FAILURE MAP
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FA-43566 / Borrow checking / Open access

Alpha-renaming splits equal bound lifetime occurrences · case 01

Alpha-renaming splits equal bound lifetime occurrences.

Verified by executionVariant 1 · 21 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The static analyzer mishandles alpha equality: alpha-renaming splits equal bound lifetime occurrences.

THE FAILURE

The static analyzer mishandles alpha equality: alpha-renaming splits equal bound lifetime occurrences.

Unsuccessful approach: The partial repair uses if any(a!=b and d['renaming'][a]!=d['renaming'][b] for a,b in d['equal_pairs'][:1]): errors.append('alpha-equality'), which still violates the stipulated analysis contract.

Case contract

Check higher-ranked borrow instantiation using de Bruijn binder depths and universes. Bound occurrence index less than binder arity; substitutions shift free lifetimes when crossing binders; fresh universes strictly exceed caller universe; existential solutions cannot mention newer universes; universally quantified input must be checked for every listed placeholder; alpha renaming preserves equal occurrences; distinct bound parameters remain distinct; dropping a binder decrements outside indices; bound lifetimes cannot be selected as static; placeholder obligations cannot be discharged by one concrete lifetime. Input is a fully explicit descriptor of the stated toy IR. Return rule identifiers in declaration order; absent optional obligations use the provided neutral defaults. No rule is an assertion about a production language.

Why this case matters

A finite offline static-analysis model of ownership and borrowing; it does not execute the analyzed program.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(d):
    errors=[]
    if any(i<0 or i>=d['arity'] for i in d['indices']): errors.append('bound-index-range')
    if d['after']!=[i+d['crossed'] for i in d['before']]: errors.append('capture-avoiding-shift')
    if d['fresh']<=d['caller']: errors.append('fresh-universe')
    if any(u>d['existential_universe'] for u in d['solution_universes']): errors.append('universe-leak')
    if not set(d['placeholders'])<=set(d['checked']): errors.append('universal-coverage')
    if False: errors.append('alpha-equality')
    if any(d['renaming'][a]==d['renaming'][b] for a,b in d['distinct_pairs']): errors.append('alpha-distinctness')
    if d['outside_after']!=[i-1 for i in d['outside_before']]: errors.append('binder-pop')
    if bool(set(d['static_choices'])&set(d['bound'])): errors.append('static-skolem')
    if d['universal_obligation'] and d['concrete_only']: errors.append('concrete-universal')
    return errors
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
base={'indices': [], 'arity': 0, 'crossed': 0, 'before': [], 'after': [], 'fresh': 1, 'caller': 0, 'solution_universes': [], 'existential_universe': 0, 'placeholders': [], 'checked': [], 'equal_pairs': [], 'renaming': {}, 'distinct_pairs': [], 'outside_before': [], 'outside_after': [], 'static_choices': [], 'bound': [], 'universal_obligation': False, 'concrete_only': False}
check('well formed empty obligations',solve(base),[])
check('bound-index-range regression 0', solve(dict(base, **({'indices':[N],'arity':N}))), ['bound-index-range'])
check('bound-index-range regression 1', solve(dict(base, **({'indices':[N+1],'arity':N+1}))), ['bound-index-range'])
check('capture-avoiding-shift regression 0', solve(dict(base, **({'before':[N],'after':[N],'crossed':1}))), ['capture-avoiding-shift'])
check('capture-avoiding-shift regression 1', solve(dict(base, **({'before':[N,N+1],'after':[N,N+1],'crossed':2}))), ['capture-avoiding-shift'])
