Nuprl Lemma : rsqrt-unique2
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[s:ℝ]. uiff((s * s) = x;¬¬((s = rsqrt(x)) ∨ (s = -(rsqrt(x)))))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
or: P ∨ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermVar: vvar,
itermMinus: "-"num,
rev_uimplies: rev_uimplies(P;Q),
stable: Stable{P}
Lemmas referenced :
not_wf,
or_wf,
req_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
real_wf,
rmul_wf,
rminus_wf,
req_witness,
set_wf,
false_wf,
rless_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rleq_weakening_rless,
rsqrt-unique,
not-rless,
rleq-implies-rleq,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf,
req-implies-req,
itermMultiply_wf,
real_term_value_mul_lemma,
req_functionality,
req_weakening,
rminus_functionality,
req_inversion,
rminus-rminus,
rmul_over_rminus,
req_transitivity,
uiff_transitivity,
rsqrt_squared,
rmul_functionality,
double-negation-hyp-elim,
stable_req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
sqequalRule,
because_Cache,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionElimination,
independent_isectElimination,
inlFormation,
inrFormation,
computeAll,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 \mleq{} x\} ]. \mforall{}[s:\mBbbR{}]. uiff((s * s) = x;\mneg{}\mneg{}((s = rsqrt(x)) \mvee{} (s = -(rsqrt(x)))))
Date html generated:
2017_10_03-AM-10_43_22
Last ObjectModification:
2017_07_28-AM-08_18_31
Theory : reals
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