### 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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