Nuprl Lemma : bag-combine-size-bound2
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[L:bag(A)]. ∀[a:A]. #(f[a]) ≤ #(⋃a∈L.f[a]) supposing a ↓∈ L
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False
Lemmas referenced :
bag_to_squash_list,
sq_stable__le,
bag-size_wf,
bag-combine_wf,
bag-member_wf,
bag-member-sq-list-member,
list-subtype-bag,
nat_wf,
bag-combine-size-bound,
le_wf,
less_than'_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
cumulativity,
applyEquality,
functionExtensionality,
hypothesis,
sqequalRule,
lambdaEquality,
independent_functionElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
rename,
dependent_functionElimination,
independent_isectElimination,
setElimination,
imageMemberEquality,
baseClosed,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
isect_memberEquality,
functionEquality,
universeEquality,
voidElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[L:bag(A)]. \mforall{}[a:A]. \#(f[a]) \mleq{} \#(\mcup{}a\mmember{}L.f[a]) supposing a \mdownarrow{}\mmember{} L
Date html generated:
2016_10_25-AM-10_28_31
Last ObjectModification:
2016_07_12-AM-06_44_21
Theory : bags
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