Nuprl Lemma : arith-geom-mean-inequality-simple
∀x:{t:ℝ| r0 ≤ t} . ∀y:ℝ. ((r(2) * x * y) ≤ (x^2 + y^2))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
radd-preserves-rleq,
rmul_wf,
int-to-real_wf,
radd_wf,
rnexp_wf,
real_wf,
set_wf,
rleq_wf,
false_wf,
le_wf,
rsub_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rnexp2-nonneg,
req_functionality,
radd_functionality,
req_weakening,
rnexp2,
rleq_functionality,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
setElimination,
rename,
because_Cache,
hypothesisEquality,
productElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
dependent_functionElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x:\{t:\mBbbR{}| r0 \mleq{} t\} . \mforall{}y:\mBbbR{}. ((r(2) * x * y) \mleq{} (x\^{}2 + y\^{}2))
Date html generated:
2017_10_03-AM-10_46_13
Last ObjectModification:
2017_07_28-AM-08_19_40
Theory : reals
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