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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 835 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 1 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 40 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x5 = -2 + x3 && x3 = x2
l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x7 = x10 && x11 = -1 + x9 && x9 = x8
l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x13 = x16 && x17 = x15 && 2 <= x15 && x15 = x14
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x19 = x22 && x23 = x21 && 1 + x21 <= 1 && x21 = x20
l3(x24, x25, x26) -> l4(x27, x28, x29) :|: x25 = x28 && x29 = x27 && 1 <= x27 && x27 <= 1 && x27 = x26
l4(x30, x31, x32) -> l1(x33, x34, x35) :|: x35 = x34 && x34 = x34 && x33 = x32
l5(x36, x37, x38) -> l3(x39, x40, x41) :|: x37 = x40 && x41 = x39 && 1 <= x39 && x39 = x38
l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x43 = x46 && x47 = x45 && x45 <= 0 && x45 = x44
l2(x48, x49, x50) -> l5(x51, x52, x53) :|: x49 = x52 && x53 = x51 && x51 = x50
l6(x54, x55, x56) -> l1(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l6(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63
l6(x66, x67, x68) -> l3(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69
l6(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75
l6(x78, x79, x80) -> l5(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81
l6(x84, x85, x86) -> l2(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87
l7(x90, x91, x92) -> l6(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
Start term: l7(oldX0HAT0, oldX1HAT0, x0HAT0)

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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(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x5 = -2 + x3 && x3 = x2
l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x7 = x10 && x11 = -1 + x9 && x9 = x8
l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x13 = x16 && x17 = x15 && 2 <= x15 && x15 = x14
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x19 = x22 && x23 = x21 && 1 + x21 <= 1 && x21 = x20
l3(x24, x25, x26) -> l4(x27, x28, x29) :|: x25 = x28 && x29 = x27 && 1 <= x27 && x27 <= 1 && x27 = x26
l4(x30, x31, x32) -> l1(x33, x34, x35) :|: x35 = x34 && x34 = x34 && x33 = x32
l5(x36, x37, x38) -> l3(x39, x40, x41) :|: x37 = x40 && x41 = x39 && 1 <= x39 && x39 = x38
l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x43 = x46 && x47 = x45 && x45 <= 0 && x45 = x44
l2(x48, x49, x50) -> l5(x51, x52, x53) :|: x49 = x52 && x53 = x51 && x51 = x50
l6(x54, x55, x56) -> l1(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l6(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63
l6(x66, x67, x68) -> l3(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69
l6(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75
l6(x78, x79, x80) -> l5(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81
l6(x84, x85, x86) -> l2(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87
l7(x90, x91, x92) -> l6(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
Start term: l7(oldX0HAT0, oldX1HAT0, x0HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x5 = -2 + x3 && x3 = x2
(3) l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x7 = x10 && x11 = -1 + x9 && x9 = x8
(4) l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x13 = x16 && x17 = x15 && 2 <= x15 && x15 = x14
(5) l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x19 = x22 && x23 = x21 && 1 + x21 <= 1 && x21 = x20
(6) l3(x24, x25, x26) -> l4(x27, x28, x29) :|: x25 = x28 && x29 = x27 && 1 <= x27 && x27 <= 1 && x27 = x26
(7) l4(x30, x31, x32) -> l1(x33, x34, x35) :|: x35 = x34 && x34 = x34 && x33 = x32
(8) l5(x36, x37, x38) -> l3(x39, x40, x41) :|: x37 = x40 && x41 = x39 && 1 <= x39 && x39 = x38
(9) l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x43 = x46 && x47 = x45 && x45 <= 0 && x45 = x44
(10) l2(x48, x49, x50) -> l5(x51, x52, x53) :|: x49 = x52 && x53 = x51 && x51 = x50
(11) l6(x54, x55, x56) -> l1(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(12) l6(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63
(13) l6(x66, x67, x68) -> l3(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69
(14) l6(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75
(15) l6(x78, x79, x80) -> l5(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81
(16) l6(x84, x85, x86) -> l2(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87
(17) l7(x90, x91, x92) -> l6(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93

Arcs:
(2) -> (10)
(3) -> (10)
(4) -> (1), (2), (3)
(5) -> (1), (2), (3)
(6) -> (7)
(8) -> (4), (6)
(9) -> (7)
(10) -> (8), (9)
(12) -> (1), (2), (3)
(13) -> (4), (5), (6)
(14) -> (7)
(15) -> (8), (9)
(16) -> (10)
(17) -> (11), (12), (13), (14), (15), (16)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x5 = -2 + x3 && x3 = x2
(2) l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x13 = x16 && x17 = x15 && 2 <= x15 && x15 = x14
(3) l5(x36, x37, x38) -> l3(x39, x40, x41) :|: x37 = x40 && x41 = x39 && 1 <= x39 && x39 = x38
(4) l2(x48, x49, x50) -> l5(x51, x52, x53) :|: x49 = x52 && x53 = x51 && x51 = x50
(5) l0(x6, x7, x8) -> l2(x9, x10, x11) :|: x7 = x10 && x11 = -1 + x9 && x9 = x8

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:
l5(x36:0, x10:0, x15:0) -> l5(-1 + x15:0, x10:0, -1 + x15:0) :|: x15:0 > 1
l5(x, x1, x2) -> l5(-2 + x2, x1, -2 + x2) :|: x2 > 1

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

   l5(x1, x2, x3) -> l5(x3)

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(8)
Obligation:
Rules:
l5(x15:0) -> l5(-1 + x15:0) :|: x15:0 > 1
l5(x2) -> l5(-2 + x2) :|: x2 > 1

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l5(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l5(x15:0) -> l5(c) :|: c = -1 + x15:0 && x15:0 > 1
l5(x2) -> l5(c1) :|: c1 = -2 + x2 && x2 > 1

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x)] = x

The following rules are decreasing:
l5(x15:0) -> l5(c) :|: c = -1 + x15:0 && x15:0 > 1
l5(x2) -> l5(c1) :|: c1 = -2 + x2 && x2 > 1
The following rules are bounded:
l5(x15:0) -> l5(c) :|: c = -1 + x15:0 && x15:0 > 1
l5(x2) -> l5(c1) :|: c1 = -2 + x2 && x2 > 1

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(12)
YES