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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 221 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 14 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 15 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(iHAT0, jHAT0, xHAT0) -> l1(iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && xHAT0 <= iHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x3 = 1 + x && x4 = 2 + x1 && 1 + x <= x2
l3(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = 0 && 2 <= x8
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x14 <= 1
l2(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l1(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33 && 1 + x31 <= 2 * x32
l1(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 2 * x38 <= x37
l6(x42, x43, x44) -> l3(x45, x46, x47) :|: x42 = x45 && x47 = 10 && x46 = 0
l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(iHAT0, jHAT0, xHAT0)

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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(iHAT0, jHAT0, xHAT0) -> l1(iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && xHAT0 <= iHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x3 = 1 + x && x4 = 2 + x1 && 1 + x <= x2
l3(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = 0 && 2 <= x8
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x14 <= 1
l2(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l1(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33 && 1 + x31 <= 2 * x32
l1(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 2 * x38 <= x37
l6(x42, x43, x44) -> l3(x45, x46, x47) :|: x42 = x45 && x47 = 10 && x46 = 0
l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(iHAT0, jHAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0, jHAT0, xHAT0) -> l1(iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && xHAT0 <= iHAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x3 = 1 + x && x4 = 2 + x1 && 1 + x <= x2
(3) l3(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = 0 && 2 <= x8
(4) l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x14 <= 1
(5) l2(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
(6) l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(7) l1(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33 && 1 + x31 <= 2 * x32
(8) l1(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 2 * x38 <= x37
(9) l6(x42, x43, x44) -> l3(x45, x46, x47) :|: x42 = x45 && x47 = 10 && x46 = 0
(10) l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51

Arcs:
(1) -> (7), (8)
(2) -> (5)
(3) -> (5)
(5) -> (1), (2)
(7) -> (6)
(8) -> (6)
(9) -> (3)
(10) -> (9)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x3 = 1 + x && x4 = 2 + x1 && 1 + x <= x2
(2) l2(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21

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:
l0(x:0, x1:0, x23:0) -> l0(1 + x:0, 2 + x1:0, x23:0) :|: x23:0 >= 1 + x: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:

   l0(x1, x2, x3) -> l0(x1, x3)

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(8)
Obligation:
Rules:
l0(x:0, x23:0) -> l0(1 + x:0, x23:0) :|: x23:0 >= 1 + x:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x:0, x23:0) -> l0(c, x23:0) :|: c = 1 + x:0 && x23:0 >= 1 + x:0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -x + x1

The following rules are decreasing:
l0(x:0, x23:0) -> l0(c, x23:0) :|: c = 1 + x:0 && x23:0 >= 1 + x:0
The following rules are bounded:
l0(x:0, x23:0) -> l0(c, x23:0) :|: c = 1 + x:0 && x23:0 >= 1 + x:0

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(12)
YES