Nuprl Lemma : arith-geom-mean-inequality
∀x,y:{t:ℝ| r0 ≤ t} . ((rsqrt(x * y) ≤ (x + y/r(2))) ∧ (x ≠ y ⇒ (rsqrt(x * y) < (x + y/r(2)))))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rdiv: (x/y),
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
rsub: x - y,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
le: A ≤ B,
false: False,
not: ¬A,
rge: x ≥ y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+
Lemmas referenced :
set_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rsub_wf,
radd_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rminus_functionality,
rmul_comm,
rminus-rminus,
req_inversion,
radd-assoc,
radd_comm,
radd-ac,
rminus-as-rmul,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-identity1,
rmul-assoc,
square-nonneg,
radd-preserves-rless,
rless_wf,
square-nonzero,
rless_transitivity2,
rless_transitivity1,
rneq_wf,
rless_functionality,
radd-zero-both,
radd-rminus-both,
rleq_functionality,
rmul_preserves_rleq,
rdiv_wf,
rless-int,
rmul_preserves_rless,
rsqrt_wf,
rmul-nonneg-case1,
sq_stable__rleq,
rmul-rdiv-cancel2,
radd-preserves-rleq,
rmul-zero-both,
rleq-int,
false_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rsqrt_functionality_wrt_rleq,
radd-non-neg,
rsqrt-of-square,
rsqrt-rmul,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
rsqrt_functionality,
rmul-int,
rsqrt_functionality_wrt_rless,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
minusEquality,
addEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
unionElimination,
inlFormation,
dependent_functionElimination,
inrFormation,
addLevel,
levelHypothesis,
promote_hyp,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
productEquality,
instantiate,
cumulativity,
intEquality,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality
Latex:
\mforall{}x,y:\{t:\mBbbR{}| r0 \mleq{} t\} . ((rsqrt(x * y) \mleq{} (x + y/r(2))) \mwedge{} (x \mneq{} y {}\mRightarrow{} (rsqrt(x * y) < (x + y/r(2)))))
Date html generated:
2016_10_26-AM-10_14_06
Last ObjectModification:
2016_09_07-AM-00_15_59
Theory : reals
Home
Index