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
Home
Index