Nuprl Lemma : integral-rnexp
∀[a,b:ℝ]. ∀[m:ℕ]. (a_∫-b x^m dx = (b^m + 1 - a^m + 1/r(m + 1)))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
rdiv: (x/y),
rnexp: x^k1,
rsub: x - y,
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,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermSubtract: left "-" right,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
nat_plus: ℕ+,
rtermMultiply: left "*" right,
rtermConstant: "const",
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
ftc-total-integral,
rnexp_wf,
rdiv_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
req_functionality,
rnexp_functionality,
req_weakening,
req_wf,
req_witness,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rsub_wf,
istype-nat,
real_wf,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermDivide_wf,
rtermVar_wf,
riiint_wf,
rmul_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-less_than,
derivative-const-mul,
derivative-rnexp,
rtermMultiply_wf,
rtermConstant_wf,
derivative_functionality,
add-subtract-cancel,
rinv_wf2,
itermMultiply_wf,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality_alt,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
closedConclusion,
inrFormation_alt,
productElimination,
lambdaFormation_alt,
setIsType,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[m:\mBbbN{}]. (a\_\mint{}\msupminus{}b x\^{}m dx = (b\^{}m + 1 - a\^{}m + 1/r(m + 1)))
Date html generated:
2019_10_30-AM-11_39_48
Last ObjectModification:
2019_04_03-AM-00_21_55
Theory : reals_2
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