check('fresh-universe regression 0', solve(dict(base, **({'fresh':N,'caller':N}))), ['fresh-universe'])
check('fresh-universe regression 1', solve(dict(base, **({'fresh':N+1,'caller':N+1}))), ['fresh-universe'])
check('universe-leak regression 0', solve(dict(base, **({'solution_universes':[N+1],'existential_universe':N}))), ['universe-leak'])
check('universe-leak regression 1', solve(dict(base, **({'solution_universes':[N+2],'existential_universe':N+1}))), ['universe-leak'])
check('universal-coverage regression 0', solve(dict(base, **({'placeholders':['a','b'],'checked':['a']}))), ['universal-coverage'])
check('universal-coverage regression 1', solve(dict(base, **({'placeholders':list(range(N+1)),'checked':[0]}))), ['universal-coverage'])
check('alpha-equality regression 0', solve(dict(base, **({'equal_pairs':[('a','a'),('a','b')],'renaming':{'a':0,'b':1}}))), ['alpha-equality'])
check('alpha-equality regression 1', solve(dict(base, **({'equal_pairs':[('a','a'),('a','b')],'renaming':{'a':N,'b':N+1}}))), ['alpha-equality'])
check('alpha-distinctness regression 0', solve(dict(base, **({'distinct_pairs':[('a','b'),('b','c')],'renaming':{'a':0,'b':1,'c':1}}))), ['alpha-distinctness'])
check('alpha-distinctness regression 1', solve(dict(base, **({'distinct_pairs':[('a','b'),('b','c')],'renaming':{'a':N,'b':N+1,'c':N+1}}))), ['alpha-distinctness'])
check('binder-pop regression 0', solve(dict(base, **({'outside_before':[N],'outside_after':[N]}))), ['binder-pop'])
check('binder-pop regression 1', solve(dict(base, **({'outside_before':[N+1],'outside_after':[N+1]}))), ['binder-pop'])
check('static-skolem regression 0', solve(dict(base, **({'static_choices':['a'],'bound':['a']}))), ['static-skolem'])
check('static-skolem regression 1', solve(dict(base, **({'static_choices':[N],'bound':[N]}))), ['static-skolem'])
check('concrete-universal regression 0', solve(dict(base, **({'universal_obligation':True,'concrete_only':True,'checked':['a']}))), ['concrete-universal'])
check('concrete-universal regression 1', solve(dict(base, **({'universal_obligation':True,'concrete_only':True,'checked':[N]}))), ['concrete-universal'])
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
well formed empty obligations[][]Passed
bound-index-range regression 0['bound-index-range']['bound-index-range']Passed
bound-index-range regression 1['bound-index-range']['bound-index-range']Passed
capture-avoiding-shift regression 0['capture-avoiding-shift']['capture-avoiding-shift']Passed
capture-avoiding-shift regression 1['capture-avoiding-shift']['capture-avoiding-shift']Passed
fresh-universe regression 0['fresh-universe']['fresh-universe']Passed
fresh-universe regression 1['fresh-universe']['fresh-universe']Passed
universe-leak regression 0['universe-leak']['universe-leak']Passed
universe-leak regression 1['universe-leak']['universe-leak']Passed
universal-coverage regression 0['universal-coverage']['universal-coverage']Passed
universal-coverage regression 1['universal-coverage']['universal-coverage']Passed
alpha-equality regression 0[]['alpha-equality']Failed
alpha-equality regression 1[]['alpha-equality']Failed
alpha-distinctness regression 0['alpha-distinctness']['alpha-distinctness']Passed
alpha-distinctness regression 1['alpha-distinctness']['alpha-distinctness']Passed
binder-pop regression 0['binder-pop']['binder-pop']Passed
binder-pop regression 1['binder-pop']['binder-pop']Passed
static-skolem regression 0['static-skolem']['static-skolem']Passed
static-skolem regression 1['static-skolem']['static-skolem']Passed
concrete-universal regression 0['concrete-universal']['concrete-universal']Passed
concrete-universal regression 1['concrete-universal']['concrete-universal']Passed

