Nuprl Lemma : l_member-iff-length-filter
∀[A:Type]. ∀eq:EqDecider(A). ∀L:A List. ∀x:A. (1 ≤ ||filter(eq x;L)|| ⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
eqof: eqof(d),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_implies: P ⇐ Q,
deq: EqDecider(T),
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T
Lemmas referenced :
less_than'_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
eta_conv,
filter_functionality,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
set_wf,
subtype_rel_self,
bool_wf,
subtype_rel_dep_function,
decidable__le,
non_neg_length,
int_subtype_base,
set_subtype_base,
nat_wf,
subtype_base_sq,
deq_wf,
list_wf,
le_wf,
iff_wf,
less_than_wf,
l_member_wf,
eqof_wf,
filter_wf5,
length_wf,
assert_of_lt_int,
deq-member-length-filter2,
assert-deq-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
hypothesis,
productElimination,
independent_pairFormation,
independent_functionElimination,
natural_numberEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
independent_isectElimination,
promote_hyp,
addLevel,
impliesFunctionality,
universeEquality,
instantiate,
cumulativity,
intEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
introduction,
imageElimination,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[A:Type]. \mforall{}eq:EqDecider(A). \mforall{}L:A List. \mforall{}x:A. (1 \mleq{} ||filter(eq x;L)|| \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2016_05_14-PM-03_21_56
Last ObjectModification:
2016_01_14-PM-11_23_56
Theory : decidable!equality
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