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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 83 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 35 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 15 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 5 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 4 ms]
        (16) IntTRS
        (17) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (18) YES


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(0)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1) -> f149_0_doSum_LT(arg1P) :|: 10 = arg1P
f149_0_doSum_LT(x) -> f163_0_factorial_GT(x1) :|: x = x1 && -1 <= x - 1
f149_0_doSum_LT(x2) -> f149_0_doSum_LT(x3) :|: x2 - 1 = x3 && -1 <= x2 - 1
f163_0_factorial_GT(x4) -> f163_0_factorial_GT(x5) :|: x4 - 1 = x5 && x4 - 1 <= x4 - 1 && 0 <= x4 - 1
__init(x6) -> f1_0_main_ConstantStackPush(x7) :|: 0 <= 0
Start term: __init(arg1)

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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:
f1_0_main_ConstantStackPush(arg1) -> f149_0_doSum_LT(arg1P) :|: 10 = arg1P
f149_0_doSum_LT(x) -> f163_0_factorial_GT(x1) :|: x = x1 && -1 <= x - 1
f149_0_doSum_LT(x2) -> f149_0_doSum_LT(x3) :|: x2 - 1 = x3 && -1 <= x2 - 1
f163_0_factorial_GT(x4) -> f163_0_factorial_GT(x5) :|: x4 - 1 = x5 && x4 - 1 <= x4 - 1 && 0 <= x4 - 1
__init(x6) -> f1_0_main_ConstantStackPush(x7) :|: 0 <= 0
Start term: __init(arg1)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_ConstantStackPush(arg1) -> f149_0_doSum_LT(arg1P) :|: 10 = arg1P
(2) f149_0_doSum_LT(x) -> f163_0_factorial_GT(x1) :|: x = x1 && -1 <= x - 1
(3) f149_0_doSum_LT(x2) -> f149_0_doSum_LT(x3) :|: x2 - 1 = x3 && -1 <= x2 - 1
(4) f163_0_factorial_GT(x4) -> f163_0_factorial_GT(x5) :|: x4 - 1 = x5 && x4 - 1 <= x4 - 1 && 0 <= x4 - 1
(5) __init(x6) -> f1_0_main_ConstantStackPush(x7) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f149_0_doSum_LT(x2) -> f149_0_doSum_LT(x3) :|: x2 - 1 = x3 && -1 <= x2 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f149_0_doSum_LT(x2:0) -> f149_0_doSum_LT(x2:0 - 1) :|: x2:0 > -1

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f149_0_doSum_LT(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f149_0_doSum_LT(x2:0) -> f149_0_doSum_LT(c) :|: c = x2:0 - 1 && x2:0 > -1

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f149_0_doSum_LT(x)] = x

The following rules are decreasing:
f149_0_doSum_LT(x2:0) -> f149_0_doSum_LT(c) :|: c = x2:0 - 1 && x2:0 > -1
The following rules are bounded:
f149_0_doSum_LT(x2:0) -> f149_0_doSum_LT(c) :|: c = x2:0 - 1 && x2:0 > -1

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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f163_0_factorial_GT(x4) -> f163_0_factorial_GT(x5) :|: x4 - 1 = x5 && x4 - 1 <= x4 - 1 && 0 <= x4 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(14)
Obligation:
Rules:
f163_0_factorial_GT(x4:0) -> f163_0_factorial_GT(x4:0 - 1) :|: x4:0 > 0

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(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f163_0_factorial_GT(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(16)
Obligation:
Rules:
f163_0_factorial_GT(x4:0) -> f163_0_factorial_GT(c) :|: c = x4:0 - 1 && x4:0 > 0

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(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f163_0_factorial_GT(x)] = x

The following rules are decreasing:
f163_0_factorial_GT(x4:0) -> f163_0_factorial_GT(c) :|: c = x4:0 - 1 && x4:0 > 0
The following rules are bounded:
f163_0_factorial_GT(x4:0) -> f163_0_factorial_GT(c) :|: c = x4:0 - 1 && x4:0 > 0

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(18)
YES