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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