Nuprl Lemma : fset-all-filter
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)]. fset-all({x ∈ s | P[x]};x.P[x])
Proof
Definitions occuring in Statement :
fset-all: fset-all(s;x.P[x]),
fset-filter: {x ∈ s | P[x]},
fset: fset(T),
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
all: ∀x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
fset-all: fset-all(s;x.P[x])
Lemmas referenced :
deq_wf,
fset_wf,
bnot_wf,
fset-null_wf,
assert_witness,
bool_subtype_base,
bool_wf,
subtype_base_sq,
assert_elim,
member-fset-filter,
assert_wf,
fset-member_wf,
isect_wf,
uall_wf,
fset-filter_wf,
fset-all_wf,
iff_weakening_uiff,
fset-all-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
productElimination,
because_Cache,
independent_isectElimination,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. fset-all(\{x \mmember{} s | P[x]\};x.P[x])
Date html generated:
2016_05_14-PM-03_41_27
Last ObjectModification:
2016_01_19-AM-10_36_02
Theory : finite!sets
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