Nuprl Lemma : ftc-example1
∀a,b:ℝ. (a_∫-b t^2 dt = ((b^3/r(3)) - (a^3/r(3))))
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,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtract: n - m,
sq_type: SQType(T),
nat_plus: ℕ+
Lemmas referenced :
real_wf,
rnexp_wf,
istype-false,
istype-le,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
req_functionality,
rnexp_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
riiint_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermSubtract_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
istype-less_than,
ftc-total-integral,
derivative-rdiv-const-alt,
derivative-rnexp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_set_memberEquality_alt,
sqequalRule,
lambdaEquality_alt,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
independent_pairFormation,
setElimination,
rename,
setIsType,
because_Cache,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
inrFormation_alt,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt
Latex:
\mforall{}a,b:\mBbbR{}. (a\_\mint{}\msupminus{}b t\^{}2 dt = ((b\^{}3/r(3)) - (a\^{}3/r(3))))
Date html generated:
2019_10_31-AM-06_17_19
Last ObjectModification:
2018_11_08-PM-05_57_17
Theory : reals_2
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