Nuprl Lemma : bag-member-iff-hd
∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T]. uiff(x ↓∈ bs;↓∃L:T List. (bs = [x / L] ∈ bag(T)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
cons: [a / b],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
bag-member: x ↓∈ bs,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
true: True,
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
rev_implies: P ⇐ Q,
cand: A c∧ B,
or: P ∨ Q
Lemmas referenced :
bag-member_wf,
squash_wf,
exists_wf,
list_wf,
equal_wf,
bag_wf,
cons_wf,
list-subtype-bag,
l_member_decomp,
append_wf,
true_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
nil_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
permutation_functionality_wrt_permutation,
cons_functionality_wrt_permutation,
permutation-rotate,
permutation_weakening,
cons_member,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
applyEquality,
because_Cache,
independent_isectElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
hyp_replacement,
natural_numberEquality,
applyLambdaEquality,
voidElimination,
voidEquality,
inlFormation,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[x:T]. uiff(x \mdownarrow{}\mmember{} bs;\mdownarrow{}\mexists{}L:T List. (bs = [x / L]))
Date html generated:
2017_10_01-AM-08_53_49
Last ObjectModification:
2017_07_26-PM-04_35_30
Theory : bags
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