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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 206 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms]
(10) AND
    (11) IRSwT
        (12) TempFilterProof [SOUND, 6 ms]
        (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 3 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(c0HAT0, deltaextHAT0, wntHAT0) -> l1(c0HATpost, deltaextHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && c0HAT0 = c0HATpost && deltaextHATpost = -1 + deltaextHAT0 && 1 + c0HAT0 + wntHAT0 <= -1 + 2 * deltaextHAT0
l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + 2 * x7 <= x6 + x8
l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 2 && x19 <= 3 && 0 <= x19 && x20 <= 3 && 0 <= x20
l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(c0HAT0, deltaextHAT0, wntHAT0)

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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(c0HAT0, deltaextHAT0, wntHAT0) -> l1(c0HATpost, deltaextHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && c0HAT0 = c0HATpost && deltaextHATpost = -1 + deltaextHAT0 && 1 + c0HAT0 + wntHAT0 <= -1 + 2 * deltaextHAT0
l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + 2 * x7 <= x6 + x8
l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 2 && x19 <= 3 && 0 <= x19 && x20 <= 3 && 0 <= x20
l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(c0HAT0, deltaextHAT0, wntHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(c0HAT0, deltaextHAT0, wntHAT0) -> l1(c0HATpost, deltaextHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && c0HAT0 = c0HATpost && deltaextHATpost = -1 + deltaextHAT0 && 1 + c0HAT0 + wntHAT0 <= -1 + 2 * deltaextHAT0
(2) l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
(3) l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + 2 * x7 <= x6 + x8
(4) l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(5) l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 2 && x19 <= 3 && 0 <= x19 && x20 <= 3 && 0 <= x20
(6) l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27

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(c0HAT0, deltaextHAT0, wntHAT0) -> l1(c0HATpost, deltaextHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && c0HAT0 = c0HATpost && deltaextHATpost = -1 + deltaextHAT0 && 1 + c0HAT0 + wntHAT0 <= -1 + 2 * deltaextHAT0
(2) l2(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(3) l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + 2 * x7 <= x6 + x8
(4) l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3

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.
----------------------------------------

(6)
Obligation:
Rules:
l0(x15:0, x7:0, x11:0) -> l0(x15:0, 1 + x7:0, x11:0) :|: x15:0 + x11:0 >= 1 + 2 * x7:0
l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, -1 + deltaextHAT0:0, wntHAT0:0) :|: 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
l0(x, x1, x2) -> l0(x, 2 + x1, x2) :|: TRUE && x + x2 + -2 * x1 >= 3
l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, -1 + deltaextHAT0:0, wntHAT0:0) :|: 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x, x1, x2) -> l0(x, 2 + x1, x2) :|: TRUE && x + x2 + -2 * x1 >= 3
(2) l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, -1 + deltaextHAT0:0, wntHAT0:0) :|: 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

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

This digraph is fully evaluated!
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(10)
Complex Obligation (AND)

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(11)
Obligation:

Termination digraph:
Nodes:
(1) l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, -1 + deltaextHAT0:0, wntHAT0:0) :|: 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(12) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(13)
Obligation:
Rules:
l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, c, wntHAT0:0) :|: c = -1 + deltaextHAT0:0 && 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

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(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = -1/2*l0_1 + -1/2*l0_3 + l0_2

The following rules are decreasing:
l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, c, wntHAT0:0) :|: c = -1 + deltaextHAT0:0 && 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0

The following rules are bounded:
l0(c0HAT0:0, deltaextHAT0:0, wntHAT0:0) -> l0(c0HAT0:0, c, wntHAT0:0) :|: c = -1 + deltaextHAT0:0 && 1 + c0HAT0:0 + wntHAT0:0 <= -1 + 2 * deltaextHAT0:0


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(15)
YES

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(16)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l0(x, 2 + x1, x2) :|: TRUE && x + x2 + -2 * x1 >= 3

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(18)
Obligation:
Rules:
l0(x:0, x1:0, x2:0) -> l0(x:0, 2 + x1:0, x2:0) :|: x:0 + x2:0 + -2 * x1:0 >= 3

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, x2:0) :|: c = 2 + x1:0 && x:0 + x2:0 + -2 * x1:0 >= 3

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1, x2)] = -1 + x - 2*x1 + x2

The following rules are decreasing:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, x2:0) :|: c = 2 + x1:0 && x:0 + x2:0 + -2 * x1:0 >= 3
The following rules are bounded:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, x2:0) :|: c = 2 + x1:0 && x:0 + x2:0 + -2 * x1:0 >= 3

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(22)
YES