|: x96 = x101 && x95 = x100 && x94 = x99 && x93 = x98 && x92 = x97

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, dobreakHAT0, nHAT0) -> l1(AHATpost, RHATpost, ___rho_1_HATpost, dobreakHATpost, nHATpost) :|: nHAT0 = nHATpost && dobreakHAT0 = dobreakHATpost && ___rho_1_HAT0 = ___rho_1_HATpost && RHAT0 = RHATpost && AHAT0 = AHATpost
(2) l3(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 <= 0 && x60 = 1 && x56 = 0 && x57 = x57 && x58 = x57 && x50 = x55 && x54 = x59
(3) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
(4) l1(x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70) :|: x64 <= 0 && x71 = 1 && x66 = 0 && x70 = x70 && 1 <= x70 && x62 = x67 && x63 = x68 && x64 = x69
(5) l3(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x44

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.
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(7)
Obligation:
Rules:
l2(AHATpost:0, x1:0, x2:0, x3:0, nHATpost:0) -> l2(0, 0, ___rho_1_HATpost:0, ___rho_1_HATpost:0, x70:0) :|: nHATpost:0 < 1 && ___rho_1_HATpost:0 < 1 && x70:0 > 0
l2(x, x1, x2, x3, x4) -> l2(x, x1, x2, x3, x4) :|: x4 > 0

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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) -> l2(x5)

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(9)
Obligation:
Rules:
l2(nHATpost:0) -> l2(x70:0) :|: nHATpost:0 < 1 && ___rho_1_HATpost:0 < 1 && x70:0 > 0
l2(x4) -> l2(x4) :|: x4 > 0

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(10) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l2(nHATpost:0) -> l2(x70:0) :|: nHATpost:0 < 1 && ___rho_1_HATpost:0 < 1 && x70:0 > 0
l2(x4) -> l2(x4) :|: x4 > 0

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

(13)
Obligation:
Rules:
l2(x4:0) -> l2(x4:0) :|: x4:0 > 0
l2(nHATpost:0:0) -> l2(x70:0:0) :|: nHATpost:0:0 < 1 && ___rho_1_HATpost:0:0 < 1 && x70:0:0 > 0

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(14) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x4:0) -> f(1, x4:0) :|: pc = 1 && x4:0 > 0
f(pc, nHATpost:0:0) -> f(1, x70:0:0) :|: pc = 1 && (nHATpost:0:0 < 1 && ___rho_1_HATpost:0:0 < 1 && x70:0:0 > 0)
Witness term starting non-terminating reduction: f(1, 1)
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(15)
NO

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(16)
Obligation:

Termination digraph:
Nodes:
(1) l6(x20, x21, x22, x23, x24) -> l7(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(2) l7(x30, x31, x32, x33, x34) -> l6(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35

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

This digraph is fully evaluated!

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

(18)
Obligation:
Rules:
l6(x20:0, x21:0, x22:0, x23:0, x24:0) -> l6(x20:0, x21:0, x22:0, x23:0, x24:0) :|: TRUE

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(19) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l6(VARIABLE, VARIABLE, VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
l6(x20:0, x21:0, x22:0, x23:0, x24:0) -> l6(x20:0, x21:0, x22:0, x23:0, x24:0) :|: TRUE

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(21) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x20:0, x21:0, x22:0, x23:0, x24:0) -> f(1, x20:0, x21:0, x22:0, x23:0, x24:0) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1, -8, -8, -8, -8, -8)
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(22)
NO