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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 299 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 31 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 14 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(jHAT0, tmp3HAT0, x1HAT0, y2HAT0) -> l1(jHATpost, tmp3HATpost, x1HATpost, y2HATpost) :|: jHAT0 = jHATpost && tmp3HATpost = tmp3HATpost && y2HATpost = 1 + jHAT0 && x1HATpost = jHAT0
l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 4 <= x8
l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 4
l4(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l6(x32, x33, x34, x35) -> l2(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l3(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l1(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = 1 + x48
l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = 0
l8(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
Start term: l8(jHAT0, tmp3HAT0, x1HAT0, y2HAT0)

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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(jHAT0, tmp3HAT0, x1HAT0, y2HAT0) -> l1(jHATpost, tmp3HATpost, x1HATpost, y2HATpost) :|: jHAT0 = jHATpost && tmp3HATpost = tmp3HATpost && y2HATpost = 1 + jHAT0 && x1HATpost = jHAT0
l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 4 <= x8
l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 4
l4(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l6(x32, x33, x34, x35) -> l2(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l3(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l1(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = 1 + x48
l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = 0
l8(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
Start term: l8(jHAT0, tmp3HAT0, x1HAT0, y2HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(jHAT0, tmp3HAT0, x1HAT0, y2HAT0) -> l1(jHATpost, tmp3HATpost, x1HATpost, y2HATpost) :|: jHAT0 = jHATpost && tmp3HATpost = tmp3HATpost && y2HATpost = 1 + jHAT0 && x1HATpost = jHAT0
(2) l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4
(3) l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 4 <= x8
(4) l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 4
(5) l4(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(6) l6(x32, x33, x34, x35) -> l2(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
(7) l3(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
(8) l1(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = 1 + x48
(9) l7(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x60 = 0
(10) l8(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(jHAT0, tmp3HAT0, x1HAT0, y2HAT0) -> l1(jHATpost, tmp3HATpost, x1HATpost, y2HATpost) :|: jHAT0 = jHATpost && tmp3HATpost = tmp3HATpost && y2HATpost = 1 + jHAT0 && x1HATpost = jHAT0
(2) l2(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 4
(3) l6(x32, x33, x34, x35) -> l2(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
(4) l1(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = 1 + x48
(5) l0(x, x1, x2, x3) -> l1(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l6(x20:0, x21:0, x22:0, x23:0) -> l6(1 + x20:0, x21:0, x22:0, x23:0) :|: x20:0 < 4
l6(x, x1, x2, x3) -> l6(1 + x, x4, x, 1 + x) :|: x < 4

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

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

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(8)
Obligation:
Rules:
l6(x20:0) -> l6(1 + x20:0) :|: x20:0 < 4

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l6(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l6(x20:0) -> l6(c) :|: c = 1 + x20:0 && x20:0 < 4

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x)] = 3 - x

The following rules are decreasing:
l6(x20:0) -> l6(c) :|: c = 1 + x20:0 && x20:0 < 4
The following rules are bounded:
l6(x20:0) -> l6(c) :|: c = 1 + x20:0 && x20:0 < 4

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(12)
YES