Nuprl Lemma : rsum_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] | n≤k≤m} ∈ ℝ)
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
rsum: Σ{x[k] | n≤k≤m},
uall: ∀[x:A]. B[x],
member: t ∈ T,
has-value: (a)↓,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
value-type-has-value,
int-value-type,
valueall-type-has-valueall,
list_wf,
real_wf,
list-valueall-type,
real-valueall-type,
map_wf,
and_wf,
le_wf,
less_than_wf,
from-upto_wf,
evalall-reduce,
valueall-type-real-list,
radd-list_wf-bag,
list-subtype-bag,
subtype_rel_self,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
callbyvalueReduce,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
because_Cache,
setEquality,
addEquality,
natural_numberEquality,
lambdaEquality,
applyEquality,
productEquality,
lambdaFormation,
setElimination,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (\mSigma{}\{x[k] | n\mleq{}k\mleq{}m\} \mmember{} \mBbbR{})
Date html generated:
2016_05_18-AM-07_41_37
Last ObjectModification:
2015_12_28-AM-00_59_22
Theory : reals
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