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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 189 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTChainingProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(12) AND
    (13) IRSwT
    (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:
f1_0_main_Load(arg1, arg2) -> f104_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f104_0_loop_EQ(x, x1) -> f104_0_loop_EQ(x2, x3) :|: -1 * x + 1 = x2 && x <= 4 && x <= -1
f104_0_loop_EQ(x4, x5) -> f104_0_loop_EQ(x6, x7) :|: -1 * x4 + 1 = x6 && x4 <= 4 && 0 <= x4 - 1
f104_0_loop_EQ(x8, x9) -> f104_0_loop_EQ(x10, x11) :|: -1 * x8 - 1 = x10 && -1 * x8 <= 1 && 4 <= x8 - 1
__init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2)

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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:
f1_0_main_Load(arg1, arg2) -> f104_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f104_0_loop_EQ(x, x1) -> f104_0_loop_EQ(x2, x3) :|: -1 * x + 1 = x2 && x <= 4 && x <= -1
f104_0_loop_EQ(x4, x5) -> f104_0_loop_EQ(x6, x7) :|: -1 * x4 + 1 = x6 && x4 <= 4 && 0 <= x4 - 1
f104_0_loop_EQ(x8, x9) -> f104_0_loop_EQ(x10, x11) :|: -1 * x8 - 1 = x10 && -1 * x8 <= 1 && 4 <= x8 - 1
__init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f104_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f104_0_loop_EQ(x, x1) -> f104_0_loop_EQ(x2, x3) :|: -1 * x + 1 = x2 && x <= 4 && x <= -1
(3) f104_0_loop_EQ(x4, x5) -> f104_0_loop_EQ(x6, x7) :|: -1 * x4 + 1 = x6 && x4 <= 4 && 0 <= x4 - 1
(4) f104_0_loop_EQ(x8, x9) -> f104_0_loop_EQ(x10, x11) :|: -1 * x8 - 1 = x10 && -1 * x8 <= 1 && 4 <= x8 - 1
(5) __init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f104_0_loop_EQ(x4, x5) -> f104_0_loop_EQ(x6, x7) :|: -1 * x4 + 1 = x6 && x4 <= 4 && 0 <= x4 - 1
(2) f104_0_loop_EQ(x, x1) -> f104_0_loop_EQ(x2, x3) :|: -1 * x + 1 = x2 && x <= 4 && x <= -1
(3) f104_0_loop_EQ(x8, x9) -> f104_0_loop_EQ(x10, x11) :|: -1 * x8 - 1 = x10 && -1 * x8 <= 1 && 4 <= x8 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f104_0_loop_EQ(x4:0, x5:0) -> f104_0_loop_EQ(-1 * x4:0 + 1, x7:0) :|: x4:0 > 0 && x4:0 < 5
f104_0_loop_EQ(x:0, x1:0) -> f104_0_loop_EQ(-1 * x:0 + 1, x3:0) :|: x:0 < 0 && x:0 < 5
f104_0_loop_EQ(x8:0, x9:0) -> f104_0_loop_EQ(-1 * x8:0 - 1, x11:0) :|: x8:0 > 4 && 1 >= -1 * x8:0

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

   f104_0_loop_EQ(x1, x2) -> f104_0_loop_EQ(x1)

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(8)
Obligation:
Rules:
f104_0_loop_EQ(x4:0) -> f104_0_loop_EQ(-1 * x4:0 + 1) :|: x4:0 > 0 && x4:0 < 5
f104_0_loop_EQ(x:0) -> f104_0_loop_EQ(-1 * x:0 + 1) :|: x:0 < 0 && x:0 < 5
f104_0_loop_EQ(x8:0) -> f104_0_loop_EQ(-1 * x8:0 - 1) :|: x8:0 > 4 && 1 >= -1 * x8:0

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(9) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(10)
Obligation:
Rules:
f104_0_loop_EQ(x:0) -> f104_0_loop_EQ(-1 * x:0 + 1) :|: x:0 < 0 && x:0 < 5
f104_0_loop_EQ(x2) -> f104_0_loop_EQ(x2) :|: TRUE && x2 <= 4 && -1 * x2 <= -2
f104_0_loop_EQ(x8:0) -> f104_0_loop_EQ(-1 * x8:0 - 1) :|: x8:0 > 4 && 1 >= -1 * x8:0

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(11) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f104_0_loop_EQ(x:0) -> f104_0_loop_EQ(-1 * x:0 + 1) :|: x:0 < 0 && x:0 < 5
(2) f104_0_loop_EQ(x2) -> f104_0_loop_EQ(x2) :|: TRUE && x2 <= 4 && -1 * x2 <= -2
(3) f104_0_loop_EQ(x8:0) -> f104_0_loop_EQ(-1 * x8:0 - 1) :|: x8:0 > 4 && 1 >= -1 * x8:0

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

This digraph is fully evaluated!
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(12)
Complex Obligation (AND)

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(13)
Obligation:

Termination digraph:
Nodes:
(1) f104_0_loop_EQ(x:0) -> f104_0_loop_EQ(-1 * x:0 + 1) :|: x:0 < 0 && x:0 < 5
(2) f104_0_loop_EQ(x8:0) -> f104_0_loop_EQ(-1 * x8:0 - 1) :|: x8:0 > 4 && 1 >= -1 * x8:0

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

This digraph is fully evaluated!

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f104_0_loop_EQ(x2) -> f104_0_loop_EQ(x2) :|: TRUE && x2 <= 4 && -1 * x2 <= -2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
f104_0_loop_EQ(x2:0) -> f104_0_loop_EQ(x2:0) :|: -2 >= -1 * x2:0 && x2:0 < 5

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f104_0_loop_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
f104_0_loop_EQ(x2:0) -> f104_0_loop_EQ(x2:0) :|: -2 >= -1 * x2:0 && x2:0 < 5

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(19) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && (-2 >= -1 * x2:0 && x2:0 < 5)
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 (((run1_0 * 1)) = ((1 * 1)) and (((1 * -2)) >= ((run1_1 * -1)) and ((run1_1 * 1)) < ((1 * 5))))) and !(((run2_0 * 1)) = ((1 * 1)) and (((1 * -2)) >= ((run2_1 * -1)) and ((run2_1 * 1)) < ((1 * 5)))))
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 (((run1_0 * 1)) = ((1 * 1)) and (((1 * -2)) >= ((run1_1 * -1)) and ((run1_1 * 1)) < ((1 * 5)))))

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(20)
NO