Nuprl Lemma : qsum-non-neg
∀[j:ℕ]. ∀[f:ℕj ⟶ ℚ]. 0 ≤ Σ0 ≤ n < j. f[n] supposing ∀n:ℕj. (0 ≤ f[n])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
subtype_rel_set,
qmul_zero_qrng,
iff_weakening_equal,
qsum-const,
true_wf,
squash_wf,
lelt_wf,
le_wf,
less_than_wf,
int-subtype-rationals,
qsum-qle,
nat_wf,
rationals_wf,
qle_wf,
all_wf,
int_seg_wf,
qsum_wf,
qle_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_isectElimination,
lambdaFormation,
intEquality,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[j:\mBbbN{}]. \mforall{}[f:\mBbbN{}j {}\mrightarrow{} \mBbbQ{}]. 0 \mleq{} \mSigma{}0 \mleq{} n < j. f[n] supposing \mforall{}n:\mBbbN{}j. (0 \mleq{} f[n])
Date html generated:
2016_05_15-PM-11_15_41
Last ObjectModification:
2016_01_16-PM-09_18_56
Theory : rationals
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