Nuprl Lemma : member-filter-witness_wf
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ∀[x:{x:T| ↑(P x)} ]. (member-filter-witness(P;L;x) ∈ (x ∈ L) ⇒ (x ∈ filter(P;L)))
Proof
Definitions occuring in Statement :
member-filter-witness: member-filter-witness(P;L;x),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
member-filter-witness: member-filter-witness(P;L;x),
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T),
le: A ≤ B,
less_than': less_than'(a;b),
it: ⋅
Lemmas referenced :
assert_wf,
list_wf,
bool_wf,
filter-index_wf,
nat_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,
le_wf,
less_than_wf,
length_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_subtype_nat,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
istype-false,
member-less_than,
filter_type,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :setIsType,
Error :universeIsType,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
independent_pairFormation,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
Error :productIsType,
Error :equalityIsType1,
applyLambdaEquality,
instantiate,
cumulativity,
because_Cache,
imageElimination,
Error :lambdaFormation_alt,
Error :dependent_pairEquality_alt,
closedConclusion,
setEquality,
independent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[x:\{x:T| \muparrow{}(P x)\} ].
(member-filter-witness(P;L;x) \mmember{} (x \mmember{} L) {}\mRightarrow{} (x \mmember{} filter(P;L)))
Date html generated:
2019_06_20-PM-01_25_12
Last ObjectModification:
2018_10_15-PM-02_35_51
Theory : list_1
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