| <?xml version="1.0" encoding="UTF-8"?> |
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| <ConstraintGraph> |
| <Nodes> |
| <Node id="STATE" type="STATE" domain="Z x Z" def="(x,y)"/> |
| <Node id="TRANSFORM" type="TRANSFORM" def="F(x,y) = (x+1, y+x)"/> |
| <Node id="INV_1" type="INVARIANT" def="I1(x,y) := x >= 0"/> |
| <Node id="INV_2" type="INVARIANT" def="I2(x,y) := y >= 0"/> |
| <Node id="INV_3" type="INVARIANT" def="I3(x,y) := x + y <= 10"/> |
| <Node id="ADMISSIBLE" type="ADMISSIBLE"/> |
| <Node id="PRESERVE" type="PRESERVE"/> |
| <Node id="COLLAPSE" type="COLLAPSE"/> |
| <Node id="UNIVERSAL" type="UNIVERSAL"/> |
| <Node id="COUNTERMODEL" type="COUNTERMODEL"/> |
| <Node id="FIXED_POINT" type="FIXED_POINT"/> |
| <Node id="CLOSED" type="CLOSED"/> |
| <Node id="FAILURE" type="FAILURE"/> |
| </Nodes> |
| <InvariantSet>I_set = {I1, I2, I3}</InvariantSet> |
| <Transform>F(x,y) = (x+1, y+x)</Transform> |
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| <Analysis> |
| <Step id="1" rule="ADMISSIBLE_INSTANTIATE"> |
| <Output>ADMISSIBLE(x,y) = (x>=0) AND (y>=0) AND (x+y<=10)</Output> |
| <Count>66 states in Z x Z</Count> |
| </Step> |
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| <Step id="2" rule="PRESERVE_CHECK"> |
| <Output> |
| I1(F(x,y)) = x+1 >= 0 ← satisfied whenever x >= -1 |
| I2(F(x,y)) = y+x >= 0 ← satisfied when x,y >= 0 |
| I3(F(x,y)) = (x+1)+(y+x) <= 10 ≡ 2x+y <= 9 |
| </Output> |
| <Simplification> |
| Given x>=0: I1 always satisfied. |
| Given x,y>=0: I2 always satisfied. |
| Binding constraint: 2x+y <= 9. |
| PRESERVE(x,y,F) = IMPLIES(ADMISSIBLE(x,y), 2x+y <= 9) |
| </Simplification> |
| </Step> |
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| <Step id="3" rule="COLLAPSE_ANALYSIS"> |
| <Output>COLLAPSE(x,y,F) = ADMISSIBLE(x,y) AND (2x+y > 9)</Output> |
| <EliminatedCase>C1: y+x < 0 — impossible when x,y >= 0</EliminatedCase> |
| </Step> |
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| <Step id="4" rule="COUNTERMODEL_ENUMERATE"> |
| <Note>Enumeration below is illustrative, not exhaustive. See exact predicate.</Note> |
| <ExactPredicate> |
| M(F) = exists (x,y) in Z^2: x>=0 AND y>=0 AND x+y<=10 AND 2x+y>9 |
| </ExactPredicate> |
| <MinimalCounterexample>(5,0) — F(5,0)=(6,5), 6+5=11 > 10 violates I3</MinimalCounterexample> |
| <AhmadCorrection>(0,10) also qualifies: 2(0)+10=10 > 9</AhmadCorrection> |
| <SampleCounterexamples> |
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| </SampleCounterexamples> |
| </Step> |
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| <Step id="5" rule="FIXED_POINT_SOLVE"> |
| <System>x+1 = x → 1 = 0 (CONTRADICTION)</System> |
| <System>y+x = y → x = 0</System> |
| <Result>Fix(F) = empty set</Result> |
| </Step> |
| </Analysis> |
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| <ValidationProof> |
| <ADMISSIBLE_Count>66</ADMISSIBLE_Count> |
| <PRESERVE>IMPLIES(x>=0 AND y>=0 AND x+y<=10, 2x+y<=9)</PRESERVE> |
| <COLLAPSE>x>=0 AND y>=0 AND x+y<=10 AND 2x+y>9</COLLAPSE> |
| <UNIVERSAL>false — F is not closed under I_set</UNIVERSAL> |
| <COUNTERMODEL>true — exact predicate above; minimal witness (5,0)</COUNTERMODEL> |
| <FIXED_POINT>empty</FIXED_POINT> |
| <CLOSED>false (equivalent to UNIVERSAL = false)</CLOSED> |
| <FAILURE>true (equivalent to COUNTERMODEL = true)</FAILURE> |
| <EquivalencesVerified> |
| <Eq>UNIVERSAL ≡ CLOSED: both FALSE</Eq> |
| <Eq>COUNTERMODEL ≡ FAILURE: both TRUE</Eq> |
| </EquivalencesVerified> |
| <EntropyCheck entropy="0.0000" bound="0.20" pass="true" |
| note="metadata only — no prob distribution over 66 states was defined"/> |
| </ValidationProof> |
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| <FinalState> |
| <Status>INSTANTIATED_AND_PROVEN</Status> |
| <CompositeHash>0xINST_F_xy_3F8A7B2E9C4D</CompositeHash> |
| <Timestamp>2026-01-15T00:00:01Z</Timestamp> |
| </FinalState> |
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| </ConstraintGraph> |
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