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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 310 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 1 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 17 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0
l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: 1 + x1 <= 10 && x10 = 1 + x1 && x6 = x6 && 2 <= x6 && x6 <= 2 && x5 = x5 && x5 <= x6 && x6 <= x5 && x2 = x7 && x3 = x8 && x4 = x9
l3(x11, x12, x13, x14, x15) -> l1(x16, x17, x18, x19, x20) :|: x15 = x20 && x14 = x19 && x13 = x18 && x11 = x16 && x17 <= 1 && 1 <= x17 && x17 = 1 + x12 && 1 + x12 <= 10
l1(x21, x22, x23, x24, x25) -> l4(x26, x27, x28, x29, x30) :|: x25 = x30 && x23 = x28 && x22 = x27 && x21 = x26 && x29 = x25 && 10 <= x22
l1(x31, x32, x33, x34, x35) -> l5(x36, x37, x38, x39, x40) :|: x35 = x40 && x34 = x39 && x31 = x36 && 1 + x38 <= 10 && 1 + x38 <= x37 && x37 <= 1 + x38 && x37 = 1 + x32 && 1 + x32 <= 10 && x38 = x38
l5(x41, x42, x43, x44, x45) -> l1(x46, x47, x48, x49, x50) :|: x45 = x50 && x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46
l6(x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60) :|: x55 = x60 && x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56
Start term: l6(a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0)

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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(a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0
l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: 1 + x1 <= 10 && x10 = 1 + x1 && x6 = x6 && 2 <= x6 && x6 <= 2 && x5 = x5 && x5 <= x6 && x6 <= x5 && x2 = x7 && x3 = x8 && x4 = x9
l3(x11, x12, x13, x14, x15) -> l1(x16, x17, x18, x19, x20) :|: x15 = x20 && x14 = x19 && x13 = x18 && x11 = x16 && x17 <= 1 && 1 <= x17 && x17 = 1 + x12 && 1 + x12 <= 10
l1(x21, x22, x23, x24, x25) -> l4(x26, x27, x28, x29, x30) :|: x25 = x30 && x23 = x28 && x22 = x27 && x21 = x26 && x29 = x25 && 10 <= x22
l1(x31, x32, x33, x34, x35) -> l5(x36, x37, x38, x39, x40) :|: x35 = x40 && x34 = x39 && x31 = x36 && 1 + x38 <= 10 && 1 + x38 <= x37 && x37 <= 1 + x38 && x37 = 1 + x32 && 1 + x32 <= 10 && x38 = x38
l5(x41, x42, x43, x44, x45) -> l1(x46, x47, x48, x49, x50) :|: x45 = x50 && x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46
l6(x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60) :|: x55 = x60 && x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56
Start term: l6(a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0
(2) l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: 1 + x1 <= 10 && x10 = 1 + x1 && x6 = x6 && 2 <= x6 && x6 <= 2 && x5 = x5 && x5 <= x6 && x6 <= x5 && x2 = x7 && x3 = x8 && x4 = x9
(3) l3(x11, x12, x13, x14, x15) -> l1(x16, x17, x18, x19, x20) :|: x15 = x20 && x14 = x19 && x13 = x18 && x11 = x16 && x17 <= 1 && 1 <= x17 && x17 = 1 + x12 && 1 + x12 <= 10
(4) l1(x21, x22, x23, x24, x25) -> l4(x26, x27, x28, x29, x30) :|: x25 = x30 && x23 = x28 && x22 = x27 && x21 = x26 && x29 = x25 && 10 <= x22
(5) l1(x31, x32, x33, x34, x35) -> l5(x36, x37, x38, x39, x40) :|: x35 = x40 && x34 = x39 && x31 = x36 && 1 + x38 <= 10 && 1 + x38 <= x37 && x37 <= 1 + x38 && x37 = 1 + x32 && 1 + x32 <= 10 && x38 = x38
(6) l5(x41, x42, x43, x44, x45) -> l1(x46, x47, x48, x49, x50) :|: x45 = x50 && x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46
(7) l6(x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60) :|: x55 = x60 && x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x31, x32, x33, x34, x35) -> l5(x36, x37, x38, x39, x40) :|: x35 = x40 && x34 = x39 && x31 = x36 && 1 + x38 <= 10 && 1 + x38 <= x37 && x37 <= 1 + x38 && x37 = 1 + x32 && 1 + x32 <= 10 && x38 = x38
(2) l5(x41, x42, x43, x44, x45) -> l1(x46, x47, x48, x49, x50) :|: x45 = x50 && x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46

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:
l1(x31:0, x32:0, x33:0, x34:0, x35:0) -> l1(x31:0, 1 + x32:0, x38:0, x34:0, x35:0) :|: x38:0 < 10 && 1 + x38:0 = 1 + x32:0 && x32:0 < 10

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

   l1(x1, x2, x3, x4, x5) -> l1(x2, x3)

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(8)
Obligation:
Rules:
l1(x32:0, x33:0) -> l1(1 + x32:0, x38:0) :|: x38:0 < 10 && 1 + x38:0 = 1 + x32:0 && x32:0 < 10

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l1(x32:0, x33:0) -> l1(c, x38:0) :|: c = 1 + x32:0 && (x38:0 < 10 && 1 + x38:0 = 1 + x32:0 && x32:0 < 10)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = 9 - x

The following rules are decreasing:
l1(x32:0, x33:0) -> l1(c, x38:0) :|: c = 1 + x32:0 && (x38:0 < 10 && 1 + x38:0 = 1 + x32:0 && x32:0 < 10)
The following rules are bounded:
l1(x32:0, x33:0) -> l1(c, x38:0) :|: c = 1 + x32:0 && (x38:0 < 10 && 1 + x38:0 = 1 + x32:0 && x32:0 < 10)

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(12)
YES