Nuprl Lemma : mem_iff_exists
∀s:DSet. ∀a:|s|. ∀bs:|s| List. (↑(a ∈b bs) ⇐⇒ ∃n:ℕ||bs||. (bs[n] = a ∈ |s|))
Proof
Definitions occuring in Statement :
mem: a ∈b as,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
mem: a ∈b as,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
dset: DSet,
so_apply: x[s],
subtype_rel: A ⊆r B,
bool: 𝔹,
grp_car: |g|,
pi1: fst(t),
bor_mon: <𝔹,∨b>,
guard: {T},
uimplies: b supposing a,
infix_ap: x f y,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
bexists_iff_exists_nth,
infix_ap_wf,
set_car_wf,
bool_wf,
set_eq_wf,
assert_wf,
mon_for_wf,
bor_mon_wf,
grp_car_wf,
subtype_rel_self,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
subtype_rel_transitivity,
abmonoid_wf,
mon_wf,
grp_sig_wf,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
dset_wf,
iff_weakening_uiff,
equal_wf,
assert_of_dset_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_functionElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
inhabitedIsType,
universeIsType,
applyEquality,
functionEquality,
instantiate,
independent_isectElimination,
independent_pairFormation,
promote_hyp,
productIsType,
natural_numberEquality,
equalityIsType1,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
imageElimination
Latex:
\mforall{}s:DSet. \mforall{}a:|s|. \mforall{}bs:|s| List. (\muparrow{}(a \mmember{}\msubb{} bs) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}||bs||. (bs[n] = a))
Date html generated:
2019_10_16-PM-01_03_35
Last ObjectModification:
2018_10_08-AM-11_21_01
Theory : list_2
Home
Index