Nuprl Lemma : alt-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-alt(f;a;b;k + 1))
Proof
Definitions occuring in Statement :
Riemann-sum-alt: Riemann-sum-alt(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: $n
Definitions unfolded in proof :
rfun: I ⟶ℝ,
label: ...$L... t,
so_apply: x[s],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
guard: {T},
rneq: x ≠ y,
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
sq_exists: ∃x:{A| B[x]},
cauchy: cauchy(n.x[n]),
member: t ∈ T,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
rccint: [l, u],
i-member: r ∈ I
Lemmas referenced :
and_wf,
continuous-implies-functional,
member_rccint_lemma,
req_inversion,
req_weakening,
Riemann-sum-alt-req,
rsub_functionality,
rabs_functionality,
rleq_functionality,
Riemann-sums-cauchy,
le_wf,
nat_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
Riemann-sum-alt_wf,
decidable__lt,
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,
less_than_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
continuous_wf,
rccint_wf,
subtype_rel_self,
rfun_wf,
real_wf,
i-member_wf,
Riemann-sum_wf,
req_wf
Rules used in proof :
setEquality,
computeAll,
int_eqEquality,
dependent_pairFormation,
inrFormation,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
applyEquality,
independent_isectElimination,
voidElimination,
independent_pairFormation,
unionElimination,
productElimination,
natural_numberEquality,
addEquality,
functionEquality,
because_Cache,
lambdaEquality,
sqequalRule,
isectElimination,
independent_functionElimination,
dependent_set_memberEquality,
introduction,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lemma_by_obid,
cut,
addLevel,
levelHypothesis,
promote_hyp,
andLevelFunctionality
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-alt(f;a;b;k + 1))
Date html generated:
2016_05_18-AM-10_45_55
Last ObjectModification:
2016_01_17-AM-00_22_33
Theory : reals
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