Nuprl Lemma : bag-filter-no-repeats
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. bag-no-repeats(T;[x∈bs|p[x]]) supposing bag-no-repeats(T;bs)
Proof
Definitions occuring in Statement :
bag-no-repeats: bag-no-repeats(T;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
bag-no-repeats: bag-no-repeats(T;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-filter: [x∈b|p[x]],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T)
Lemmas referenced :
bag_to_squash_list,
list_wf,
equal_wf,
bag_wf,
list-subtype-bag,
no_repeats_wf,
permutation_wf,
squash_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
istype-assert,
bool_wf,
filter_wf5,
l_member_wf,
no_repeats_filter,
no_repeats_functionality_wrt_permutation,
permutation_inversion,
respects-equality-list-bag,
respects-equality-trivial
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
productEquality,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
universeIsType,
rename,
pertypeElimination,
imageMemberEquality,
baseClosed,
productIsType,
equalityIstype,
inhabitedIsType,
setEquality,
setElimination,
setIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
dependent_pairFormation_alt,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. bag-no-repeats(T;[x\mmember{}bs|p[x]]) supposing bag-no-repeats(T;bs)
Date html generated:
2020_05_20-AM-08_01_45
Last ObjectModification:
2020_01_04-PM-11_16_57
Theory : bags
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