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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 134 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (11) IRSwT
        (12) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (13) IRSwT
        (14) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (15) IRSwT
        (16) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (17) IRSwT
        (18) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (19) IRSwT
        (20) TempFilterProof [SOUND, 47 ms]
        (21) IntTRS
        (22) PolynomialOrderProcessor [EQUIVALENT, 6 ms]
        (23) YES
    (24) IRSwT
        (25) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (26) IRSwT
        (27) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (28) IRSwT
        (29) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (30) IRSwT
        (31) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (32) IRSwT
        (33) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (34) IRSwT
        (35) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (36) IRSwT
        (37) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (38) IRSwT
        (39) TempFilterProof [SOUND, 15 ms]
        (40) IntTRS
        (41) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (42) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f223_0_main_LE(arg1P, arg2P) :|: 0 = arg1P && 1 = arg2 && -1 <= arg2P - 1 && 0 <= arg1 - 1
f1_0_main_Load(x, x1) -> f259_0_main_LE(x2, x3) :|: 0 <= x - 1 && -1 <= x3 - 1 && 1 <= x1 - 1 && -1 <= x2 - 1
f223_0_main_LE(x4, x5) -> f223_0_main_LE(x6, x7) :|: x5 + x4 = x7 && x4 - 1 = x6 && 0 <= x5 - 1
f259_0_main_LE(x8, x9) -> f259_0_main_LE(x10, x11) :|: x9 + x8 = x11 && x8 - 1 = x10 && 0 <= x9 - 1
__init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 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) -> f223_0_main_LE(arg1P, arg2P) :|: 0 = arg1P && 1 = arg2 && -1 <= arg2P - 1 && 0 <= arg1 - 1
f1_0_main_Load(x, x1) -> f259_0_main_LE(x2, x3) :|: 0 <= x - 1 && -1 <= x3 - 1 && 1 <= x1 - 1 && -1 <= x2 - 1
f223_0_main_LE(x4, x5) -> f223_0_main_LE(x6, x7) :|: x5 + x4 = x7 && x4 - 1 = x6 && 0 <= x5 - 1
f259_0_main_LE(x8, x9) -> f259_0_main_LE(x10, x11) :|: x9 + x8 = x11 && x8 - 1 = x10 && 0 <= x9 - 1
__init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f223_0_main_LE(arg1P, arg2P) :|: 0 = arg1P && 1 = arg2 && -1 <= arg2P - 1 && 0 <= arg1 - 1
(2) f1_0_main_Load(x, x1) -> f259_0_main_LE(x2, x3) :|: 0 <= x - 1 && -1 <= x3 - 1 && 1 <= x1 - 1 && -1 <= x2 - 1
(3) f223_0_main_LE(x4, x5) -> f223_0_main_LE(x6, x7) :|: x5 + x4 = x7 && x4 - 1 = x6 && 0 <= x5 - 1
(4) f259_0_main_LE(x8, x9) -> f259_0_main_LE(x10, x11) :|: x9 + x8 = x11 && x8 - 1 = x10 && 0 <= x9 - 1
(5) __init(x12, x13) -> f1_0_main_Load(x14, x15) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f259_0_main_LE(x8, x9) -> f259_0_main_LE(x10, x11) :|: x9 + x8 = x11 && x8 - 1 = x10 && 0 <= x9 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f259_0_main_LE(x8:0, x9:0) -> f259_0_main_LE(x8:0 - 1, x9:0 + x8:0) :|: x9:0 > 0

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

(8) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(9)
Obligation:
Rules:
f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

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

(10) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

Arcs:
(1) -> (1)

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

(11)
Obligation:

Termination digraph:
Nodes:
(1) f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(13)
Obligation:
Rules:
f259_0_main_LE(x:0, x1:0) -> f259_0_main_LE(x:0 - 2, x1:0 + 2 * x:0 - 1) :|: x1:0 + x:0 >= 1 && x1:0 > 0

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

(14) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(15)
Obligation:
Rules:
f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

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

(16) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

Arcs:
(1) -> (1)

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

(17)
Obligation:

Termination digraph:
Nodes:
(1) f259_0_main_LE(x, x1) -> f259_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(19)
Obligation:
Rules:
f259_0_main_LE(x:0, x1:0) -> f259_0_main_LE(x:0 - 4, x1:0 + 4 * x:0 - 6) :|: x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0

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

(20) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f259_0_main_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(21)
Obligation:
Rules:
f259_0_main_LE(x:0, x1:0) -> f259_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)

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

(22) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f259_0_main_LE(x, x1)] = -2 + 2*x + x^2 + 2*x1

The following rules are decreasing:
f259_0_main_LE(x:0, x1:0) -> f259_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)
The following rules are bounded:
f259_0_main_LE(x:0, x1:0) -> f259_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)

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

(23)
YES

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

(24)
Obligation:

Termination digraph:
Nodes:
(1) f223_0_main_LE(x4, x5) -> f223_0_main_LE(x6, x7) :|: x5 + x4 = x7 && x4 - 1 = x6 && 0 <= x5 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(26)
Obligation:
Rules:
f223_0_main_LE(x4:0, x5:0) -> f223_0_main_LE(x4:0 - 1, x5:0 + x4:0) :|: x5:0 > 0

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

(27) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(28)
Obligation:
Rules:
f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

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

(29) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

Arcs:
(1) -> (1)

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

(30)
Obligation:

Termination digraph:
Nodes:
(1) f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -2, x1 + 2 * x + -1) :|: TRUE && x1 >= 1 && x1 + x >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(32)
Obligation:
Rules:
f223_0_main_LE(x:0, x1:0) -> f223_0_main_LE(x:0 - 2, x1:0 + 2 * x:0 - 1) :|: x1:0 + x:0 >= 1 && x1:0 > 0

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

(33) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(34)
Obligation:
Rules:
f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

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

(35) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

Arcs:
(1) -> (1)

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

(36)
Obligation:

Termination digraph:
Nodes:
(1) f223_0_main_LE(x, x1) -> f223_0_main_LE(x + -4, x1 + 4 * x + -6) :|: TRUE && x1 + x >= 1 && x1 >= 1 && x1 + 3 * x >= 4 && x1 + 2 * x >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(38)
Obligation:
Rules:
f223_0_main_LE(x:0, x1:0) -> f223_0_main_LE(x:0 - 4, x1:0 + 4 * x:0 - 6) :|: x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0

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

(39) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f223_0_main_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(40)
Obligation:
Rules:
f223_0_main_LE(x:0, x1:0) -> f223_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)

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

(41) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f223_0_main_LE(x, x1)] = -2 + 2*x + x^2 + 2*x1

The following rules are decreasing:
f223_0_main_LE(x:0, x1:0) -> f223_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)
The following rules are bounded:
f223_0_main_LE(x:0, x1:0) -> f223_0_main_LE(c, c1) :|: c1 = x1:0 + 4 * x:0 - 6 && c = x:0 - 4 && (x1:0 + 3 * x:0 >= 4 && x1:0 + 2 * x:0 >= 2 && x1:0 + x:0 >= 1 && x1:0 > 0)

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

(42)
YES