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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 176 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 66 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) AND
    (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 2 ms]
        (13) YES
    (14) IntTRS
        (15) PolynomialOrderProcessor [EQUIVALENT, 3 ms]
        (16) YES


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(0)
Obligation:
Rules:
l0(eHAT0, nHAT0) -> l1(eHATpost, nHATpost) :|: eHATpost = 1 + eHAT0 && nHATpost = 11 + nHAT0 && nHAT0 <= 100 && 1 <= eHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x6 = -1 + x4 && x7 = -10 + x5 && 101 <= x5 && 1 <= x4
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l3(x12, x13) -> l0(x14, x15) :|: x14 = 1 && x15 = x15
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18
Start term: l4(eHAT0, nHAT0)

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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(eHAT0, nHAT0) -> l1(eHATpost, nHATpost) :|: eHATpost = 1 + eHAT0 && nHATpost = 11 + nHAT0 && nHAT0 <= 100 && 1 <= eHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x6 = -1 + x4 && x7 = -10 + x5 && 101 <= x5 && 1 <= x4
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l3(x12, x13) -> l0(x14, x15) :|: x14 = 1 && x15 = x15
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18
Start term: l4(eHAT0, nHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(eHAT0, nHAT0) -> l1(eHATpost, nHATpost) :|: eHATpost = 1 + eHAT0 && nHATpost = 11 + nHAT0 && nHAT0 <= 100 && 1 <= eHAT0
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l0(x4, x5) -> l2(x6, x7) :|: x6 = -1 + x4 && x7 = -10 + x5 && 101 <= x5 && 1 <= x4
(4) l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(5) l3(x12, x13) -> l0(x14, x15) :|: x14 = 1 && x15 = x15
(6) l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(eHAT0, nHAT0) -> l1(eHATpost, nHATpost) :|: eHATpost = 1 + eHAT0 && nHATpost = 11 + nHAT0 && nHAT0 <= 100 && 1 <= eHAT0
(2) l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(3) l0(x4, x5) -> l2(x6, x7) :|: x6 = -1 + x4 && x7 = -10 + x5 && 101 <= x5 && 1 <= x4
(4) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(eHAT0:0, nHAT0:0) -> l0(1 + eHAT0:0, 11 + nHAT0:0) :|: eHAT0:0 > 0 && nHAT0:0 < 101
l0(x4:0, x5:0) -> l0(-1 + x4:0, -10 + x5:0) :|: x4:0 > 0 && x5:0 > 100

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l0(eHAT0:0, nHAT0:0) -> l0(c, c1) :|: c1 = 11 + nHAT0:0 && c = 1 + eHAT0:0 && (eHAT0:0 > 0 && nHAT0:0 < 101)
l0(x4:0, x5:0) -> l0(c2, c3) :|: c3 = -10 + x5:0 && c2 = -1 + x4:0 && (x4:0 > 0 && x5:0 > 100)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = 89 + 11*x - x1

The following rules are decreasing:
l0(x4:0, x5:0) -> l0(c2, c3) :|: c3 = -10 + x5:0 && c2 = -1 + x4:0 && (x4:0 > 0 && x5:0 > 100)
The following rules are bounded:
l0(eHAT0:0, nHAT0:0) -> l0(c, c1) :|: c1 = 11 + nHAT0:0 && c = 1 + eHAT0:0 && (eHAT0:0 > 0 && nHAT0:0 < 101)

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(10)
Complex Obligation (AND)

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(11)
Obligation:
Rules:
l0(eHAT0:0, nHAT0:0) -> l0(c, c1) :|: c1 = 11 + nHAT0:0 && c = 1 + eHAT0:0 && (eHAT0:0 > 0 && nHAT0:0 < 101)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = 90 + 10*x - x1

The following rules are decreasing:
l0(eHAT0:0, nHAT0:0) -> l0(c, c1) :|: c1 = 11 + nHAT0:0 && c = 1 + eHAT0:0 && (eHAT0:0 > 0 && nHAT0:0 < 101)
The following rules are bounded:
l0(eHAT0:0, nHAT0:0) -> l0(c, c1) :|: c1 = 11 + nHAT0:0 && c = 1 + eHAT0:0 && (eHAT0:0 > 0 && nHAT0:0 < 101)

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(13)
YES

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(14)
Obligation:
Rules:
l0(x4:0, x5:0) -> l0(c2, c3) :|: c3 = -10 + x5:0 && c2 = -1 + x4:0 && (x4:0 > 0 && x5:0 > 100)

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(15) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -1 + x

The following rules are decreasing:
l0(x4:0, x5:0) -> l0(c2, c3) :|: c3 = -10 + x5:0 && c2 = -1 + x4:0 && (x4:0 > 0 && x5:0 > 100)
The following rules are bounded:
l0(x4:0, x5:0) -> l0(c2, c3) :|: c3 = -10 + x5:0 && c2 = -1 + x4:0 && (x4:0 > 0 && x5:0 > 100)

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(16)
YES