Nuprl Lemma : Riemann-sums-converge-no-mc
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} . Riemann-sum(f;a;b;k + 1)↓ as k→∞
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
converges: x[n]↓ as n→∞,
rleq: x ≤ y,
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cauchy: cauchy(n.x[n]),
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
guard: {T},
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
rless: x < y,
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
real: ℝ,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
i-length: |I|
Lemmas referenced :
converges-iff-cauchy-ext,
Riemann-sum_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
istype-nat,
general-partition-sum-no-mc,
rccint_wf,
rccint-icompact,
sq_stable__rleq,
rdiv_wf,
rless-int,
nat_plus_properties,
rless_wf,
int-to-real_wf,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
small-reciprocal-real,
r-archimedean-implies2,
i-length_wf,
nat_plus_wf,
rfun_wf,
ifun_wf,
rleq_wf,
real_wf,
nat_plus_subtype_nat,
istype-le,
rabs_wf,
rsub_wf,
default-partition-choice_wf,
full-partition_wf,
uniform-partition_wf,
rleq-int,
sq_stable__less_than,
decidable__le,
rmul_preserves_rleq,
rless_transitivity1,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
rinv-mul-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul-rinv3,
rmul_preserves_rleq2,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
radd-preserves-rleq,
radd_wf,
real_term_value_add_lemma,
rleq_transitivity,
partition-mesh_wf,
mesh-uniform-partition,
full-partition-non-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality_alt,
isectElimination,
hypothesisEquality,
setElimination,
rename,
hypothesis,
dependent_set_memberEquality_alt,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
productElimination,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation_alt,
multiplyEquality,
setIsType,
inhabitedIsType,
dependent_set_memberFormation_alt,
applyEquality,
functionIsType,
closedConclusion,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
promote_hyp
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} . Riemann-sum(f;a;b;k + 1)\mdownarrow{} as k\mrightarrow{}\minfty{}
Date html generated:
2019_10_30-AM-11_38_33
Last ObjectModification:
2019_01_27-PM-03_29_13
Theory : reals_2
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