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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 321 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 2 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 16 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(__const_10HAT0, iHAT0, jHAT0, xHAT0) -> l1(__const_10HATpost, iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost && xHAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x = x4 && x5 = 1 + x1 && x6 = 2 + x2 && 1 + x1 <= x3
l3(x8, x9, x10, x11) -> l2(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0 && 2 <= x11
l3(x16, x17, x18, x19) -> l4(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && x19 <= 1
l2(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l1(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 && 1 + x42 <= 2 * x43
l1(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 2 * x51 <= x50
l6(x56, x57, x58, x59) -> l3(x60, x61, x62, x63) :|: x57 = x61 && x56 = x60 && x63 = x56 && x62 = 0
l7(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
Start term: l7(__const_10HAT0, iHAT0, jHAT0, xHAT0)

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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(__const_10HAT0, iHAT0, jHAT0, xHAT0) -> l1(__const_10HATpost, iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost && xHAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x = x4 && x5 = 1 + x1 && x6 = 2 + x2 && 1 + x1 <= x3
l3(x8, x9, x10, x11) -> l2(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0 && 2 <= x11
l3(x16, x17, x18, x19) -> l4(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && x19 <= 1
l2(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
l1(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 && 1 + x42 <= 2 * x43
l1(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 2 * x51 <= x50
l6(x56, x57, x58, x59) -> l3(x60, x61, x62, x63) :|: x57 = x61 && x56 = x60 && x63 = x56 && x62 = 0
l7(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
Start term: l7(__const_10HAT0, iHAT0, jHAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, iHAT0, jHAT0, xHAT0) -> l1(__const_10HATpost, iHATpost, jHATpost, xHATpost) :|: xHAT0 = xHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost && xHAT0 <= iHAT0
(2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x = x4 && x5 = 1 + x1 && x6 = 2 + x2 && 1 + x1 <= x3
(3) l3(x8, x9, x10, x11) -> l2(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0 && 2 <= x11
(4) l3(x16, x17, x18, x19) -> l4(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && x19 <= 1
(5) l2(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
(6) l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
(7) l1(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 && 1 + x42 <= 2 * x43
(8) l1(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 2 * x51 <= x50
(9) l6(x56, x57, x58, x59) -> l3(x60, x61, x62, x63) :|: x57 = x61 && x56 = x60 && x63 = x56 && x62 = 0
(10) l7(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x = x4 && x5 = 1 + x1 && x6 = 2 + x2 && 1 + x1 <= x3
(2) l2(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(x28:0, x1:0, x2:0, x31:0) -> l0(x28:0, 1 + x1:0, 2 + x2:0, x31:0) :|: x31:0 >= 1 + x1:0

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

   l0(x1, x2, x3, x4) -> l0(x2, x4)

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(8)
Obligation:
Rules:
l0(x1:0, x31:0) -> l0(1 + x1:0, x31:0) :|: x31:0 >= 1 + x1:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x1:0, x31:0) -> l0(c, x31:0) :|: c = 1 + x1:0 && x31:0 >= 1 + x1:0

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

The following rules are decreasing:
l0(x1:0, x31:0) -> l0(c, x31:0) :|: c = 1 + x1:0 && x31:0 >= 1 + x1:0
The following rules are bounded:
l0(x1:0, x31:0) -> l0(c, x31:0) :|: c = 1 + x1:0 && x31:0 >= 1 + x1:0

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(12)
YES