Nuprl Lemma : bag-max-ub
∀[A:Type]. ∀[f:A ⟶ ℤ]. ∀[bs:bag(A)]. ∀[x:A]. (f x) ≤ bag-max(f;bs) supposing x ↓∈ bs
Proof
Definitions occuring in Statement :
bag-max: bag-max(f;bs),
bag-member: x ↓∈ bs,
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
bag-max: bag-max(f;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
le: A ≤ B,
not: ¬A,
false: False,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
equal_wf,
and_wf,
bag-member-map,
bag_wf,
bag-max_wf,
less_than'_wf,
bag-member_wf,
bag-size-bag-member,
bag-map_wf,
imax-bag-ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
hypothesis,
applyEquality,
independent_functionElimination,
because_Cache,
productElimination,
dependent_pairFormation,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
isect_memberFormation,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
voidElimination,
universeEquality,
independent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbZ{}]. \mforall{}[bs:bag(A)]. \mforall{}[x:A]. (f x) \mleq{} bag-max(f;bs) supposing x \mdownarrow{}\mmember{} bs
Date html generated:
2016_05_15-PM-02_51_13
Last ObjectModification:
2016_01_16-AM-08_41_53
Theory : bags
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