Nuprl Lemma : mem_iff_count_nzero
∀s:DSet. ∀a:|s|. ∀bs:|s| List. (↑(a ∈b bs) ⇐⇒ (a #∈ bs) > 0)
Proof
Definitions occuring in Statement :
count: a #∈ as,
mem: a ∈b as,
list: T List,
assert: ↑b,
gt: i > j,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
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,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
gt: i > j,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
infix_ap: x f y,
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
b2i: b2i(b),
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt
Lemmas referenced :
list_induction,
iff_wf,
assert_wf,
mem_wf,
gt_wf,
count_wf,
list_wf,
set_car_wf,
mem_nil_lemma,
count_nil_lemma,
mem_cons_lemma,
count_cons_lemma,
dset_wf,
false_wf,
bor_wf,
set_eq_wf,
or_wf,
equal_wf,
b2i_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
bool_wf,
equal-wf-T-base,
non_neg_length,
count_bounds,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
bnot_wf,
not_wf,
add-is-int-iff,
uiff_transitivity,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
productElimination,
applyEquality,
addEquality,
addLevel,
impliesFunctionality,
orFunctionality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
inlFormation,
inrFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
equalityElimination
Latex:
\mforall{}s:DSet. \mforall{}a:|s|. \mforall{}bs:|s| List. (\muparrow{}(a \mmember{}\msubb{} bs) \mLeftarrow{}{}\mRightarrow{} (a \#\mmember{} bs) > 0)
Date html generated:
2017_10_01-AM-09_56_21
Last ObjectModification:
2017_03_03-PM-00_57_55
Theory : list_2
Home
Index