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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1108 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 10 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 2 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 29 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 10 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 && jHAT0 = jHATpost && iHAT0 = iHATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 100 && 100 <= x
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x11 = x16 && x10 = x15 && x17 = x10 && 1 + x10 <= 100
l4(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21
l5(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l6(x40, x41, x42, x43, x44) -> l4(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l7(x50, x51, x52, x53, x54) -> l8(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && 200 <= x51
l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x69 = x61 && 1 + x61 <= 200
l3(x70, x71, x72, x73, x74) -> l7(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75
l9(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80
l10(x90, x91, x92, x93, x94) -> l9(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
l1(x100, x101, x102, x103, x104) -> l10(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = x100
l11(x110, x111, x112, x113, x114) -> l5(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x115 = 0
l12(x120, x121, x122, x123, x124) -> l11(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125
Start term: l12(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 %).
----------------------------------------

(2)
Obligation:
Rules:
l0(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 100 && 100 <= x
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x11 = x16 && x10 = x15 && x17 = x10 && 1 + x10 <= 100
l4(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21
l5(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l6(x40, x41, x42, x43, x44) -> l4(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l7(x50, x51, x52, x53, x54) -> l8(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && 200 <= x51
l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x69 = x61 && 1 + x61 <= 200
l3(x70, x71, x72, x73, x74) -> l7(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75
l9(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80
l10(x90, x91, x92, x93, x94) -> l9(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
l1(x100, x101, x102, x103, x104) -> l10(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = x100
l11(x110, x111, x112, x113, x114) -> l5(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x115 = 0
l12(x120, x121, x122, x123, x124) -> l11(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125
Start term: l12(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 && jHAT0 = jHATpost && iHAT0 = iHATpost
(2) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 100 && 100 <= x
(3) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x11 = x16 && x10 = x15 && x17 = x10 && 1 + x10 <= 100
(4) l4(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21
(5) l5(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
(6) l6(x40, x41, x42, x43, x44) -> l4(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(7) l7(x50, x51, x52, x53, x54) -> l8(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && 200 <= x51
(8) l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x69 = x61 && 1 + x61 <= 200
(9) l3(x70, x71, x72, x73, x74) -> l7(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75
(10) l9(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80
(11) l10(x90, x91, x92, x93, x94) -> l9(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
(12) l1(x100, x101, x102, x103, x104) -> l10(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = x100
(13) l11(x110, x111, x112, x113, x114) -> l5(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x115 = 0
(14) l12(x120, x121, x122, x123, x124) -> l11(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125

Arcs:
(1) -> (12)
(2) -> (9)
(3) -> (1)
(4) -> (9)
(5) -> (2), (3)
(6) -> (4)
(8) -> (6)
(9) -> (7), (8)
(10) -> (5)
(11) -> (10)
(12) -> (11)
(13) -> (5)
(14) -> (13)

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost
(2) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x11 = x16 && x10 = x15 && x17 = x10 && 1 + x10 <= 100
(3) l5(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
(4) l9(x80, x81, x82, x83, x84) -> l5(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80
(5) l10(x90, x91, x92, x93, x94) -> l9(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
(6) l1(x100, x101, x102, x103, x104) -> l10(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = x100

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l10(x90:0, jHATpost:0, x37:0, x18:0, x109:0) -> l10(1 + x90:0, jHATpost:0, 1 + x90:0, 1 + x90:0, x109:0) :|: x90:0 < 99

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

   l10(x1, x2, x3, x4, x5) -> l10(x1)

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(9)
Obligation:
Rules:
l10(x90:0) -> l10(1 + x90:0) :|: x90:0 < 99

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l10(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l10(x90:0) -> l10(c) :|: c = 1 + x90:0 && x90:0 < 99

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l10(x)] = 98 - x

The following rules are decreasing:
l10(x90:0) -> l10(c) :|: c = 1 + x90:0 && x90:0 < 99
The following rules are bounded:
l10(x90:0) -> l10(c) :|: c = 1 + x90:0 && x90:0 < 99

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(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l3(x70, x71, x72, x73, x74) -> l7(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75
(2) l4(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21
(3) l6(x40, x41, x42, x43, x44) -> l4(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(4) l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && x69 = x61 && 1 + x61 <= 200

Arcs:
(1) -> (4)
(2) -> (1)
(3) -> (2)
(4) -> (3)

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
l6(x25:0, x41:0, x27:0, x28:0, x29:0) -> l6(x25:0, 1 + x41:0, x27:0, x28:0, 1 + x41:0) :|: x41:0 < 199

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

   l6(x1, x2, x3, x4, x5) -> l6(x2)

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(18)
Obligation:
Rules:
l6(x41:0) -> l6(1 + x41:0) :|: x41:0 < 199

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l6(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l6(x41:0) -> l6(c) :|: c = 1 + x41:0 && x41:0 < 199

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x)] = 198 - x

The following rules are decreasing:
l6(x41:0) -> l6(c) :|: c = 1 + x41:0 && x41:0 < 199
The following rules are bounded:
l6(x41:0) -> l6(c) :|: c = 1 + x41:0 && x41:0 < 199

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(22)
YES