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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 403 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 32 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(__const_500HAT0, i_13HAT0, j_15HAT0, rt_11HAT0, s_16HAT0, s_17HAT0, st_14HAT0) -> l1(__const_500HATpost, i_13HATpost, j_15HATpost, rt_11HATpost, s_16HATpost, s_17HATpost, st_14HATpost) :|: i_13HAT1 = s_17HAT0 && j_15HATpost = s_16HAT0 && i_13HATpost = j_15HATpost && __const_500HAT0 = __const_500HATpost && rt_11HAT0 = rt_11HATpost && s_16HAT0 = s_16HATpost && s_17HAT0 = s_17HATpost && st_14HAT0 = st_14HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x6 && x <= x1
l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x14
l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l3(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = x48 && x42 <= x43
l1(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x56 = x63 && x64 = 1 + x57 && 1 + x57 <= x56
l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77
Start term: l5(__const_500HAT0, i_13HAT0, j_15HAT0, rt_11HAT0, s_16HAT0, s_17HAT0, 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(__const_500HAT0, i_13HAT0, j_15HAT0, rt_11HAT0, s_16HAT0, s_17HAT0, st_14HAT0) -> l1(__const_500HATpost, i_13HATpost, j_15HATpost, rt_11HATpost, s_16HATpost, s_17HATpost, st_14HATpost) :|: i_13HAT1 = s_17HAT0 && j_15HATpost = s_16HAT0 && i_13HATpost = j_15HATpost && __const_500HAT0 = __const_500HATpost && rt_11HAT0 = rt_11HATpost && s_16HAT0 = s_16HATpost && s_17HAT0 = s_17HATpost && st_14HAT0 = st_14HATpost
l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x6 && x <= x1
l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x14
l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l3(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = x48 && x42 <= x43
l1(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x56 = x63 && x64 = 1 + x57 && 1 + x57 <= x56
l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77
Start term: l5(__const_500HAT0, i_13HAT0, j_15HAT0, rt_11HAT0, s_16HAT0, s_17HAT0, st_14HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_500HAT0, i_13HAT0, j_15HAT0, rt_11HAT0, s_16HAT0, s_17HAT0, st_14HAT0) -> l1(__const_500HATpost, i_13HATpost, j_15HATpost, rt_11HATpost, s_16HATpost, s_17HATpost, st_14HATpost) :|: i_13HAT1 = s_17HAT0 && j_15HATpost = s_16HAT0 && i_13HATpost = j_15HATpost && __const_500HAT0 = __const_500HATpost && rt_11HAT0 = rt_11HATpost && s_16HAT0 = s_16HATpost && s_17HAT0 = s_17HATpost && st_14HAT0 = st_14HATpost
(2) l2(x, x1, x2, x3, x4, x5, x6) -> l3(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = x6 && x <= x1
(3) l2(x14, x15, x16, x17, x18, x19, x20) -> l4(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x14 = x21 && x22 = 1 + x15 && 1 + x15 <= x14
(4) l4(x28, x29, x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
(5) l1(x42, x43, x44, x45, x46, x47, x48) -> l3(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = x48 && x42 <= x43
(6) l1(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x56 = x63 && x64 = 1 + x57 && 1 + x57 <= x56
(7) l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77

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

This digraph is fully evaluated!
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(4)
Obligation:

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

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:
l2(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) -> l2(x14:0, 1 + x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) :|: x14:0 >= 1 + x15:0

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

   l2(x1, x2, x3, x4, x5, x6, x7) -> l2(x1, x2)

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(8)
Obligation:
Rules:
l2(x14:0, x15:0) -> l2(x14:0, 1 + x15:0) :|: x14:0 >= 1 + x15:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l2(x14:0, x15:0) -> l2(x14:0, c) :|: c = 1 + x15:0 && x14:0 >= 1 + x15:0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = x - x1

The following rules are decreasing:
l2(x14:0, x15:0) -> l2(x14:0, c) :|: c = 1 + x15:0 && x14:0 >= 1 + x15:0
The following rules are bounded:
l2(x14:0, x15:0) -> l2(x14:0, c) :|: c = 1 + x15:0 && x14:0 >= 1 + x15:0

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(12)
YES