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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 337 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 15 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


----------------------------------------

(0)
Obligation:
Rules:
l0(Fnew5HAT0, Fold6HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && 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) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x8 = x17 && x6 = x15 && x5 = x14 && x3 = x12 && x2 = x11 && x13 = 1 + x4 && x10 = x16 && x9 = x + x1 && x16 = x && x4 <= x5
l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x26 = x35 && x25 = x34 && x24 = x33 && x23 = x32 && x22 = x31 && x21 = x30 && x20 = x29 && x19 = x28 && x18 = x27
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x44 = x53 && x43 = x52 && x42 = x51 && x39 = x48 && x49 = 2 && x46 = 0 && x45 = 1 && x50 = x47 && x47 = 30
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x62 = x71 && x61 = x70 && x60 = x69 && x59 = x68 && x58 = x67 && x57 = x66 && x56 = x65 && x55 = x64 && x54 = x63
Start term: l4(Fnew5HAT0, Fold6HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(Fnew5HAT0, Fold6HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && 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) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x8 = x17 && x6 = x15 && x5 = x14 && x3 = x12 && x2 = x11 && x13 = 1 + x4 && x10 = x16 && x9 = x + x1 && x16 = x && x4 <= x5
l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x26 = x35 && x25 = x34 && x24 = x33 && x23 = x32 && x22 = x31 && x21 = x30 && x20 = x29 && x19 = x28 && x18 = x27
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x44 = x53 && x43 = x52 && x42 = x51 && x39 = x48 && x49 = 2 && x46 = 0 && x45 = 1 && x50 = x47 && x47 = 30
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x62 = x71 && x61 = x70 && x60 = x69 && x59 = x68 && x58 = x67 && x57 = x66 && x56 = x65 && x55 = x64 && x54 = x63
Start term: l4(Fnew5HAT0, Fold6HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Fnew5HAT0, Fold6HAT0, aHAT0, ans8HAT0, i4HAT0, n3HAT0, ret_fib9HAT0, temp7HAT0, tmpHAT0) -> l1(Fnew5HATpost, Fold6HATpost, aHATpost, ans8HATpost, i4HATpost, n3HATpost, ret_fib9HATpost, temp7HATpost, tmpHATpost) :|: temp7HAT0 = temp7HATpost && n3HAT0 = n3HATpost && i4HAT0 = i4HATpost && aHAT0 = aHATpost && 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) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x8 = x17 && x6 = x15 && x5 = x14 && x3 = x12 && x2 = x11 && x13 = 1 + x4 && x10 = x16 && x9 = x + x1 && x16 = x && x4 <= x5
(3) l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x26 = x35 && x25 = x34 && x24 = x33 && x23 = x32 && x22 = x31 && x21 = x30 && x20 = x29 && x19 = x28 && x18 = x27
(4) l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x44 = x53 && x43 = x52 && x42 = x51 && x39 = x48 && x49 = 2 && x46 = 0 && x45 = 1 && x50 = x47 && x47 = 30
(5) l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x62 = x71 && x61 = x70 && x60 = x69 && x59 = x68 && x58 = x67 && x57 = x66 && x56 = x65 && x55 = x64 && x54 = x63

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

This digraph is fully evaluated!
----------------------------------------

(4)
Obligation:

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

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

This digraph is fully evaluated!

----------------------------------------

(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
l0(x10:0, x1:0, x11:0, x12:0, x4:0, x14:0, x15:0, x7:0, x17:0) -> l0(x10:0 + x1:0, x10:0, x11:0, x12:0, 1 + x4:0, x14:0, x15:0, x10:0, x17:0) :|: x4:0 <= x14:0

----------------------------------------

(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) -> l0(x5, x6)

----------------------------------------

(8)
Obligation:
Rules:
l0(x4:0, x14:0) -> l0(1 + x4:0, x14:0) :|: x4:0 <= x14:0

----------------------------------------

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l0(x4:0, x14:0) -> l0(c, x14:0) :|: c = 1 + x4:0 && x4:0 <= x14:0

----------------------------------------

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

The following rules are decreasing:
l0(x4:0, x14:0) -> l0(c, x14:0) :|: c = 1 + x4:0 && x4:0 <= x14:0
The following rules are bounded:
l0(x4:0, x14:0) -> l0(c, x14:0) :|: c = 1 + x4:0 && x4:0 <= x14:0

----------------------------------------

(12)
YES