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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 316 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 4 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 8 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 21 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(__const_10HATpost, i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost && __const_10HAT0 = __const_10HATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x <= x2
l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 1 + x14 && 1 + x14 <= x12
l4(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
l1(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x40 = x46 && x39 = x45 && x37 = x43 && x36 = x42 && x44 = 0 && x47 = x47 && x36 <= x37
l1(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x48 = x54 && x55 = 1 + x49 && 1 + x49 <= x48
l5(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x62 = x68 && x60 = x66 && x67 = 0 && x70 = x70 && x69 = x69
l6(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78
Start term: l6(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(__const_10HATpost, i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost && __const_10HAT0 = __const_10HATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x <= x2
l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 1 + x14 && 1 + x14 <= x12
l4(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
l1(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x40 = x46 && x39 = x45 && x37 = x43 && x36 = x42 && x44 = 0 && x47 = x47 && x36 <= x37
l1(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x48 = x54 && x55 = 1 + x49 && 1 + x49 <= x48
l5(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x62 = x68 && x60 = x66 && x67 = 0 && x70 = x70 && x69 = x69
l6(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78
Start term: l6(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(__const_10HATpost, i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost && __const_10HAT0 = __const_10HATpost
(2) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x <= x2
(3) l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 1 + x14 && 1 + x14 <= x12
(4) l4(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
(5) l1(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x40 = x46 && x39 = x45 && x37 = x43 && x36 = x42 && x44 = 0 && x47 = x47 && x36 <= x37
(6) l1(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x48 = x54 && x55 = 1 + x49 && 1 + x49 <= x48
(7) l5(x60, x61, x62, x63, x64, x65) -> l0(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x62 = x68 && x60 = x66 && x67 = 0 && x70 = x70 && x69 = x69
(8) l6(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78

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(__const_10HAT0, i57HAT0, i911HAT0, tmpHAT0, x35HAT0, x79HAT0) -> l1(__const_10HATpost, i57HATpost, i911HATpost, tmpHATpost, x35HATpost, x79HATpost) :|: x79HAT0 = x79HATpost && x35HAT0 = x35HATpost && tmpHAT0 = tmpHATpost && i911HAT0 = i911HATpost && i57HAT0 = i57HATpost && __const_10HAT0 = __const_10HATpost
(2) l1(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x48 = x54 && x55 = 1 + x49 && 1 + x49 <= x48

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
l0(__const_10HAT0:0, i57HAT0:0, i911HAT0:0, tmpHAT0:0, x35HAT0:0, x59:0) -> l0(__const_10HAT0:0, 1 + i57HAT0:0, i911HAT0:0, tmpHAT0:0, x35HAT0:0, x59:0) :|: __const_10HAT0:0 >= 1 + i57HAT0:0

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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, x6) -> l0(x1, x2)

----------------------------------------

(9)
Obligation:
Rules:
l0(__const_10HAT0:0, i57HAT0:0) -> l0(__const_10HAT0:0, 1 + i57HAT0:0) :|: __const_10HAT0:0 >= 1 + i57HAT0:0

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

(11)
Obligation:
Rules:
l0(__const_10HAT0:0, i57HAT0:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + i57HAT0:0 && __const_10HAT0:0 >= 1 + i57HAT0:0

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = l0_1 + -1*l0_2

The following rules are decreasing:
l0(__const_10HAT0:0, i57HAT0:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + i57HAT0:0 && __const_10HAT0:0 >= 1 + i57HAT0:0

The following rules are bounded:
l0(__const_10HAT0:0, i57HAT0:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + i57HAT0:0 && __const_10HAT0:0 >= 1 + i57HAT0:0


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(13)
YES

----------------------------------------

(14)
Obligation:

Termination digraph:
Nodes:
(1) l4(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30
(2) l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 1 + x14 && 1 + x14 <= x12

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
l4(x18:0, x19:0, x26:0, x21:0, x22:0, x23:0) -> l4(x18:0, x19:0, 1 + x26:0, x21:0, x22:0, x23:0) :|: x18:0 >= 1 + x26:0

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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, x6) -> l4(x1, x3)

----------------------------------------

(18)
Obligation:
Rules:
l4(x18:0, x26:0) -> l4(x18:0, 1 + x26:0) :|: x18:0 >= 1 + x26:0

----------------------------------------

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

(20)
Obligation:
Rules:
l4(x18:0, x26:0) -> l4(x18:0, c) :|: c = 1 + x26:0 && x18:0 >= 1 + x26:0

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l4 ] = l4_1 + -1*l4_2

The following rules are decreasing:
l4(x18:0, x26:0) -> l4(x18:0, c) :|: c = 1 + x26:0 && x18:0 >= 1 + x26:0

The following rules are bounded:
l4(x18:0, x26:0) -> l4(x18:0, c) :|: c = 1 + x26:0 && x18:0 >= 1 + x26:0


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(22)
YES