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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 450 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 22 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 6 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0) -> l1(i10HATpost, i13HATpost, n12HATpost, n9HATpost, nHATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && n9HAT0 = n9HATpost && n12HAT0 = n12HATpost && nHAT0 = nHATpost && i13HAT0 = i13HATpost && i10HAT0 = i10HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x4 = x11 && x1 = x8 && x = x7 && x2 <= x1
l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x18 = x25 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x16
l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x31 = x38 && x30 = x37 && x32 = x39 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x45 = x52 && x46 = x53 && x42 = x49 && x50 = 0 && x51 = x46 && x55 = x55 && x45 <= x42
l1(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x59 = x66 && x58 = x65 && x60 = x67 && x57 = x64 && x63 = 1 + x56 && 1 + x56 <= x59
l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x72 = x79 && x71 = x78 && x77 = 0 && x80 = x80 && x82 = x82 && x81 = x81
l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x87 = x94 && x86 = x93 && x88 = x95 && x85 = x92 && x84 = x91
Start term: l6(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0)

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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(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0) -> l1(i10HATpost, i13HATpost, n12HATpost, n9HATpost, nHATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && n9HAT0 = n9HATpost && n12HAT0 = n12HATpost && nHAT0 = nHATpost && i13HAT0 = i13HATpost && i10HAT0 = i10HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x4 = x11 && x1 = x8 && x = x7 && x2 <= x1
l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x18 = x25 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x16
l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x31 = x38 && x30 = x37 && x32 = x39 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x45 = x52 && x46 = x53 && x42 = x49 && x50 = 0 && x51 = x46 && x55 = x55 && x45 <= x42
l1(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x59 = x66 && x58 = x65 && x60 = x67 && x57 = x64 && x63 = 1 + x56 && 1 + x56 <= x59
l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x72 = x79 && x71 = x78 && x77 = 0 && x80 = x80 && x82 = x82 && x81 = x81
l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x87 = x94 && x86 = x93 && x88 = x95 && x85 = x92 && x84 = x91
Start term: l6(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0) -> l1(i10HATpost, i13HATpost, n12HATpost, n9HATpost, nHATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && n9HAT0 = n9HATpost && n12HAT0 = n12HATpost && nHAT0 = nHATpost && i13HAT0 = i13HATpost && i10HAT0 = i10HATpost
(2) l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x4 = x11 && x1 = x8 && x = x7 && x2 <= x1
(3) l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x18 = x25 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x16
(4) l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x31 = x38 && x30 = x37 && x32 = x39 && x29 = x36 && x28 = x35
(5) l1(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x45 = x52 && x46 = x53 && x42 = x49 && x50 = 0 && x51 = x46 && x55 = x55 && x45 <= x42
(6) l1(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x59 = x66 && x58 = x65 && x60 = x67 && x57 = x64 && x63 = 1 + x56 && 1 + x56 <= x59
(7) l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x72 = x79 && x71 = x78 && x77 = 0 && x80 = x80 && x82 = x82 && x81 = x81
(8) l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x87 = x94 && x86 = x93 && x88 = x95 && x85 = x92 && x84 = x91

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(i10HAT0, i13HAT0, n12HAT0, n9HAT0, nHAT0, tmpHAT0, tmp___0HAT0) -> l1(i10HATpost, i13HATpost, n12HATpost, n9HATpost, nHATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && n9HAT0 = n9HATpost && n12HAT0 = n12HATpost && nHAT0 = nHATpost && i13HAT0 = i13HATpost && i10HAT0 = i10HATpost
(2) l1(x56, x57, x58, x59, x60, x61, x62) -> l0(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x59 = x66 && x58 = x65 && x60 = x67 && x57 = x64 && x63 = 1 + x56 && 1 + x56 <= x59

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
l0(i10HAT0:0, i13HAT0:0, n12HAT0:0, n9HAT0:0, nHAT0:0, tmpHAT0:0, tmp___0HAT0:0) -> l0(1 + i10HAT0:0, i13HAT0:0, n12HAT0:0, n9HAT0:0, nHAT0:0, tmpHAT0:0, tmp___0HAT0:0) :|: n9HAT0:0 >= 1 + i10HAT0:0

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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, x7) -> l0(x1, x4)

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(9)
Obligation:
Rules:
l0(i10HAT0:0, n9HAT0:0) -> l0(1 + i10HAT0:0, n9HAT0:0) :|: n9HAT0:0 >= 1 + i10HAT0:0

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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(i10HAT0:0, n9HAT0:0) -> l0(c, n9HAT0:0) :|: c = 1 + i10HAT0:0 && n9HAT0:0 >= 1 + i10HAT0:0

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -x + x1

The following rules are decreasing:
l0(i10HAT0:0, n9HAT0:0) -> l0(c, n9HAT0:0) :|: c = 1 + i10HAT0:0 && n9HAT0:0 >= 1 + i10HAT0:0
The following rules are bounded:
l0(i10HAT0:0, n9HAT0:0) -> l0(c, n9HAT0:0) :|: c = 1 + i10HAT0:0 && n9HAT0:0 >= 1 + i10HAT0:0

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(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x31 = x38 && x30 = x37 && x32 = x39 && x29 = x36 && x28 = x35
(2) l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x17 = x24 && x16 = x23 && x18 = x25 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x16

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
l4(x21:0, x29:0, x23:0, x24:0, x25:0, x26:0, x27:0) -> l4(x21:0, 1 + x29:0, x23:0, x24:0, x25:0, x26:0, x27:0) :|: x23:0 >= 1 + x29:0

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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, x7) -> l4(x2, x3)

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

(18)
Obligation:
Rules:
l4(x29:0, x23:0) -> l4(1 + x29:0, x23:0) :|: x23:0 >= 1 + x29:0

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

(20)
Obligation:
Rules:
l4(x29:0, x23:0) -> l4(c, x23:0) :|: c = 1 + x29:0 && x23:0 >= 1 + x29:0

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1)] = -x + x1

The following rules are decreasing:
l4(x29:0, x23:0) -> l4(c, x23:0) :|: c = 1 + x29:0 && x23:0 >= 1 + x29:0
The following rules are bounded:
l4(x29:0, x23:0) -> l4(c, x23:0) :|: c = 1 + x29:0 && x23:0 >= 1 + x29:0

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(22)
YES