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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 158 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 36 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTChainingProof [EQUIVALENT, 2 ms]
(10) IRSwT
(11) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(14) IRSwT
(15) IRSwTChainingProof [EQUIVALENT, 0 ms]
(16) IRSwT
(17) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(18) IRSwT
(19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(20) IRSwT
(21) TempFilterProof [SOUND, 17 ms]
(22) IntTRS
(23) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(24) YES


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(0)
Obligation:
Rules:
l0(rt_11HAT0, st_14HAT0, x_13HAT0, y_15HAT0) -> l1(rt_11HATpost, st_14HATpost, x_13HATpost, y_15HATpost) :|: y_15HAT0 = y_15HATpost && x_13HAT0 = x_13HATpost && st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost
l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x1 && x2 <= 0
l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x9 = x13 && x8 = x12 && x15 = -1 + x11 && x14 = x10 + x11 && 1 <= x10
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
Start term: l4(rt_11HAT0, st_14HAT0, x_13HAT0, y_15HAT0)

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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(rt_11HAT0, st_14HAT0, x_13HAT0, y_15HAT0) -> l1(rt_11HATpost, st_14HATpost, x_13HATpost, y_15HATpost) :|: y_15HAT0 = y_15HATpost && x_13HAT0 = x_13HATpost && st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost
l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x1 && x2 <= 0
l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x9 = x13 && x8 = x12 && x15 = -1 + x11 && x14 = x10 + x11 && 1 <= x10
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
Start term: l4(rt_11HAT0, st_14HAT0, x_13HAT0, y_15HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(rt_11HAT0, st_14HAT0, x_13HAT0, y_15HAT0) -> l1(rt_11HATpost, st_14HATpost, x_13HATpost, y_15HATpost) :|: y_15HAT0 = y_15HATpost && x_13HAT0 = x_13HATpost && st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost
(2) l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x1 && x2 <= 0
(3) l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x9 = x13 && x8 = x12 && x15 = -1 + x11 && x14 = x10 + x11 && 1 <= x10
(4) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
(5) l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x9 = x13 && x8 = x12 && x15 = -1 + x11 && x14 = x10 + x11 && 1 <= x10
(2) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l1(x12:0, x13:0, x10:0, x11:0) -> l1(x12:0, x13:0, x10:0 + x11:0, -1 + x11:0) :|: x10:0 > 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:

   l1(x1, x2, x3, x4) -> l1(x3, x4)

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(8)
Obligation:
Rules:
l1(x10:0, x11:0) -> l1(x10:0 + x11:0, -1 + x11:0) :|: x10:0 > 0

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(9) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(10)
Obligation:
Rules:
l1(x, x1) -> l1(x + 2 * x1 + -1, -2 + x1) :|: TRUE && x >= 1 && x + x1 >= 1

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(11) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x, x1) -> l1(x + 2 * x1 + -1, -2 + x1) :|: TRUE && x >= 1 && x + x1 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(12)
Obligation:

Termination digraph:
Nodes:
(1) l1(x, x1) -> l1(x + 2 * x1 + -1, -2 + x1) :|: TRUE && x >= 1 && x + x1 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
l1(x:0, x1:0) -> l1(x:0 + 2 * x1:0 - 1, -2 + x1:0) :|: x:0 + x1:0 >= 1 && x:0 > 0

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(15) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(16)
Obligation:
Rules:
l1(x, x1) -> l1(x + 4 * x1 + -6, -4 + x1) :|: TRUE && x + x1 >= 1 && x >= 1 && x + 3 * x1 >= 4 && x + 2 * x1 >= 2

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(17) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x, x1) -> l1(x + 4 * x1 + -6, -4 + x1) :|: TRUE && x + x1 >= 1 && x >= 1 && x + 3 * x1 >= 4 && x + 2 * x1 >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(18)
Obligation:

Termination digraph:
Nodes:
(1) l1(x, x1) -> l1(x + 4 * x1 + -6, -4 + x1) :|: TRUE && x + x1 >= 1 && x >= 1 && x + 3 * x1 >= 4 && x + 2 * x1 >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(19) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(20)
Obligation:
Rules:
l1(x:0, x1:0) -> l1(x:0 + 4 * x1:0 - 6, -4 + x1:0) :|: x:0 + 3 * x1:0 >= 4 && x:0 + 2 * x1:0 >= 2 && x:0 + x1:0 >= 1 && x:0 > 0

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(21) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(22)
Obligation:
Rules:
l1(x:0, x1:0) -> l1(c, c1) :|: c1 = -4 + x1:0 && c = x:0 + 4 * x1:0 - 6 && (x:0 + 3 * x1:0 >= 4 && x:0 + 2 * x1:0 >= 2 && x:0 + x1:0 >= 1 && x:0 > 0)

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(23) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -2 + 2*x + 2*x1 + x1^2

The following rules are decreasing:
l1(x:0, x1:0) -> l1(c, c1) :|: c1 = -4 + x1:0 && c = x:0 + 4 * x1:0 - 6 && (x:0 + 3 * x1:0 >= 4 && x:0 + 2 * x1:0 >= 2 && x:0 + x1:0 >= 1 && x:0 > 0)
The following rules are bounded:
l1(x:0, x1:0) -> l1(c, c1) :|: c1 = -4 + x1:0 && c = x:0 + 4 * x1:0 - 6 && (x:0 + 3 * x1:0 >= 4 && x:0 + 2 * x1:0 >= 2 && x:0 + x1:0 >= 1 && x:0 > 0)

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(24)
YES