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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 409 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 36 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, __cil_tmp4_8HAT0, __const_1000HAT0, __const_101HAT0, __const_9HAT0, __retres3_7HAT0, i_5HAT0, x_6HAT0) -> l1(Result_4HATpost, __cil_tmp4_8HATpost, __const_1000HATpost, __const_101HATpost, __const_9HATpost, __retres3_7HATpost, i_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && i_5HAT0 = i_5HATpost && __const_9HAT0 = __const_9HATpost && __const_101HAT0 = __const_101HATpost && __const_1000HAT0 = __const_1000HATpost && Result_4HATpost = __cil_tmp4_8HATpost && __cil_tmp4_8HATpost = __retres3_7HATpost && __retres3_7HATpost = 0 && 1 + __const_9HAT0 - x_6HAT0 <= 0
l0(x, x1, x2, x3, x4, x5, x6, x7) -> l1(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x4 = x12 && x3 = x11 && x2 = x10 && x8 = x9 && x9 = x13 && x13 = 0 && 1 - x3 + x6 <= 0 && 0 <= x4 - x7
l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && x30 = -1 + x22 && 0 <= -1 * x19 + x22 && 0 <= x20 - x23
l2(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 && x32 = x40
l3(x48, x49, x50, x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56 && x62 = x50
l4(x64, x65, x66, x67, x68, x69, x70, x71) -> l3(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x65 = x73 && x64 = x72
Start term: l4(Result_4HAT0, __cil_tmp4_8HAT0, __const_1000HAT0, __const_101HAT0, __const_9HAT0, __retres3_7HAT0, i_5HAT0, x_6HAT0)

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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(Result_4HAT0, __cil_tmp4_8HAT0, __const_1000HAT0, __const_101HAT0, __const_9HAT0, __retres3_7HAT0, i_5HAT0, x_6HAT0) -> l1(Result_4HATpost, __cil_tmp4_8HATpost, __const_1000HATpost, __const_101HATpost, __const_9HATpost, __retres3_7HATpost, i_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && i_5HAT0 = i_5HATpost && __const_9HAT0 = __const_9HATpost && __const_101HAT0 = __const_101HATpost && __const_1000HAT0 = __const_1000HATpost && Result_4HATpost = __cil_tmp4_8HATpost && __cil_tmp4_8HATpost = __retres3_7HATpost && __retres3_7HATpost = 0 && 1 + __const_9HAT0 - x_6HAT0 <= 0
l0(x, x1, x2, x3, x4, x5, x6, x7) -> l1(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x4 = x12 && x3 = x11 && x2 = x10 && x8 = x9 && x9 = x13 && x13 = 0 && 1 - x3 + x6 <= 0 && 0 <= x4 - x7
l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && x30 = -1 + x22 && 0 <= -1 * x19 + x22 && 0 <= x20 - x23
l2(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 && x32 = x40
l3(x48, x49, x50, x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56 && x62 = x50
l4(x64, x65, x66, x67, x68, x69, x70, x71) -> l3(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x65 = x73 && x64 = x72
Start term: l4(Result_4HAT0, __cil_tmp4_8HAT0, __const_1000HAT0, __const_101HAT0, __const_9HAT0, __retres3_7HAT0, i_5HAT0, x_6HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, __cil_tmp4_8HAT0, __const_1000HAT0, __const_101HAT0, __const_9HAT0, __retres3_7HAT0, i_5HAT0, x_6HAT0) -> l1(Result_4HATpost, __cil_tmp4_8HATpost, __const_1000HATpost, __const_101HATpost, __const_9HATpost, __retres3_7HATpost, i_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && i_5HAT0 = i_5HATpost && __const_9HAT0 = __const_9HATpost && __const_101HAT0 = __const_101HATpost && __const_1000HAT0 = __const_1000HATpost && Result_4HATpost = __cil_tmp4_8HATpost && __cil_tmp4_8HATpost = __retres3_7HATpost && __retres3_7HATpost = 0 && 1 + __const_9HAT0 - x_6HAT0 <= 0
(2) l0(x, x1, x2, x3, x4, x5, x6, x7) -> l1(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x4 = x12 && x3 = x11 && x2 = x10 && x8 = x9 && x9 = x13 && x13 = 0 && 1 - x3 + x6 <= 0 && 0 <= x4 - x7
(3) l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && x30 = -1 + x22 && 0 <= -1 * x19 + x22 && 0 <= x20 - x23
(4) l2(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 && x32 = x40
(5) l3(x48, x49, x50, x51, x52, x53, x54, x55) -> l0(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56 && x62 = x50
(6) l4(x64, x65, x66, x67, x68, x69, x70, x71) -> l3(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x65 = x73 && x64 = x72

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && x30 = -1 + x22 && 0 <= -1 * x19 + x22 && 0 <= x20 - x23
(2) l2(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 && x32 = x40

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(x16:0, x17:0, x18:0, x19:0, x20:0, x21:0, x22:0, x23:0) -> l0(x16:0, x17:0, x18:0, x19:0, x20:0, x21:0, -1 + x22:0, x23:0) :|: x20:0 - x23:0 >= 0 && 0 <= -1 * x19:0 + x22:0

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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, x5, x6, x7, x8) -> l0(x4, x5, x7, x8)

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(8)
Obligation:
Rules:
l0(x19:0, x20:0, x22:0, x23:0) -> l0(x19:0, x20:0, -1 + x22:0, x23:0) :|: x20:0 - x23:0 >= 0 && 0 <= -1 * x19:0 + x22:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x19:0, x20:0, x22:0, x23:0) -> l0(x19:0, x20:0, c, x23:0) :|: c = -1 + x22:0 && (x20:0 - x23:0 >= 0 && 0 <= -1 * x19:0 + x22:0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1, x2, x3)] = -x + x2

The following rules are decreasing:
l0(x19:0, x20:0, x22:0, x23:0) -> l0(x19:0, x20:0, c, x23:0) :|: c = -1 + x22:0 && (x20:0 - x23:0 >= 0 && 0 <= -1 * x19:0 + x22:0)
The following rules are bounded:
l0(x19:0, x20:0, x22:0, x23:0) -> l0(x19:0, x20:0, c, x23:0) :|: c = -1 + x22:0 && (x20:0 - x23:0 >= 0 && 0 <= -1 * x19:0 + x22:0)

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(12)
YES