Nuprl Lemma : bag-member-filter-implies1
∀[T:Type]. ∀[x:T]. ∀[bs:bag(T)]. ∀[P:{x:T| x ↓∈ bs} ⟶ 𝔹]. x ↓∈ [x∈bs|P[x]] supposing x ↓∈ bs ∧ (↑P[x])
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
bag-member: x ↓∈ bs,
squash: ↓T
Lemmas referenced :
bag_wf,
bool_wf,
assert_wf,
subtype_rel_bag,
bag-filter-wf2,
bag-subtype2,
bag-subtype,
bag-member_wf,
bag-member-filter
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
productElimination,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
lambdaEquality,
applyEquality,
setElimination,
rename,
productEquality,
cumulativity,
dependent_set_memberEquality,
functionEquality,
universeEquality,
isect_memberFormation,
introduction,
imageElimination,
imageMemberEquality,
baseClosed,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. \mforall{}[P:\{x:T| x \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} \mBbbB{}].
x \mdownarrow{}\mmember{} [x\mmember{}bs|P[x]] supposing x \mdownarrow{}\mmember{} bs \mwedge{} (\muparrow{}P[x])
Date html generated:
2016_05_15-PM-02_47_32
Last ObjectModification:
2016_01_16-AM-08_43_37
Theory : bags
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