Nuprl Lemma : derivative-rsqrt
d(rsqrt(x))/dx = λx.(r1/r(2) * rsqrt(x)) on (r0, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rsqrt: rsqrt(x),
roiint: (l, ∞),
rdiv: (x/y),
rmul: a * b,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
top: Top,
so_apply: x[s],
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
cand: A c∧ B,
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rtermConstant: "const",
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
derivative-rexp-function,
roiint_wf,
int-to-real_wf,
rmul_wf,
rdiv_wf,
rless-int,
rless_wf,
rlog_wf,
member_roiint_lemma,
istype-void,
real_wf,
i-member_wf,
sq_stable__rless,
iproper-roiint,
req_functionality,
rmul_functionality,
req_weakening,
rdiv_functionality,
req_wf,
derivative-const-mul,
derivative-rlog,
rexp_wf,
rsqrt_wf,
rleq_weakening_rless,
rleq_wf,
derivative_functionality,
rsqrt-positive,
rmul-is-positive,
rexp_functionality,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
istype-int,
rsqrt-as-rexp,
req-rdiv,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_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,
rsqrt_squared,
rmul-rinv
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
closedConclusion,
independent_isectElimination,
inrFormation_alt,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
universeIsType,
isect_memberEquality_alt,
voidElimination,
setIsType,
setElimination,
rename,
imageElimination,
lambdaFormation_alt,
inhabitedIsType,
dependent_set_memberEquality_alt,
applyEquality,
inlFormation_alt,
productIsType,
int_eqEquality,
approximateComputation,
equalityTransitivity,
equalitySymmetry
Latex:
d(rsqrt(x))/dx = \mlambda{}x.(r1/r(2) * rsqrt(x)) on (r0, \minfty{})
Date html generated:
2019_10_31-AM-06_11_42
Last ObjectModification:
2019_04_03-AM-00_27_02
Theory : reals_2
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