Nuprl Lemma : integral-rnexp-from-0
∀[b:ℝ]. ∀[m:ℕ]. (r0_∫-b x^m dx = (b^m + 1/r(m + 1)))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
rdiv: (x/y),
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
prop: ℙ,
ifun: ifun(f;I),
all: ∀x:A. B[x],
top: Top,
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
rneq: x ≠ y,
guard: {T},
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
le: A ≤ B,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_witness,
rnexp_wf,
real_wf,
i-member_wf,
rccint_wf,
rmin_wf,
int-to-real_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rnexp_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rdiv_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
nat_wf,
rsub_wf,
integral-rnexp,
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,
itermSubtract_wf,
req-iff-rsub-is-0,
rdiv_functionality,
rsub_functionality,
rnexp0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
setEquality,
natural_numberEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
addEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
inrFormation,
applyEquality,
minusEquality
Latex:
\mforall{}[b:\mBbbR{}]. \mforall{}[m:\mBbbN{}]. (r0\_\mint{}\msupminus{}b x\^{}m dx = (b\^{}m + 1/r(m + 1)))
Date html generated:
2018_05_22-PM-02_58_33
Last ObjectModification:
2017_10_23-PM-04_15_44
Theory : reals_2
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