Nuprl Lemma : rsum-zero-req
∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ ℝ]. Σ{f[k] | n≤k≤m} = r0 supposing ∀k:{n..m + 1-}. (f[k] = r0)
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
int-to-real: r(n),
real: ℝ,
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],
add: n + m,
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],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
all: ∀x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rsum-zero,
req_witness,
rsum_wf,
int_seg_wf,
int-to-real_wf,
all_wf,
req_wf,
real_wf,
rsum_functionality,
le_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
req_transitivity
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
independent_functionElimination,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
intEquality,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mSigma{}\{f[k] | n\mleq{}k\mleq{}m\} = r0 supposing \mforall{}k:\{n..m + 1\msupminus{}\}. (f[k] = r0)
Date html generated:
2016_10_26-AM-09_16_54
Last ObjectModification:
2016_10_10-PM-01_24_23
Theory : reals
Home
Index