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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 306 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 23 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 13 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 25 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 16 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 10 <= x1
l2(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = 1 + x11 && 1 + x11 <= 10
l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 0 && x39 = x39 && 10 <= x30
l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x45 = 1 + x40 && 1 + x40 <= 10
l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x51 = x56 && x55 = 0 && x58 = x58 && x57 = x57
l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l6(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0)

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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(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 10 <= x1
l2(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = 1 + x11 && 1 + x11 <= 10
l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 0 && x39 = x39 && 10 <= x30
l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x45 = 1 + x40 && 1 + x40 <= 10
l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x51 = x56 && x55 = 0 && x58 = x58 && x57 = x57
l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l6(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost
(2) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 10 <= x1
(3) l2(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = 1 + x11 && 1 + x11 <= 10
(4) l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(5) l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 0 && x39 = x39 && 10 <= x30
(6) l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x45 = 1 + x40 && 1 + x40 <= 10
(7) l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x51 = x56 && x55 = 0 && x58 = x58 && x57 = x57
(8) l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65

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

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost
(2) l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x45 = 1 + x40 && 1 + x40 <= 10

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
l0(i57HAT0:0, i911HAT0:0, tmpHAT0:0, x35HAT0:0, x49:0) -> l0(1 + i57HAT0:0, i911HAT0:0, tmpHAT0:0, x35HAT0:0, x49:0) :|: i57HAT0:0 < 10

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(8) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l0(x1, x2, x3, x4, x5) -> l0(x1)

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(9)
Obligation:
Rules:
l0(i57HAT0:0) -> l0(1 + i57HAT0:0) :|: i57HAT0:0 < 10

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l0(i57HAT0:0) -> l0(c) :|: c = 1 + i57HAT0:0 && i57HAT0:0 < 10

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = -1*l0_1

The following rules are decreasing:
l0(i57HAT0:0) -> l0(c) :|: c = 1 + i57HAT0:0 && i57HAT0:0 < 10

The following rules are bounded:
l0(i57HAT0:0) -> l0(c) :|: c = 1 + i57HAT0:0 && i57HAT0:0 < 10


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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(2) l2(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = 1 + x11 && 1 + x11 <= 10

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
l4(x15:0, x21:0, x17:0, x18:0, x19:0) -> l4(x15:0, 1 + x21:0, x17:0, x18:0, x19:0) :|: x21:0 < 10

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(17) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l4(x1, x2, x3, x4, x5) -> l4(x2)

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(18)
Obligation:
Rules:
l4(x21:0) -> l4(1 + x21:0) :|: x21:0 < 10

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l4(x21:0) -> l4(c) :|: c = 1 + x21:0 && x21:0 < 10

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l4 ] = -1*l4_1

The following rules are decreasing:
l4(x21:0) -> l4(c) :|: c = 1 + x21:0 && x21:0 < 10

The following rules are bounded:
l4(x21:0) -> l4(c) :|: c = 1 + x21:0 && x21:0 < 10


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(22)
YES