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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 341 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 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(Fnew5HAT0, Fold6HAT0, __const_30HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, __const_30HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && __const_30HAT0 = __const_30HATpost && Fold6HAT0 = Fold6HATpost && Fnew5HAT0 = Fnew5HATpost && tmpHATpost = ret_fib9HATpost && ret_fib9HATpost = ans8HATpost && ans8HATpost = Fnew5HAT0 && 1 + n3HAT0 <= i4HAT0
l0(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l2(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x9 = x19 && x7 = x17 && x6 = x16 && x4 = x14 && x3 = x13 && x2 = x12 && x15 = 1 + x5 && x11 = x18 && x10 = x + x1 && x18 = x && x5 <= x6
l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x29 = x39 && x28 = x38 && x27 = x37 && x26 = x36 && x25 = x35 && x24 = x34 && x23 = x33 && x22 = x32 && x21 = x31 && x20 = x30
l3(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l2(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x49 = x59 && x48 = x58 && x47 = x57 && x44 = x54 && x42 = x52 && x55 = 2 && x51 = 0 && x50 = 1 && x56 = x53 && x53 = x42
l4(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l3(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x69 = x79 && x68 = x78 && x67 = x77 && x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x62 = x72 && x61 = x71 && x60 = x70
Start term: l4(Fnew5HAT0, Fold6HAT0, __const_30HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0)

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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(Fnew5HAT0, Fold6HAT0, __const_30HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, __const_30HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && __const_30HAT0 = __const_30HATpost && Fold6HAT0 = Fold6HATpost && Fnew5HAT0 = Fnew5HATpost && tmpHATpost = ret_fib9HATpost && ret_fib9HATpost = ans8HATpost && ans8HATpost = Fnew5HAT0 && 1 + n3HAT0 <= i4HAT0
l0(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l2(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x9 = x19 && x7 = x17 && x6 = x16 && x4 = x14 && x3 = x13 && x2 = x12 && x15 = 1 + x5 && x11 = x18 && x10 = x + x1 && x18 = x && x5 <= x6
l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x29 = x39 && x28 = x38 && x27 = x37 && x26 = x36 && x25 = x35 && x24 = x34 && x23 = x33 && x22 = x32 && x21 = x31 && x20 = x30
l3(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l2(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x49 = x59 && x48 = x58 && x47 = x57 && x44 = x54 && x42 = x52 && x55 = 2 && x51 = 0 && x50 = 1 && x56 = x53 && x53 = x42
l4(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l3(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x69 = x79 && x68 = x78 && x67 = x77 && x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x62 = x72 && x61 = x71 && x60 = x70
Start term: l4(Fnew5HAT0, Fold6HAT0, __const_30HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Fnew5HAT0, Fold6HAT0, __const_30HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, __const_30HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && __const_30HAT0 = __const_30HATpost && Fold6HAT0 = Fold6HATpost && Fnew5HAT0 = Fnew5HATpost && tmpHATpost = ret_fib9HATpost && ret_fib9HATpost = ans8HATpost && ans8HATpost = Fnew5HAT0 && 1 + n3HAT0 <= i4HAT0
(2) l0(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l2(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x9 = x19 && x7 = x17 && x6 = x16 && x4 = x14 && x3 = x13 && x2 = x12 && x15 = 1 + x5 && x11 = x18 && x10 = x + x1 && x18 = x && x5 <= x6
(3) l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x29 = x39 && x28 = x38 && x27 = x37 && x26 = x36 && x25 = x35 && x24 = x34 && x23 = x33 && x22 = x32 && x21 = x31 && x20 = x30
(4) l3(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l2(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x49 = x59 && x48 = x58 && x47 = x57 && x44 = x54 && x42 = x52 && x55 = 2 && x51 = 0 && x50 = 1 && x56 = x53 && x53 = x42
(5) l4(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l3(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x69 = x79 && x68 = x78 && x67 = x77 && x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x62 = x72 && x61 = x71 && x60 = x70

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l2(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x9 = x19 && x7 = x17 && x6 = x16 && x4 = x14 && x3 = x13 && x2 = x12 && x15 = 1 + x5 && x11 = x18 && x10 = x + x1 && x18 = x && x5 <= x6
(2) l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x29 = x39 && x28 = x38 && x27 = x37 && x26 = x36 && x25 = x35 && x24 = x34 && x23 = x33 && x22 = x32 && x21 = x31 && x20 = x30

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(x11:0, x1:0, x12:0, x13:0, x14:0, x5:0, x16:0, x17:0, x8:0, x19:0) -> l0(x11:0 + x1:0, x11:0, x12:0, x13:0, x14:0, 1 + x5:0, x16:0, x17:0, x11:0, x19:0) :|: x5:0 <= x16:0

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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, x9, x10) -> l0(x6, x7)

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(8)
Obligation:
Rules:
l0(x5:0, x16:0) -> l0(1 + x5:0, x16:0) :|: x5:0 <= x16:0

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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(x5:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x5:0 && x5:0 <= x16:0

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

The following rules are decreasing:
l0(x5:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x5:0 && x5:0 <= x16:0
The following rules are bounded:
l0(x5:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x5:0 && x5:0 <= x16:0

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(12)
YES