Nuprl Lemma : bag-sum-nonneg
∀[A:Type]. ∀[f:A ⟶ ℤ]. ∀[ba:bag(A)]. 0 ≤ bag-sum(ba;x.f[x]) supposing ∀x:A. (0 ≤ f[x])
Proof
Definitions occuring in Statement :
bag-sum: bag-sum(ba;x.f[x]),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
zero-le-nat,
bag-sum_wf_nat,
le_wf,
less_than'_wf,
bag-sum_wf,
all_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
applyEquality,
natural_numberEquality,
hypothesis,
because_Cache,
productElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbZ{}]. \mforall{}[ba:bag(A)]. 0 \mleq{} bag-sum(ba;x.f[x]) supposing \mforall{}x:A. (0 \mleq{} f[x])
Date html generated:
2016_05_15-PM-02_29_35
Last ObjectModification:
2015_12_27-AM-09_49_30
Theory : bags
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