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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 304 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 18 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 5 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && jHATpost = 100 && 100 <= iHAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x1 = x6 && x5 = 1 + x && x8 = x && x7 = x && 1 + x <= 100
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l3(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 200 <= x21
l3(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 1 + x31 && x39 = x31 && 1 + x31 <= 200
l1(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l5(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = 0
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(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0)

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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, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && jHATpost = 100 && 100 <= iHAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x1 = x6 && x5 = 1 + x && x8 = x && x7 = x && 1 + x <= 100
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l3(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 200 <= x21
l3(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 1 + x31 && x39 = x31 && 1 + x31 <= 200
l1(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l5(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = 0
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(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && jHATpost = 100 && 100 <= iHAT0
(2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x1 = x6 && x5 = 1 + x && x8 = x && x7 = x && 1 + x <= 100
(3) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
(4) l3(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 200 <= x21
(5) l3(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 1 + x31 && x39 = x31 && 1 + x31 <= 200
(6) l1(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(7) l5(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = 0
(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) -> (6)
(2) -> (3)
(3) -> (1), (2)
(5) -> (6)
(6) -> (4), (5)
(7) -> (3)
(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(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x1 = x6 && x5 = 1 + x && x8 = x && x7 = x && 1 + x <= 100
(2) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l0(x17:0, x16:0, x2:0, x3:0, x19:0) -> l0(1 + x17:0, x16:0, x17:0, x17:0, x19:0) :|: x17:0 < 100

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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(x17:0) -> l0(1 + x17:0) :|: x17:0 < 100

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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(x17:0) -> l0(c) :|: c = 1 + x17:0 && x17:0 < 100

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x)] = 99 - x

The following rules are decreasing:
l0(x17:0) -> l0(c) :|: c = 1 + x17:0 && x17:0 < 100
The following rules are bounded:
l0(x17:0) -> l0(c) :|: c = 1 + x17:0 && x17:0 < 100

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(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l1(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(2) l3(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x30 = x35 && x36 = 1 + x31 && x39 = x31 && 1 + x31 <= 200

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

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l1(x35:0, x39:0, x37:0, x38:0, x44:0) -> l1(x35:0, 1 + x39:0, x37:0, x38:0, x39:0) :|: x39:0 < 200

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

   l1(x1, x2, x3, x4, x5) -> l1(x2)

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(18)
Obligation:
Rules:
l1(x39:0) -> l1(1 + x39:0) :|: x39:0 < 200

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l1(x39:0) -> l1(c) :|: c = 1 + x39:0 && x39:0 < 200

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x)] = 199 - x

The following rules are decreasing:
l1(x39:0) -> l1(c) :|: c = 1 + x39:0 && x39:0 < 200
The following rules are bounded:
l1(x39:0) -> l1(c) :|: c = 1 + x39:0 && x39:0 < 200

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(22)
YES