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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