Nuprl Lemma : rsqrt-positive
∀x:{x:ℝ| r0 < x} . (r0 < rsqrt(x))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rless: x < y,
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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
false: False,
nat_plus: ℕ+,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
rsqrt_wf,
rleq_weakening_rless,
rleq_wf,
set_wf,
real_wf,
rless_wf,
rmul_wf,
rless_functionality,
req_weakening,
rsqrt_squared,
rmul-is-positive,
rsqrt_nonneg,
rless_transitivity2,
rless_transitivity1,
nat_plus_properties,
satisfiable-full-omega-tt,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
dependent_set_memberEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 < x\} . (r0 < rsqrt(x))
Date html generated:
2016_10_26-AM-10_09_33
Last ObjectModification:
2016_09_06-AM-11_54_07
Theory : reals
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