Nuprl Lemma : Riemann-integral-rless
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} . ∀c:ℝ.
(((∀x:ℝ. ((x ∈ [a, b]) ⇒ (f[x] ≤ c))) ∧ (∃x:ℝ. ((x ∈ [a, b]) ∧ (f[x] < c))))
⇒ (∫ f[x] dx on [a, b] < (c * (b - a))))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
guard: {T},
iff: P ⇐⇒ Q,
uimplies: b supposing a,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
real-fun: real-fun(f;a;b),
top: Top,
ifun: ifun(f;I),
true: True,
less_than': less_than'(a;b),
less_than: a < b,
nat_plus: ℕ+,
rccint: [l, u],
i-approx: i-approx(I;n),
continuous: f[x] continuous for x ∈ I,
cand: A c∧ B,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rless: x < y,
rev_implies: P ⇐ Q,
or: P ∨ Q,
rneq: x ≠ y,
sq_exists: ∃x:{A| B[x]},
i-member: r ∈ I,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
label: ...$L... t,
rsub: x - y
Lemmas referenced :
rleq_weakening_rless,
rccint-icompact,
ifun_wf,
rfun_wf,
set_wf,
rless_wf,
exists_wf,
rleq_wf,
rccint_wf,
i-member_wf,
real_wf,
all_wf,
sq_stable__rless,
req_wf,
function-is-continuous,
member_rccint_lemma,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
sq_stable__req,
icompact_wf,
less_than_wf,
ravg_wf,
ravg-between,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
rsub_wf,
int-to-real_wf,
rless-implies-rless,
small-reciprocal-real,
squash_wf,
nat_plus_wf,
less_than'_wf,
sq_stable__rleq,
sq_stable__all,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
rabs_wf,
sq_stable__and,
rabs-difference-bound-rleq,
real_term_value_add_lemma,
itermAdd_wf,
rleq_weakening,
rleq_weakening_equal,
radd_functionality_wrt_rless1,
rleq_functionality_wrt_implies,
radd_wf,
req_weakening,
rless_functionality,
rleq_transitivity,
rless-cases,
rfun_subtype_3,
sq_stable__ifun,
real_term_value_mul_lemma,
itermMultiply_wf,
real_term_polynomial,
rmul_wf,
ifun_subtype_3,
Riemann-integral_wf,
iff_weakening_equal,
eta_conv,
interval_wf,
subtype_rel_sets,
Riemann-integral-additive,
equal_wf,
trivial-rsub-rless,
rless_transitivity2,
rless_transitivity1,
Riemann-integral-upper-bound,
radd-zero-both,
radd-rminus-both,
radd_functionality,
radd-ac,
radd_comm,
uiff_transitivity,
rabs-of-nonneg,
rabs-difference-symmetry,
rleq_functionality,
rminus_wf,
radd-preserves-rleq,
radd-rminus-assoc,
real_term_value_minus_lemma,
itermMinus_wf,
rmul_preserves_rless,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
radd-preserves-rless,
radd_comm_eq,
true_wf,
trivial-rless-radd
Rules used in proof :
independent_isectElimination,
dependent_set_memberEquality,
applyEquality,
because_Cache,
functionEquality,
lambdaEquality,
isectElimination,
productEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
rename,
setElimination,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_pairFormation,
intEquality,
int_eqEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
independent_pairEquality,
unionElimination,
inrFormation,
computeAll,
universeEquality,
setEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} . \mforall{}c:\mBbbR{}.
(((\mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (f[x] \mleq{} c))) \mwedge{} (\mexists{}x:\mBbbR{}. ((x \mmember{} [a, b]) \mwedge{} (f[x] < c))))
{}\mRightarrow{} (\mint{} f[x] dx on [a, b] < (c * (b - a))))
Date html generated:
2017_10_04-PM-10_15_45
Last ObjectModification:
2017_08_02-PM-00_31_27
Theory : reals_2
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