Nuprl Lemma : alpha-rename-alist-property
∀[opr:Type]
  ∀t:term(opr). ∀L:varname() List.
    ((∀x,x':varname().  ((<x, x'> ∈ alpha-rename-alist(t;L)) 
⇒ ((x ∈ L) ∧ (¬(x' ∈ L @ all-vars(t))))))
    ∧ (∀x,x',y,y':varname().
         ((<x, x'> ∈ alpha-rename-alist(t;L))
         
⇒ (<y, y'> ∈ alpha-rename-alist(t;L))
         
⇒ (x' = y' ∈ varname())
         
⇒ (x = y ∈ varname()))))
Proof
Definitions occuring in Statement : 
alpha-rename-alist: alpha-rename-alist(t;L)
, 
all-vars: all-vars(t)
, 
term: term(opr)
, 
varname: varname()
, 
l_member: (x ∈ l)
, 
append: as @ bs
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
pair: <a, b>
, 
product: x:A × B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
alpha-rename-alist: alpha-rename-alist(t;L)
, 
member: t ∈ T
, 
and: P ∧ Q
, 
so_lambda: λ2x y.t[x; y]
, 
has-value: (a)↓
, 
uimplies: b supposing a
, 
varname: varname()
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
so_apply: x[s1;s2]
, 
guard: {T}
, 
prop: ℙ
, 
implies: P 
⇒ Q
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
cand: A c∧ B
, 
not: ¬A
, 
false: False
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
squash: ↓T
, 
true: True
, 
subtype_rel: A ⊆r B
, 
rev_implies: P 
⇐ Q
, 
l_member: (x ∈ l)
, 
exists: ∃x:A. B[x]
, 
nat: ℕ
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
select: L[n]
, 
cons: [a / b]
, 
nat_plus: ℕ+
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
uiff: uiff(P;Q)
, 
ge: i ≥ j 
, 
l_contains: A ⊆ B
Lemmas referenced : 
list_wf, 
varname_wf, 
term_wf, 
istype-universe, 
list_accum_invariant2, 
value-type-has-value, 
bunion-value-type, 
nat_wf, 
atom-value-type, 
product-value-type, 
istype-atom, 
maybe_new_var_wf, 
cons_wf, 
l_contains_wf, 
append_wf, 
all-vars_wf, 
pi1_wf, 
l_member_wf, 
pi2_wf, 
not_wf, 
equal_wf, 
nil_wf, 
l_contains_weakening, 
null_nil_lemma, 
btrue_wf, 
member-implies-null-eq-bfalse, 
btrue_neq_bfalse, 
cons-l_contains, 
cons_member, 
squash_wf, 
true_wf, 
subtype_rel_self, 
iff_weakening_equal, 
istype-void, 
istype-le, 
length_of_cons_lemma, 
add_nat_plus, 
length_wf_nat, 
decidable__lt, 
full-omega-unsat, 
intformnot_wf, 
intformless_wf, 
itermConstant_wf, 
istype-int, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_wf, 
istype-less_than, 
nat_plus_properties, 
add-is-int-iff, 
intformand_wf, 
itermVar_wf, 
itermAdd_wf, 
intformeq_wf, 
int_formula_prop_and_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_formula_prop_eq_lemma, 
false_wf, 
length_wf, 
select_wf, 
nat_properties, 
decidable__le, 
intformle_wf, 
int_formula_prop_le_lemma, 
maybe_new_var-distinct, 
l_all_iff
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
universeIsType, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesis, 
hypothesisEquality, 
instantiate, 
universeEquality, 
productElimination, 
productEquality, 
dependent_functionElimination, 
sqequalRule, 
lambdaEquality_alt, 
callbyvalueReduce, 
independent_isectElimination, 
atomEquality, 
because_Cache, 
independent_pairEquality, 
closedConclusion, 
productIsType, 
inhabitedIsType, 
functionEquality, 
independent_functionElimination, 
independent_pairFormation, 
equalityTransitivity, 
equalitySymmetry, 
voidElimination, 
equalityIstype, 
unionElimination, 
applyLambdaEquality, 
applyEquality, 
imageElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
dependent_pairFormation_alt, 
dependent_set_memberEquality_alt, 
Error :memTop, 
approximateComputation, 
setElimination, 
rename, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
int_eqEquality, 
inrFormation_alt, 
setIsType
Latex:
\mforall{}[opr:Type]
    \mforall{}t:term(opr).  \mforall{}L:varname()  List.
        ((\mforall{}x,x':varname().
                ((<x,  x'>  \mmember{}  alpha-rename-alist(t;L))  {}\mRightarrow{}  ((x  \mmember{}  L)  \mwedge{}  (\mneg{}(x'  \mmember{}  L  @  all-vars(t))))))
        \mwedge{}  (\mforall{}x,x',y,y':varname().
                  ((<x,  x'>  \mmember{}  alpha-rename-alist(t;L))
                  {}\mRightarrow{}  (<y,  y'>  \mmember{}  alpha-rename-alist(t;L))
                  {}\mRightarrow{}  (x'  =  y')
                  {}\mRightarrow{}  (x  =  y))))
Date html generated:
2020_05_19-PM-09_57_13
Last ObjectModification:
2020_03_11-PM-07_41_19
Theory : terms
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