Nuprl Lemma : rsqrt-as-rexp
∀[x:{x:ℝ| r0 < x} ]. (rsqrt(x) = e^(rlog(x)/r(2)))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rsqrt: rsqrt(x),
rexp: e^x,
rdiv: (x/y),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rsqrt-unique,
subtype_rel_sets,
rless_wf,
int-to-real_wf,
real_wf,
rleq_wf,
rleq_weakening_rless,
rexp-positive,
rdiv_wf,
rless-int,
rexp_wf,
req_inversion,
rsqrt_wf,
set_wf,
rmul_wf,
rlog_wf,
radd_wf,
rmul_preserves_req,
req_wf,
req_weakening,
req_functionality,
rexp-radd,
uiff_transitivity,
rmul_functionality,
req_transitivity,
radd_functionality,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-rdiv-cancel,
rmul_comm,
rexp-rlog,
rexp_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
lambdaFormation,
dependent_functionElimination,
inrFormation,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
addEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. (rsqrt(x) = e\^{}(rlog(x)/r(2)))
Date html generated:
2016_10_26-PM-00_40_17
Last ObjectModification:
2016_09_12-PM-05_44_57
Theory : reals_2
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