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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 227 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 6 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 62 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 5 ms]
        (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 13 ms]
        (18) IRSwT
        (19) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (20) IRSwT
        (21) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (22) IRSwT
        (23) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (24) IRSwT
        (25) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (26) IRSwT
        (27) FilterProof [EQUIVALENT, 0 ms]
        (28) IntTRS
        (29) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (30) NO


----------------------------------------

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f98_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f98_0_loop_EQ(x, x1) -> f98_0_loop_EQ(x2, x3) :|: x + 2 = x2 && x <= -1 && x <= 0 && -3 <= x - 1
f98_0_loop_EQ(x4, x5) -> f98_0_loop_EQ(x6, x7) :|: x4 - 2 = x6 && x4 <= 2 && 0 <= x4 - 1
f98_0_loop_EQ(x8, x9) -> f98_0_loop_EQ(x10, x11) :|: -1 * x8 + 2 = x10 && -1 * x8 <= -1 - 1 && 2 <= x8 - 1
f98_0_loop_EQ(x12, x13) -> f98_0_loop_EQ(x14, x15) :|: -1 * x12 - 2 = x14 && x12 <= 0 && 3 <= -1 * x12 && x12 <= -2 - 1 && x12 <= -1 && x12 <= -1 - 1
__init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 0 <= 0
Start term: __init(arg1, arg2)

----------------------------------------

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

(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f98_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f98_0_loop_EQ(x, x1) -> f98_0_loop_EQ(x2, x3) :|: x + 2 = x2 && x <= -1 && x <= 0 && -3 <= x - 1
f98_0_loop_EQ(x4, x5) -> f98_0_loop_EQ(x6, x7) :|: x4 - 2 = x6 && x4 <= 2 && 0 <= x4 - 1
f98_0_loop_EQ(x8, x9) -> f98_0_loop_EQ(x10, x11) :|: -1 * x8 + 2 = x10 && -1 * x8 <= -1 - 1 && 2 <= x8 - 1
f98_0_loop_EQ(x12, x13) -> f98_0_loop_EQ(x14, x15) :|: -1 * x12 - 2 = x14 && x12 <= 0 && 3 <= -1 * x12 && x12 <= -2 - 1 && x12 <= -1 && x12 <= -1 - 1
__init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 0 <= 0
Start term: __init(arg1, arg2)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f98_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f98_0_loop_EQ(x, x1) -> f98_0_loop_EQ(x2, x3) :|: x + 2 = x2 && x <= -1 && x <= 0 && -3 <= x - 1
(3) f98_0_loop_EQ(x4, x5) -> f98_0_loop_EQ(x6, x7) :|: x4 - 2 = x6 && x4 <= 2 && 0 <= x4 - 1
(4) f98_0_loop_EQ(x8, x9) -> f98_0_loop_EQ(x10, x11) :|: -1 * x8 + 2 = x10 && -1 * x8 <= -1 - 1 && 2 <= x8 - 1
(5) f98_0_loop_EQ(x12, x13) -> f98_0_loop_EQ(x14, x15) :|: -1 * x12 - 2 = x14 && x12 <= 0 && 3 <= -1 * x12 && x12 <= -2 - 1 && x12 <= -1 && x12 <= -1 - 1
(6) __init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 0 <= 0

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

This digraph is fully evaluated!
----------------------------------------

(4)
Complex Obligation (AND)

----------------------------------------

(5)
Obligation:

Termination digraph:
Nodes:
(1) f98_0_loop_EQ(x8, x9) -> f98_0_loop_EQ(x10, x11) :|: -1 * x8 + 2 = x10 && -1 * x8 <= -1 - 1 && 2 <= x8 - 1
(2) f98_0_loop_EQ(x12, x13) -> f98_0_loop_EQ(x14, x15) :|: -1 * x12 - 2 = x14 && x12 <= 0 && 3 <= -1 * x12 && x12 <= -2 - 1 && x12 <= -1 && x12 <= -1 - 1

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

This digraph is fully evaluated!

----------------------------------------

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

(7)
Obligation:
Rules:
f98_0_loop_EQ(x8:0, x9:0) -> f98_0_loop_EQ(-1 * x8:0 + 2, x11:0) :|: x8:0 > 2 && -2 >= -1 * x8:0
f98_0_loop_EQ(x12:0, x13:0) -> f98_0_loop_EQ(-1 * x12:0 - 2, x15:0) :|: x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0

----------------------------------------

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

   f98_0_loop_EQ(x1, x2) -> f98_0_loop_EQ(x1)

----------------------------------------

(9)
Obligation:
Rules:
f98_0_loop_EQ(x8:0) -> f98_0_loop_EQ(-1 * x8:0 + 2) :|: x8:0 > 2 && -2 >= -1 * x8:0
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(-1 * x12:0 - 2) :|: x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0

