Nuprl Lemma : non_neg_sum-map
∀[T:Type]. ∀[f:T ⟶ ℤ]. ∀[L:T List]. 0 ≤ Σf[x] for x ∈ L supposing (∀x∈L.0 ≤ f[x])
Proof
Definitions occuring in Statement :
sum-map: Σf[x] for x ∈ L,
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
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,
sum-map: Σf[x] for x ∈ L,
so_lambda: λ2x.t[x],
l_all: (∀x∈L.P[x]),
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
guard: {T}
Lemmas referenced :
non_neg_sum,
length_wf_nat,
int_seg_wf,
length_wf,
less_than'_wf,
sum-map_wf,
l_all_wf2,
le_wf,
l_member_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
natural_numberEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
independent_pairEquality,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setEquality,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbZ{}]. \mforall{}[L:T List]. 0 \mleq{} \mSigma{}f[x] for x \mmember{} L supposing (\mforall{}x\mmember{}L.0 \mleq{} f[x])
Date html generated:
2016_05_15-PM-06_26_06
Last ObjectModification:
2015_12_27-PM-00_02_05
Theory : general
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