Nuprl Lemma : alpha-rename_wf
∀[opr:Type]. ∀[t:term(opr)].
∀f:{v:varname()| (v ∈ all-vars(t))} ⟶ varname()
alpha-rename(f;t) ∈ term(opr)
supposing ∀x:{v:varname()| (v ∈ all-vars(t))} . (((f x) = nullvar() ∈ varname()) ⇒ (x = nullvar() ∈ varname()))
Proof
Definitions occuring in Statement :
alpha-rename: alpha-rename(f;t),
all-vars: all-vars(t),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
l_member: (x ∈ l),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
alpha-rename: alpha-rename(f;t),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
alpha-rename-aux_wf,
nil_wf,
varname_wf,
list_ind_nil_lemma,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
nullvar_wf,
l_member_wf,
append_wf,
all-vars_wf,
term_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
Error :memTop,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
because_Cache,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
equalityIstype,
setIsType,
axiomEquality,
functionIsType,
setElimination,
rename,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
universeEquality
Latex:
\mforall{}[opr:Type]. \mforall{}[t:term(opr)].
\mforall{}f:\{v:varname()| (v \mmember{} all-vars(t))\} {}\mrightarrow{} varname()
alpha-rename(f;t) \mmember{} term(opr)
supposing \mforall{}x:\{v:varname()| (v \mmember{} all-vars(t))\} . (((f x) = nullvar()) {}\mRightarrow{} (x = nullvar()))
Date html generated:
2020_05_19-PM-09_56_42
Last ObjectModification:
2020_03_09-PM-04_09_28
Theory : terms
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