Nuprl Lemma : inv-sinh-domain
∀x:ℝ. ((r0 ≤ ((x * x) + r1)) ∧ (r0 < (x + rsqrt((x * x) + r1))))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
rless: x < y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
subtype_rel: A ⊆r B,
cand: A c∧ B,
uimplies: b supposing a,
nat: ℕ,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
true: True,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
square-nonneg,
radd-non-neg,
rmul_wf,
int-to-real_wf,
rleq-int,
false_wf,
rsqrt_nonneg,
radd_wf,
rleq_wf,
square-rless-implies,
rminus_wf,
rsqrt_wf,
real_wf,
req_wf,
rless-implies-rless,
rnexp_wf,
le_wf,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
req-iff-rsub-is-0,
trivial-rless-radd,
rless-int,
rsub_wf,
itermAdd_wf,
itermConstant_wf,
rless_functionality,
req_weakening,
rsqrt-rnexp-2,
req_functionality,
rnexp2,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x:\mBbbR{}. ((r0 \mleq{} ((x * x) + r1)) \mwedge{} (r0 < (x + rsqrt((x * x) + r1))))
Date html generated:
2017_10_04-PM-10_41_47
Last ObjectModification:
2017_06_24-AM-10_43_48
Theory : reals_2
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