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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 118 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 33 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 13 ms]
(10) YES


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(0)
Obligation:
Rules:
l0(iHAT0) -> l1(iHATpost) :|: iHAT0 = iHATpost
l2(x) -> l0(x1) :|: x = x1 && 20 <= x
l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 20
l4(x4) -> l0(x5) :|: x4 = x5 && x4 <= 0
l4(x6) -> l3(x7) :|: x6 = x7 && 1 <= x6
l3(x8) -> l2(x9) :|: x8 = x9
l5(x10) -> l4(x11) :|: x12 = 0 && x11 = x11
l6(x13) -> l5(x14) :|: x13 = x14
Start term: l6(iHAT0)

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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) -> l1(iHATpost) :|: iHAT0 = iHATpost
l2(x) -> l0(x1) :|: x = x1 && 20 <= x
l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 20
l4(x4) -> l0(x5) :|: x4 = x5 && x4 <= 0
l4(x6) -> l3(x7) :|: x6 = x7 && 1 <= x6
l3(x8) -> l2(x9) :|: x8 = x9
l5(x10) -> l4(x11) :|: x12 = 0 && x11 = x11
l6(x13) -> l5(x14) :|: x13 = x14
Start term: l6(iHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0) -> l1(iHATpost) :|: iHAT0 = iHATpost
(2) l2(x) -> l0(x1) :|: x = x1 && 20 <= x
(3) l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 20
(4) l4(x4) -> l0(x5) :|: x4 = x5 && x4 <= 0
(5) l4(x6) -> l3(x7) :|: x6 = x7 && 1 <= x6
(6) l3(x8) -> l2(x9) :|: x8 = x9
(7) l5(x10) -> l4(x11) :|: x12 = 0 && x11 = x11
(8) l6(x13) -> l5(x14) :|: x13 = x14

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 20
(2) l3(x8) -> l2(x9) :|: x8 = x9

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:
l2(x2:0) -> l2(1 + x2:0) :|: x2:0 < 20

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 20

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x)] = 19 - x

The following rules are decreasing:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 20
The following rules are bounded:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 20

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(10)
YES