Nuprl Lemma : Riemann-sums-cauchy
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. cauchy(k.Riemann-sum(f;a;b;k + 1))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
cauchy: cauchy(n.x[n]),
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
cauchy: cauchy(n.x[n]),
all: ∀x:A. B[x],
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
or: P ∨ Q,
nat_plus: ℕ+,
squash: ↓T,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
implies: P ⇒ Q,
sq_stable: SqStable(P),
rsub: x - y,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
guard: {T},
rneq: x ≠ y,
i-member: r ∈ I,
rccint: [l, u],
i-approx: i-approx(I;n),
true: True,
less_than': less_than'(a;b),
less_than: a < b,
continuous: f[x] continuous for x ∈ I,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
ge: i ≥ j ,
subtract: n - m,
real: ℝ,
nat: ℕ,
rge: x ≥ y,
i-length: |I|
Lemmas referenced :
r-triangle-inequality2,
rabs-difference-symmetry,
radd_functionality_wrt_rleq,
Riemann-sum-rleq,
rsub_functionality,
rabs-Riemann-sum,
Riemann-sum-rsub,
rabs_functionality,
rleq_weakening,
rleq_transitivity,
uimplies_transitivity,
Riemann-sum-constant,
i-member-diff-bound,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
rless_transitivity2,
rmul-one-both,
rmul-rdiv-cancel,
rmul-ac,
rmul-assoc,
req_functionality,
req_wf,
equal_wf,
rmul-int,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul_comm,
itermAdd_wf,
int_term_value_add_lemma,
rmul_preserves_rleq2,
Riemann-sums-near,
less-iff-le,
minus-minus,
add-swap,
radd-int,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
req_inversion,
radd-assoc,
rminus-as-rmul,
req_transitivity,
rmul-zero-both,
rmul-rdiv-cancel2,
rabs-of-nonneg,
rless_functionality,
radd-preserves-rless,
rmul_preserves_rleq,
rleq_weakening_rless,
subtract_wf,
sq_stable__less_than,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
nat_wf,
Riemann-sum_wf,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
nat_properties,
rless-cases,
sq_stable__all,
sq_stable__rless,
sq_stable__and,
less_than_wf,
rccint-icompact,
icompact_wf,
i-approx_wf,
all_wf,
sq_exists_wf,
rabs_wf,
member_rccint_lemma,
rdiv_wf,
rless-int,
decidable__lt,
rless_wf,
sq_exists_subtype_rel,
less_than'_wf,
squash_wf,
radd-zero-both,
radd-rminus-assoc,
req_weakening,
radd_functionality,
radd_comm,
rleq_functionality,
uiff_transitivity,
sq_stable__rleq,
int-to-real_wf,
rsub_wf,
radd-preserves-rleq,
rmul-nonneg,
rleq-int,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
and_wf,
integer-bound,
rmul_wf,
radd_wf,
rminus_wf,
nat_plus_wf,
continuous_wf,
rccint_wf,
subtype_rel_self,
rfun_wf,
real_wf,
i-member_wf,
set_wf,
rleq_wf
Rules used in proof :
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
cut,
rename,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
inlFormation,
multiplyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
productElimination,
because_Cache,
introduction,
independent_functionElimination,
natural_numberEquality,
minusEquality,
equalitySymmetry,
equalityTransitivity,
inrFormation,
functionEquality,
productEquality,
dependent_set_memberEquality,
independent_pairEquality,
axiomEquality,
dependent_set_memberFormation,
addEquality,
promote_hyp,
isect_memberFormation,
equalityEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} [a, b].
cauchy(k.Riemann-sum(f;a;b;k + 1))
Date html generated:
2016_05_18-AM-10_42_44
Last ObjectModification:
2016_01_17-AM-00_32_09
Theory : reals
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