SHA-256 / 9e1f1bf364d9d34826cd2fb7e31e258c9b7e15c2b771a705bec89b139a14f5a7

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(d):
    errors=[]
    if any(i<0 or i>=d['arity'] for i in d['indices']): errors.append('bound-index-range')
    if d['after']!=[i+d['crossed'] for i in d['before']]: errors.append('capture-avoiding-shift')
    if d['fresh']<=d['caller']: errors.append('fresh-universe')
    if any(u>d['existential_universe'] for u in d['solution_universes']): errors.append('universe-leak')
    if not set(d['placeholders'])<=set(d['checked']): errors.append('universal-coverage')
    if any(a!=b and d['renaming'][a]!=d['renaming'][b] for a,b in d['equal_pairs'][:1]): errors.append('alpha-equality')
    if any(d['renaming'][a]==d['renaming'][b] for a,b in d['distinct_pairs']): errors.append('alpha-distinctness')
    if d['outside_after']!=[i-1 for i in d['outside_before']]: errors.append('binder-pop')
    if bool(set(d['static_choices'])&set(d['bound'])): errors.append('static-skolem')
    if d['universal_obligation'] and d['concrete_only']: errors.append('concrete-universal')
    return errors
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
base={'indices': [], 'arity': 0, 'crossed': 0, 'before': [], 'after': [], 'fresh': 1, 'caller': 0, 'solution_universes': [], 'existential_universe': 0, 'placeholders': [], 'checked': [], 'equal_pairs': [], 'renaming': {}, 'distinct_pairs': [], 'outside_before': [], 'outside_after': [], 'static_choices': [], 'bound': [], 'universal_obligation': False, 'concrete_only': False}
check('well formed empty obligations',solve(base),[])
check('bound-index-range regression 0', solve(dict(base, **({'indices':[N],'arity':N}))), ['bound-index-range'])
check('bound-index-range regression 1', solve(dict(base, **({'indices':[N+1],'arity':N+1}))), ['bound-index-range'])
check('capture-avoiding-shift regression 0', solve(dict(base, **({'before':[N],'after':[N],'crossed':1}))), ['capture-avoiding-shift'])
check('capture-avoiding-shift regression 1', solve(dict(base, **({'before':[N,N+1],'after':[N,N+1],'crossed':2}))), ['capture-avoiding-shift'])
check('fresh-universe regression 0', solve(dict(base, **({'fresh':N,'caller':N}))), ['fresh-universe'])
check('fresh-universe regression 1', solve(dict(base, **({'fresh':N+1,'caller':N+1}))), ['fresh-universe'])
check('universe-leak regression 0', solve(dict(base, **({'solution_universes':[N+1],'existential_universe':N}))), ['universe-leak'])
check('universe-leak regression 1', solve(dict(base, **({'solution_universes':[N+2],'existential_universe':N+1}))), ['universe-leak'])
check('universal-coverage regression 0', solve(dict(base, **({'placeholders':['a','b'],'checked':['a']}))), ['universal-coverage'])
check('universal-coverage regression 1', solve(dict(base, **({'placeholders':list(range(N+1)),'checked':[0]}))), ['universal-coverage'])
check('alpha-equality regression 0', solve(dict(base, **({'equal_pairs':[('a','a'),('a','b')],'renaming':{'a':0,'b':1}}))), ['alpha-equality'])
check('alpha-equality regression 1', solve(dict(base, **({'equal_pairs':[('a','a'),('a','b')],'renaming':{'a':N,'b':N+1}}))), ['alpha-equality'])
check('alpha-distinctness regression 0', solve(dict(base, **({'distinct_pairs':[('a','b'),('b','c')],'renaming':{'a':0,'b':1,'c':1}}))), ['alpha-distinctness'])
check('alpha-distinctness regression 1', solve(dict(base, **({'distinct_pairs':[('a','b'),('b','c')],'renaming':{'a':N,'b':N+1,'c':N+1}}))), ['alpha-distinctness'])
check('binder-pop regression 0', solve(dict(base, **({'outside_before':[N],'outside_after':[N]}))), ['binder-pop'])
check('binder-pop regression 1', solve(dict(base, **({'outside_before':[N+1],'outside_after':[N+1]}))), ['binder-pop'])
check('static-skolem regression 0', solve(dict(base, **({'static_choices':['a'],'bound':['a']}))), ['static-skolem'])
check('static-skolem regression 1', solve(dict(base, **({'static_choices':[N],'bound':[N]}))), ['static-skolem'])
check('concrete-universal regression 0', solve(dict(base, **({'universal_obligation':True,'concrete_only':True,'checked':['a']}))), ['concrete-universal'])
check('concrete-universal regression 1', solve(dict(base, **({'universal_obligation':True,'concrete_only':True,'checked':[N]}))), ['concrete-universal'])
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
well formed empty obligations[][]Passed
bound-index-range regression 0['bound-index-range']['bound-index-range']Passed
bound-index-range regression 1['bound-index-range']['bound-index-range']Passed
capture-avoiding-shift regression 0['capture-avoiding-shift']['capture-avoiding-shift']Passed
capture-avoiding-shift regression 1['capture-avoiding-shift']['capture-avoiding-shift']Passed
fresh-universe regression 0['fresh-universe']['fresh-universe']Passed
fresh-universe regression 1['fresh-universe']['fresh-universe']Passed
universe-leak regression 0['universe-leak']['universe-leak']Passed
universe-leak regression 1['universe-leak']['universe-leak']Passed
universal-coverage regression 0['universal-coverage']['universal-coverage']Passed
universal-coverage regression 1['universal-coverage']['universal-coverage']Passed
alpha-equality regression 0[]['alpha-equality']Failed
alpha-equality regression 1[]['alpha-equality']Failed
alpha-distinctness regression 0['alpha-distinctness']['alpha-distinctness']Passed
alpha-distinctness regression 1['alpha-distinctness']['alpha-distinctness']Passed
binder-pop regression 0['binder-pop']['binder-pop']Passed
binder-pop regression 1['binder-pop']['binder-pop']Passed
static-skolem regression 0['static-skolem']['static-skolem']Passed
static-skolem regression 1['static-skolem']['static-skolem']Passed
concrete-universal regression 0['concrete-universal']['concrete-universal']Passed
concrete-universal regression 1['concrete-universal']['concrete-universal']Passed

SHA-256 / 996ec99372d0faddd57610900b8f9403170979cb844177c593226266bab49eab

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 21 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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Verification & scope

The explicitly stated toy language is the complete scope; this is not a production compiler or a claim about Rust semantics. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:44:03.545568+00:00.

Case digest / d7e6496e32fa104ddce2e76f2f198560df98223683cc7cc7b6e5c9a9a22d0b10