Nuprl Lemma : integral-rsum
∀[n,m:ℤ]. ∀[a,b:ℝ]. ∀[f:{f:{n..m + 1-} ⟶ [rmin(a;b), rmax(a;b)] ⟶ℝ|
∀i:{n..m + 1-}. ifun(λx.f[i;x];[rmin(a;b), rmax(a;b)])} ].
(a_∫-b Σ{f[i;x] | n≤i≤m} dx = Σ{a_∫-b f[i;x] dx | n≤i≤m})
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rsum: Σ{x[k] | n≤k≤m},
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
so_lambda: λ2x y.t[x; y],
guard: {T},
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
ge: i ≥ j ,
nat: ℕ,
iff: P ⇐⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
pointwise-req: x[k] = y[k] for k ∈ [n,m],
uimplies: b supposing a,
real-fun: real-fun(f;a;b),
top: Top,
ifun: ifun(f;I),
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
integral-zero,
rsum-empty,
trivial-int-eq1,
sq_stable__req,
integral-radd,
decidable__equal_int,
subtype_base_sq,
rsum-split-last,
radd_functionality,
radd_wf,
rsum-single,
integral_functionality,
req_functionality,
req_weakening,
rleq_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
equal-wf-base,
decidable__le,
add-zero,
nat_wf,
nat_properties,
primrec-wf2,
less_than_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
uall_wf,
all_wf,
rfun_wf,
integral_wf,
rmin-rleq-rmax,
rccint-icompact,
ifun_wf,
set_wf,
req_wf,
le_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
member_rccint_lemma,
subtype_rel_self,
rsum_functionality,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
rmax_wf,
rmin_wf,
rccint_wf,
i-member_wf,
real_wf,
int_seg_wf,
rsum_wf,
req_witness
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
cumulativity,
instantiate,
productEquality,
functionExtensionality,
equalitySymmetry,
equalityTransitivity,
productElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
unionElimination,
independent_pairFormation,
independent_isectElimination,
functionEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
setEquality,
natural_numberEquality,
because_Cache,
hypothesis,
applyEquality,
addEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
dependent_set_memberEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
isect_memberFormation,
setElimination,
rename,
thin,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} [rmin(a;b), rmax(a;b)] {}\mrightarrow{}\mBbbR{}|
\mforall{}i:\{n..m + 1\msupminus{}\}. ifun(\mlambda{}x.f[i;x];[rmin(a;b), rmax(a;b)])\} ].
(a\_\mint{}\msupminus{}b \mSigma{}\{f[i;x] | n\mleq{}i\mleq{}m\} dx = \mSigma{}\{a\_\mint{}\msupminus{}b f[i;x] dx | n\mleq{}i\mleq{}m\})
Date html generated:
2018_05_22-PM-02_58_04
Last ObjectModification:
2018_05_20-PM-11_02_06
Theory : reals_2
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