Nuprl Lemma : member_map_filter
∀[T:Type]
∀P:T ⟶ 𝔹
∀[T':Type]
∀f:{x:T| ↑(P x)} ⟶ T'. ∀L:T List. ∀x:T'.
((x ∈ mapfilter(f;P;L)) ⇐⇒ ∃y:T. ((y ∈ L) ∧ ((↑(P y)) c∧ (x = (f y) ∈ T'))))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
mapfilter: mapfilter(f;P;L),
member: t ∈ T,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
l_member: (x ∈ l),
subtype_rel: A ⊆r B,
guard: {T},
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
filter_type,
list_wf,
assert_wf,
l_member_wf,
equal_wf,
less_than_wf,
length_wf,
subtype_rel_list,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
member_filter,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
exists_wf,
l_member_set2,
member-map,
map_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
setEquality,
independent_pairFormation,
productElimination,
dependent_pairFormation,
because_Cache,
equalityElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
equalityTransitivity,
lambdaEquality,
sqequalRule,
productEquality,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
addLevel,
levelHypothesis,
instantiate,
impliesFunctionality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}
\mforall{}[T':Type]
\mforall{}f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} T'. \mforall{}L:T List. \mforall{}x:T'.
((x \mmember{} mapfilter(f;P;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} L) \mwedge{} ((\muparrow{}(P y)) c\mwedge{} (x = (f y)))))
Date html generated:
2017_04_17-AM-07_25_40
Last ObjectModification:
2017_02_27-PM-04_04_25
Theory : list_1
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