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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 142 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 13 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(__const_99HAT0, hiHAT0, loHAT0) -> l1(__const_99HATpost, hiHATpost, loHATpost) :|: loHAT0 = loHATpost && __const_99HAT0 = __const_99HATpost && hiHATpost = hiHATpost
l0(x, x1, x2) -> l1(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = x5
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x8 <= x7
l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= x14
l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l4(x24, x25, x26) -> l1(x27, x28, x29) :|: x24 = x27 && x29 = x24 && x28 = 0
l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l5(__const_99HAT0, hiHAT0, loHAT0)

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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(__const_99HAT0, hiHAT0, loHAT0) -> l1(__const_99HATpost, hiHATpost, loHATpost) :|: loHAT0 = loHATpost && __const_99HAT0 = __const_99HATpost && hiHATpost = hiHATpost
l0(x, x1, x2) -> l1(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = x5
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x8 <= x7
l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= x14
l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l4(x24, x25, x26) -> l1(x27, x28, x29) :|: x24 = x27 && x29 = x24 && x28 = 0
l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l5(__const_99HAT0, hiHAT0, loHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_99HAT0, hiHAT0, loHAT0) -> l1(__const_99HATpost, hiHATpost, loHATpost) :|: loHAT0 = loHATpost && __const_99HAT0 = __const_99HATpost && hiHATpost = hiHATpost
(2) l0(x, x1, x2) -> l1(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = x5
(3) l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x8 <= x7
(4) l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= x14
(5) l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
(6) l4(x24, x25, x26) -> l1(x27, x28, x29) :|: x24 = x27 && x29 = x24 && x28 = 0
(7) l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33

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(__const_99HAT0, hiHAT0, loHAT0) -> l1(__const_99HATpost, hiHATpost, loHATpost) :|: loHAT0 = loHATpost && __const_99HAT0 = __const_99HATpost && hiHATpost = hiHATpost
(2) l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= x14
(3) l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
(4) l0(x, x1, x2) -> l1(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = x5

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(__const_99HATpost:0, x13:0, loHATpost:0) -> l2(__const_99HATpost:0, hiHATpost:0, loHATpost:0) :|: loHATpost:0 >= 1 + x13:0
l2(x, x1, x2) -> l2(x, x1, x3) :|: x2 >= 1 + x1

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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) -> l2(x2, x3)

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(8)
Obligation:
Rules:
l2(x13:0, loHATpost:0) -> l2(hiHATpost:0, loHATpost:0) :|: loHATpost:0 >= 1 + x13:0
l2(x1, x2) -> l2(x1, x3) :|: x2 >= 1 + x1

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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(x13:0, loHATpost:0) -> l2(hiHATpost:0, loHATpost:0) :|: loHATpost:0 >= 1 + x13:0
l2(x1, x2) -> l2(x1, x3) :|: x2 >= 1 + x1

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(11) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x13:0, loHATpost:0) -> f(1, hiHATpost:0, loHATpost:0) :|: pc = 1 && loHATpost:0 >= 1 + x13:0
f(pc, x1, x2) -> f(1, x1, x3) :|: pc = 1 && x2 >= 1 + x1
Witness term starting non-terminating reduction: f(1, -4, 0)
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(12)
NO