Nuprl Lemma : mem_iff_mem_f
∀s:DSet. ∀a:|s|. ∀bs:|s| List. (↑(a ∈b bs) ⇐⇒ mem_f(|s|;a;bs))
Proof
Definitions occuring in Statement :
mem: a ∈b as,
mem_f: mem_f(T;a;bs),
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
dset: DSet,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
mem_f: mem_f(T;a;bs),
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
prop: ℙ,
or: P ∨ Q,
infix_ap: x f y
Lemmas referenced :
list_induction,
iff_wf,
assert_wf,
mem_wf,
mem_f_wf,
set_car_wf,
mem_nil_lemma,
istype-void,
list_ind_nil_lemma,
mem_cons_lemma,
list_ind_cons_lemma,
list_wf,
dset_wf,
bor_wf,
set_eq_wf,
or_wf,
equal_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
universeIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productIsType,
functionIsType,
inhabitedIsType,
productElimination,
unionIsType,
equalityIsType1,
unionElimination,
inlFormation_alt,
inrFormation_alt,
promote_hyp,
applyEquality
Latex:
\mforall{}s:DSet. \mforall{}a:|s|. \mforall{}bs:|s| List. (\muparrow{}(a \mmember{}\msubb{} bs) \mLeftarrow{}{}\mRightarrow{} mem\_f(|s|;a;bs))
Date html generated:
2019_10_16-PM-01_03_24
Last ObjectModification:
2018_10_08-AM-11_21_53
Theory : list_2
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