Nuprl Lemma : free-vars-all-vars
∀[opr:Type]. ∀t:term(opr). ∀x:varname().  ((x ∈ free-vars(t)) 
⇒ (x ∈ all-vars(t)))
Proof
Definitions occuring in Statement : 
all-vars: all-vars(t)
, 
free-vars: free-vars(t)
, 
term: term(opr)
, 
varname: varname()
, 
l_member: (x ∈ l)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
so_apply: x[s]
, 
all-vars: all-vars(t)
, 
varterm: varterm(v)
, 
cons: [a / b]
, 
free-vars: free-vars(t)
, 
free-vars-aux: free-vars-aux(bnds;t)
, 
ifthenelse: if b then t else f fi 
, 
deq-member: x ∈b L
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
nil: []
, 
it: ⋅
, 
bfalse: ff
, 
bound-term: bound-term(opr)
, 
pi2: snd(t)
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
or: P ∨ Q
, 
pi1: fst(t)
Lemmas referenced : 
term-induction, 
varname_wf, 
l_member_wf, 
free-vars_wf, 
subtype_rel_list, 
not_wf, 
equal-wf-T-base, 
nullvar_wf, 
istype-void, 
all-vars_wf, 
term_wf, 
varterm_wf, 
mkterm_wf, 
bound-term_wf, 
list_wf, 
istype-universe, 
member-all-vars-mkterm, 
member-free-vars-mkterm
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality_alt, 
functionEquality, 
hypothesis, 
applyEquality, 
setEquality, 
baseClosed, 
independent_isectElimination, 
setElimination, 
rename, 
setIsType, 
universeIsType, 
because_Cache, 
functionIsType, 
equalityIstype, 
independent_functionElimination, 
lambdaFormation_alt, 
inhabitedIsType, 
productElimination, 
instantiate, 
universeEquality, 
dependent_functionElimination, 
dependent_pairFormation_alt, 
independent_pairFormation, 
inrFormation_alt, 
productIsType, 
unionIsType
Latex:
\mforall{}[opr:Type].  \mforall{}t:term(opr).  \mforall{}x:varname().    ((x  \mmember{}  free-vars(t))  {}\mRightarrow{}  (x  \mmember{}  all-vars(t)))
Date html generated:
2020_05_19-PM-09_56_30
Last ObjectModification:
2020_03_09-PM-04_09_23
Theory : terms
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