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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 224 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) FilterProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
(12) NO


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(0)
Obligation:
Rules:
l0(NHAT0, pHAT0, qHAT0, rHAT0) -> l1(NHATpost, pHATpost, qHATpost, rHATpost) :|: qHAT0 = qHATpost && pHAT0 = pHATpost && NHAT0 = NHATpost && rHATpost = pHAT0
l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x1
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= 1 + x11
l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x16 = x20 && x21 = x21 && 2 + x19 <= x18
l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l4(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x33 = x37 && x32 = x36 && x38 = x32 && x39 = 1
l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
Start term: l5(NHAT0, pHAT0, qHAT0, rHAT0)

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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(NHAT0, pHAT0, qHAT0, rHAT0) -> l1(NHATpost, pHATpost, qHATpost, rHATpost) :|: qHAT0 = qHATpost && pHAT0 = pHATpost && NHAT0 = NHATpost && rHATpost = pHAT0
l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x1
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= 1 + x11
l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x16 = x20 && x21 = x21 && 2 + x19 <= x18
l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l4(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x33 = x37 && x32 = x36 && x38 = x32 && x39 = 1
l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
Start term: l5(NHAT0, pHAT0, qHAT0, rHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(NHAT0, pHAT0, qHAT0, rHAT0) -> l1(NHATpost, pHATpost, qHATpost, rHATpost) :|: qHAT0 = qHATpost && pHAT0 = pHATpost && NHAT0 = NHATpost && rHATpost = pHAT0
(2) l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x1
(3) l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x10 <= 1 + x11
(4) l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x16 = x20 && x21 = x21 && 2 + x19 <= x18
(5) l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(6) l4(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x33 = x37 && x32 = x36 && x38 = x32 && x39 = 1
(7) l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44

Arcs:
(1) -> (5)
(2) -> (5)
(4) -> (1), (2)
(5) -> (3), (4)
(6) -> (5)
(7) -> (6)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(NHAT0, pHAT0, qHAT0, rHAT0) -> l1(NHATpost, pHATpost, qHATpost, rHATpost) :|: qHAT0 = qHATpost && pHAT0 = pHATpost && NHAT0 = NHATpost && rHATpost = pHAT0
(2) l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x16 = x20 && x21 = x21 && 2 + x19 <= x18
(3) l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(4) l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l2(x16:0, x17:0, x18:0, x19:0) -> l2(x16:0, x21:0, x21:0, x19:0) :|: x18:0 >= 2 + x19:0
l2(x, x1, x2, x3) -> l2(x, x4, x2, x4) :|: x2 >= 2 + x3

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

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

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(8)
Obligation:
Rules:
l2(x18:0, x19:0) -> l2(x21:0, x19:0) :|: x18:0 >= 2 + x19:0
l2(x2, x3) -> l2(x2, x4) :|: x2 >= 2 + x3

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(9) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l2(x18:0, x19:0) -> l2(x21:0, x19:0) :|: x18:0 >= 2 + x19:0
l2(x2, x3) -> l2(x2, x4) :|: x2 >= 2 + x3

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(11) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x18:0, x19:0) -> f(1, x21:0, x19:0) :|: pc = 1 && x18:0 >= 2 + x19:0
f(pc, x2, x3) -> f(1, x2, x4) :|: pc = 1 && x2 >= 2 + x3
Witness term starting non-terminating reduction: f(1, 4, -1)
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(12)
NO