----------------------------------------

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f98_0_loop_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f98_0_loop_EQ(x8:0) -> f98_0_loop_EQ(c) :|: c = -1 * x8:0 + 2 && (x8:0 > 2 && -2 >= -1 * x8:0)
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(c1) :|: c1 = -1 * x12:0 - 2 && (x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0)

----------------------------------------

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f98_0_loop_EQ(x)] = 2*x + x^2

The following rules are decreasing:
f98_0_loop_EQ(x8:0) -> f98_0_loop_EQ(c) :|: c = -1 * x8:0 + 2 && (x8:0 > 2 && -2 >= -1 * x8:0)
The following rules are bounded:
f98_0_loop_EQ(x8:0) -> f98_0_loop_EQ(c) :|: c = -1 * x8:0 + 2 && (x8:0 > 2 && -2 >= -1 * x8:0)
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(c1) :|: c1 = -1 * x12:0 - 2 && (x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0)

----------------------------------------

(13)
Obligation:
Rules:
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(c1) :|: c1 = -1 * x12:0 - 2 && (x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0)

----------------------------------------

(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f98_0_loop_EQ ] = -1/4*f98_0_loop_EQ_1

The following rules are decreasing:
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(c1) :|: c1 = -1 * x12:0 - 2 && (x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0)

The following rules are bounded:
f98_0_loop_EQ(x12:0) -> f98_0_loop_EQ(c1) :|: c1 = -1 * x12:0 - 2 && (x12:0 < 0 && x12:0 < -1 && x12:0 < -2 && x12:0 < 1 && 3 <= -1 * x12:0)


----------------------------------------

(15)
YES

----------------------------------------

(16)
Obligation:

Termination digraph:
Nodes:
(1) f98_0_loop_EQ(x4, x5) -> f98_0_loop_EQ(x6, x7) :|: x4 - 2 = x6 && x4 <= 2 && 0 <= x4 - 1
(2) f98_0_loop_EQ(x, x1) -> f98_0_loop_EQ(x2, x3) :|: x + 2 = x2 && x <= -1 && x <= 0 && -3 <= x - 1

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

This digraph is fully evaluated!

----------------------------------------

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

(18)
Obligation:
Rules:
f98_0_loop_EQ(x:0, x1:0) -> f98_0_loop_EQ(x:0 + 2, x3:0) :|: x:0 < 1 && x:0 < 0 && x:0 > -3
f98_0_loop_EQ(x4:0, x5:0) -> f98_0_loop_EQ(x4:0 - 2, x7:0) :|: x4:0 > 0 && x4:0 < 3

----------------------------------------

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

   f98_0_loop_EQ(x1, x2) -> f98_0_loop_EQ(x1)

----------------------------------------

(20)
Obligation:
Rules:
f98_0_loop_EQ(x:0) -> f98_0_loop_EQ(x:0 + 2) :|: x:0 < 1 && x:0 < 0 && x:0 > -3
f98_0_loop_EQ(x4:0) -> f98_0_loop_EQ(x4:0 - 2) :|: x4:0 > 0 && x4:0 < 3

----------------------------------------

(21) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(22)
Obligation:
Rules:
f98_0_loop_EQ(x4:0) -> f98_0_loop_EQ(x4:0 - 2) :|: x4:0 > 0 && x4:0 < 3
f98_0_loop_EQ(x2) -> f98_0_loop_EQ(x2) :|: TRUE && x2 <= -1 && x2 >= -1

----------------------------------------

(23) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f98_0_loop_EQ(x4:0) -> f98_0_loop_EQ(x4:0 - 2) :|: x4:0 > 0 && x4:0 < 3
(2) f98_0_loop_EQ(x2) -> f98_0_loop_EQ(x2) :|: TRUE && x2 <= -1 && x2 >= -1

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

This digraph is fully evaluated!
----------------------------------------

(24)
Obligation:

Termination digraph:
Nodes:
(1) f98_0_loop_EQ(x2) -> f98_0_loop_EQ(x2) :|: TRUE && x2 <= -1 && x2 >= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

----------------------------------------

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

(26)
Obligation:
Rules:
f98_0_loop_EQ(x2:0) -> f98_0_loop_EQ(x2:0) :|: x2:0 > -2 && x2:0 < 0

----------------------------------------

(27) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f98_0_loop_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(28)
Obligation:
Rules:
f98_0_loop_EQ(x2:0) -> f98_0_loop_EQ(x2:0) :|: x2:0 > -2 && x2:0 < 0

----------------------------------------

(29) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && (x2:0 > -2 && x2:0 < 0)
Witness term starting non-terminating reduction: f(1, -1)
----------------------------------------

(30)
NO