Nuprl Lemma : qsum-non-neg2
∀[i,j:ℤ]. ∀[f:{i..j-} ⟶ ℚ]. 0 ≤ Σi ≤ n < j. f[n] supposing ∀n:{i..j-}. (0 ≤ f[n])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
lelt_wf,
all_wf,
qle_witness,
int-subtype-rationals,
subtract_wf,
rationals_wf,
le_int_wf,
ifthenelse_wf,
qmul_zero_qrng,
iff_weakening_equal,
qsum_wf,
qsum-const2,
qle_wf,
le_wf,
less_than_wf,
int_seg_wf,
qsum-qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
isect_memberEquality,
functionEquality,
intEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination
Latex:
\mforall{}[i,j:\mBbbZ{}]. \mforall{}[f:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. 0 \mleq{} \mSigma{}i \mleq{} n < j. f[n] supposing \mforall{}n:\{i..j\msupminus{}\}. (0 \mleq{} f[n])
Date html generated:
2016_05_15-PM-11_15_48
Last ObjectModification:
2016_01_16-PM-09_18_49
Theory : rationals
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