Nuprl Lemma : member-free-vars-aux
∀[opr:Type]
∀t:term(opr). ∀v:varname(). ∀bnds:varname() List.
((v ∈ free-vars-aux(bnds;t)) ⇐⇒ (v ∈ free-vars-aux([];t)) ∧ (¬(v ∈ bnds)))
Proof
Definitions occuring in Statement :
free-vars-aux: free-vars-aux(bnds;t),
term: term(opr),
varname: varname(),
l_member: (x ∈ l),
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: 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],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bound-term: bound-term(opr),
pi2: snd(t),
guard: {T},
free-vars-aux: free-vars-aux(bnds;t),
varterm: varterm(v),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
btrue: tt,
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
mkterm: mkterm(opr;bts),
respects-equality: respects-equality(S;T),
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
term-induction,
varname_wf,
list_wf,
iff_wf,
l_member_wf,
free-vars-aux_wf,
subtype_rel_list,
not_wf,
equal-wf-T-base,
nullvar_wf,
istype-void,
nil_wf,
term_wf,
mkterm_wf,
bound-term_wf,
istype-universe,
deq_member_nil_lemma,
deq-member_wf,
var-deq_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
assert_wf,
bnot_wf,
istype-assert,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
varterm_wf,
member-free-vars-aux-not-bound,
member_singleton,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
respects-equality-set-trivial2,
respects-equality-list,
member-map,
rev-append_wf,
map_wf,
member-l-union-list,
member-rev-append
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,
productEquality,
independent_functionElimination,
lambdaFormation_alt,
independent_pairFormation,
voidElimination,
productElimination,
productIsType,
instantiate,
universeEquality,
dependent_functionElimination,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
unionElimination,
cumulativity,
inhabitedIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
equalityElimination,
promote_hyp,
dependent_pairFormation_alt,
independent_pairEquality,
spreadEquality,
hyp_replacement,
dependent_set_memberEquality_alt,
applyLambdaEquality,
inrFormation_alt,
inlFormation_alt,
unionIsType
Latex:
\mforall{}[opr:Type]
\mforall{}t:term(opr). \mforall{}v:varname(). \mforall{}bnds:varname() List.
((v \mmember{} free-vars-aux(bnds;t)) \mLeftarrow{}{}\mRightarrow{} (v \mmember{} free-vars-aux([];t)) \mwedge{} (\mneg{}(v \mmember{} bnds)))
Date html generated:
2020_05_19-PM-09_56_05
Last ObjectModification:
2020_03_09-PM-04_09_11
Theory : terms
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