YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 199 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 22 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 6 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 7 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 34 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f142_0_main_LE(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f142_0_main_LE(x, x1) -> f194_0_main_LE(x2, x3) :|: x = x3 && x1 = x2 && 0 <= x - 1 && 0 <= x1 - 1 && x1 <= x
f142_0_main_LE(x4, x5) -> f209_0_main_LE(x6, x7) :|: x5 = x7 && x4 = x6 && 0 <= x4 - 1 && 0 <= x5 - 1 && x4 <= x5 - 1
f194_0_main_LE(x8, x9) -> f142_0_main_LE(x10, x11) :|: x8 = x11 && 0 = x10 && 0 = x9
f194_0_main_LE(x12, x13) -> f194_0_main_LE(x14, x15) :|: x13 - 1 = x15 && x12 = x14 && 0 <= x13 - 1
f209_0_main_LE(x16, x17) -> f142_0_main_LE(x18, x19) :|: 0 = x19 && x16 = x18 && 0 = x17
f209_0_main_LE(x20, x21) -> f209_0_main_LE(x22, x23) :|: x21 - 1 = x23 && x20 = x22 && 0 <= x21 - 1
__init(x24, x25) -> f1_0_main_Load(x26, x27) :|: 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) -> f142_0_main_LE(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f142_0_main_LE(x, x1) -> f194_0_main_LE(x2, x3) :|: x = x3 && x1 = x2 && 0 <= x - 1 && 0 <= x1 - 1 && x1 <= x
f142_0_main_LE(x4, x5) -> f209_0_main_LE(x6, x7) :|: x5 = x7 && x4 = x6 && 0 <= x4 - 1 && 0 <= x5 - 1 && x4 <= x5 - 1
f194_0_main_LE(x8, x9) -> f142_0_main_LE(x10, x11) :|: x8 = x11 && 0 = x10 && 0 = x9
f194_0_main_LE(x12, x13) -> f194_0_main_LE(x14, x15) :|: x13 - 1 = x15 && x12 = x14 && 0 <= x13 - 1
f209_0_main_LE(x16, x17) -> f142_0_main_LE(x18, x19) :|: 0 = x19 && x16 = x18 && 0 = x17
f209_0_main_LE(x20, x21) -> f209_0_main_LE(x22, x23) :|: x21 - 1 = x23 && x20 = x22 && 0 <= x21 - 1
__init(x24, x25) -> f1_0_main_Load(x26, x27) :|: 0 <= 0
Start term: __init(arg1, arg2)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f142_0_main_LE(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
(2) f142_0_main_LE(x, x1) -> f194_0_main_LE(x2, x3) :|: x = x3 && x1 = x2 && 0 <= x - 1 && 0 <= x1 - 1 && x1 <= x
(3) f142_0_main_LE(x4, x5) -> f209_0_main_LE(x6, x7) :|: x5 = x7 && x4 = x6 && 0 <= x4 - 1 && 0 <= x5 - 1 && x4 <= x5 - 1
(4) f194_0_main_LE(x8, x9) -> f142_0_main_LE(x10, x11) :|: x8 = x11 && 0 = x10 && 0 = x9
(5) f194_0_main_LE(x12, x13) -> f194_0_main_LE(x14, x15) :|: x13 - 1 = x15 && x12 = x14 && 0 <= x13 - 1
(6) f209_0_main_LE(x16, x17) -> f142_0_main_LE(x18, x19) :|: 0 = x19 && x16 = x18 && 0 = x17
(7) f209_0_main_LE(x20, x21) -> f209_0_main_LE(x22, x23) :|: x21 - 1 = x23 && x20 = x22 && 0 <= x21 - 1
(8) __init(x24, x25) -> f1_0_main_Load(x26, x27) :|: 0 <= 0

Arcs:
(1) -> (2), (3)
(2) -> (5)
(3) -> (7)
(5) -> (4), (5)
(7) -> (6), (7)
(8) -> (1)

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f209_0_main_LE(x20, x21) -> f209_0_main_LE(x22, x23) :|: x21 - 1 = x23 && x20 = x22 && 0 <= x21 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f209_0_main_LE(x20:0, x21:0) -> f209_0_main_LE(x20:0, x21:0 - 1) :|: x21:0 > 0

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

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

   f209_0_main_LE(x1, x2) -> f209_0_main_LE(x2)

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

(9)
Obligation:
Rules:
f209_0_main_LE(x21:0) -> f209_0_main_LE(x21:0 - 1) :|: x21:0 > 0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f209_0_main_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f209_0_main_LE(x21:0) -> f209_0_main_LE(c) :|: c = x21:0 - 1 && x21:0 > 0

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f209_0_main_LE(x)] = x

The following rules are decreasing:
f209_0_main_LE(x21:0) -> f209_0_main_LE(c) :|: c = x21:0 - 1 && x21:0 > 0
The following rules are bounded:
f209_0_main_LE(x21:0) -> f209_0_main_LE(c) :|: c = x21:0 - 1 && x21:0 > 0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f194_0_main_LE(x12, x13) -> f194_0_main_LE(x14, x15) :|: x13 - 1 = x15 && x12 = x14 && 0 <= x13 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
f194_0_main_LE(x12:0, x13:0) -> f194_0_main_LE(x12:0, x13:0 - 1) :|: x13:0 > 0

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

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

   f194_0_main_LE(x1, x2) -> f194_0_main_LE(x2)

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

(18)
Obligation:
Rules:
f194_0_main_LE(x13:0) -> f194_0_main_LE(x13:0 - 1) :|: x13:0 > 0

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f194_0_main_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
f194_0_main_LE(x13:0) -> f194_0_main_LE(c) :|: c = x13:0 - 1 && x13:0 > 0

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

(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f194_0_main_LE ] = f194_0_main_LE_1

The following rules are decreasing:
f194_0_main_LE(x13:0) -> f194_0_main_LE(c) :|: c = x13:0 - 1 && x13:0 > 0

The following rules are bounded:
f194_0_main_LE(x13:0) -> f194_0_main_LE(c) :|: c = x13:0 - 1 && x13:0 > 0


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

(22)
YES