Nuprl Lemma : qsum_unroll
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. (Σa ≤ j < b. E[j] ~ if a <z b then Σa ≤ j < b - 1. E[j] + E[b - 1] else 0 fi )
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qadd: r + s,
rationals: ℚ,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
qrng: <ℚ+*>,
rng_plus: +r,
rng_zero: 0,
uall: ∀[x:A]. B[x],
member: t ∈ T,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
infix_ap: x f y
Lemmas referenced :
int_seg_wf,
rationals_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
functionEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
intEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}].
(\mSigma{}a \mleq{} j < b. E[j] \msim{} if a <z b then \mSigma{}a \mleq{} j < b - 1. E[j] + E[b - 1] else 0 fi )
Date html generated:
2016_05_15-PM-11_06_20
Last ObjectModification:
2015_12_27-PM-07_45_08
Theory : rationals
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