Nuprl Lemma : bag-member-filter-implies2
∀[T:Type]. ∀[x:T]. ∀[bs:bag(T)]. ∀[P:{x:T| x ↓∈ bs} ⟶ 𝔹]. x ↓∈ bs ∧ (↑P[x]) supposing x ↓∈ [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,
guard: {T},
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
quotient: x,y:A//B[x; y],
bag: bag(T),
exists: ∃x:A. B[x],
squash: ↓T,
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
sq_stable: SqStable(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
respects-equality: respects-equality(S;T)
Lemmas referenced :
bag-member_wf,
bool_wf,
bag_wf,
istype-universe,
list_wf,
assert_wf,
subtype_rel_bag,
list-subtype-bag,
bag-filter-wf2,
permutation_wf,
l_member-bag-member,
filter_wf5,
bag-member-evidence,
list-member-bag-member,
l_member_wf,
member_filter_2,
member-permutation,
assert_witness,
sq_stable__uall,
istype-assert,
sq_stable__and,
sq_stable__bag-member,
sq_stable_from_decidable,
decidable__assert,
quotient-member-eq,
permutation-equiv,
iff_weakening_equal,
subtype-respects-equality,
subtype_rel_sets_simple,
squash_wf,
true_wf,
subtype_rel_self,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionIsType,
setIsType,
because_Cache,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
universeEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
dependent_set_memberEquality_alt,
rename,
setElimination,
inhabitedIsType,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
sqequalRule,
lambdaFormation_alt,
independent_pairFormation,
baseClosed,
imageMemberEquality,
equalityIsType1,
productIsType,
independent_functionElimination,
pertypeElimination,
productElimination,
imageElimination,
productEquality,
isect_memberEquality_alt,
independent_pairEquality,
functionIsTypeImplies,
pointwiseFunctionalityForEquality,
promote_hyp,
equalityIstype,
sqequalBase,
functionEquality,
natural_numberEquality,
isectEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. \mforall{}[P:\{x:T| x \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} \mBbbB{}].
x \mdownarrow{}\mmember{} bs \mwedge{} (\muparrow{}P[x]) supposing x \mdownarrow{}\mmember{} [x\mmember{}bs|P[x]]
Date html generated:
2019_10_15-AM-11_02_35
Last ObjectModification:
2018_11_27-AM-00_30_22
Theory : bags
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