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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 132 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 34 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
(12) YES


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

(0)
Obligation:
Rules:
l0(__const_20HAT0, iHAT0, jHAT0) -> l1(__const_20HATpost, iHATpost, jHATpost) :|: jHAT0 = jHATpost && iHAT0 = iHATpost && __const_20HAT0 = __const_20HATpost && __const_20HAT0 <= jHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= x
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 2 + x11 && x11 = 2 + x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = 0
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
Start term: l4(__const_20HAT0, iHAT0, jHAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_20HAT0, iHAT0, jHAT0) -> l1(__const_20HATpost, iHATpost, jHATpost) :|: jHAT0 = jHATpost && iHAT0 = iHATpost && __const_20HAT0 = __const_20HATpost && __const_20HAT0 <= jHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= x
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 2 + x11 && x11 = 2 + x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = 0
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
Start term: l4(__const_20HAT0, iHAT0, jHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_20HAT0, iHAT0, jHAT0) -> l1(__const_20HATpost, iHATpost, jHATpost) :|: jHAT0 = jHATpost && iHAT0 = iHATpost && __const_20HAT0 = __const_20HATpost && __const_20HAT0 <= jHAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= x
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 2 + x11 && x11 = 2 + x8
(4) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = 0
(5) l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21

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) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= x
(2) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 2 + x11 && x11 = 2 + x8

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l0(x3:0, x1:0, x2:0) -> l0(x3:0, 2 + (2 + x2:0), 2 + x2:0) :|: x3:0 >= 1 + x2:0

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

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

   l0(x1, x2, x3) -> l0(x1, x3)

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

(8)
Obligation:
Rules:
l0(x3:0, x2:0) -> l0(x3:0, 2 + x2:0) :|: x3:0 >= 1 + x2:0

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

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

(10)
Obligation:
Rules:
l0(x3:0, x2:0) -> l0(x3:0, c) :|: c = 2 + x2:0 && x3:0 >= 1 + x2:0

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

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

The following rules are decreasing:
l0(x3:0, x2:0) -> l0(x3:0, c) :|: c = 2 + x2:0 && x3:0 >= 1 + x2:0
The following rules are bounded:
l0(x3:0, x2:0) -> l0(x3:0, c) :|: c = 2 + x2:0 && x3:0 >= 1 + x2:0

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

(12)
YES