Nuprl Lemma : member-free-vars-mkterm
∀[opr:Type]
  ∀f:opr. ∀v:varname(). ∀bts:bound-term(opr) List.
    ((v ∈ free-vars(mkterm(f;bts))) 
⇐⇒ ∃bt:bound-term(opr). ((bt ∈ bts) ∧ (v ∈ free-vars(snd(bt))) ∧ (¬(v ∈ fst(bt)))))
Proof
Definitions occuring in Statement : 
free-vars: free-vars(t)
, 
bound-term: bound-term(opr)
, 
mkterm: mkterm(opr;bts)
, 
varname: varname()
, 
l_member: (x ∈ l)
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
and: P ∧ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
free-vars: free-vars(t)
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
cand: A c∧ B
, 
prop: ℙ
, 
bound-term: bound-term(opr)
, 
pi2: snd(t)
, 
subtype_rel: A ⊆r B
, 
not: ¬A
, 
pi1: fst(t)
, 
false: False
, 
rev_implies: P 
⇐ Q
, 
uimplies: b supposing a
, 
free-vars-aux: free-vars-aux(bnds;t)
, 
mkterm: mkterm(opr;bts)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
respects-equality: respects-equality(S;T)
, 
squash: ↓T
, 
true: True
, 
guard: {T}
, 
or: P ∨ Q
Lemmas referenced : 
l_member_wf, 
bound-term_wf, 
free-vars-aux_wf, 
istype-void, 
varname_wf, 
rev-append_wf, 
nil_wf, 
mkterm_wf, 
subtype_rel_list, 
not_wf, 
equal-wf-T-base, 
nullvar_wf, 
list_wf, 
istype-universe, 
member-map, 
respects-equality-list, 
respects-equality-set-trivial2, 
map_wf, 
member-l-union-list, 
var-deq_wf, 
l-union-list_wf, 
squash_wf, 
true_wf, 
subtype_rel_self, 
iff_weakening_equal, 
member-free-vars-aux, 
member-rev-append, 
null_nil_lemma, 
btrue_wf, 
member-implies-null-eq-bfalse, 
btrue_neq_bfalse
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
sqequalRule, 
cut, 
independent_pairFormation, 
sqequalHypSubstitution, 
productElimination, 
thin, 
dependent_pairFormation_alt, 
hypothesisEquality, 
promote_hyp, 
hypothesis, 
productIsType, 
universeIsType, 
introduction, 
extract_by_obid, 
isectElimination, 
because_Cache, 
applyEquality, 
functionIsType, 
independent_functionElimination, 
dependent_functionElimination, 
setEquality, 
baseClosed, 
independent_isectElimination, 
lambdaEquality_alt, 
setElimination, 
rename, 
setIsType, 
equalityIstype, 
instantiate, 
universeEquality, 
spreadEquality, 
independent_pairEquality, 
inhabitedIsType, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
inlFormation_alt, 
voidElimination, 
unionElimination
Latex:
\mforall{}[opr:Type]
    \mforall{}f:opr.  \mforall{}v:varname().  \mforall{}bts:bound-term(opr)  List.
        ((v  \mmember{}  free-vars(mkterm(f;bts)))
        \mLeftarrow{}{}\mRightarrow{}  \mexists{}bt:bound-term(opr).  ((bt  \mmember{}  bts)  \mwedge{}  (v  \mmember{}  free-vars(snd(bt)))  \mwedge{}  (\mneg{}(v  \mmember{}  fst(bt)))))
Date html generated:
2020_05_19-PM-09_56_28
Last ObjectModification:
2020_03_09-PM-04_09_22
Theory : terms
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