Nuprl Lemma : qsqrt-property
∀[r:{r:ℚ| 0 ≤ r} ]. ∀[n:ℕ+]. |(qsqrt(r;n) * qsqrt(r;n)) - r| < (1/n)
Proof
Definitions occuring in Statement :
qsqrt: qsqrt(r;n),
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qsqrt: qsqrt(r;n),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
nat_plus: ℕ+,
so_apply: x[s],
uimplies: b supposing a,
int_nzero: ℤ-o,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
approximate-qsqrt-ext,
subtype_rel_self,
rationals_wf,
qle_wf,
all_wf,
nat_plus_wf,
sq_exists_wf,
qless_wf,
qabs_wf,
qsub_wf,
qmul_wf,
qdiv_wf,
subtype_rel_set,
less_than_wf,
int-subtype-rationals,
int_nzero-rational,
subtype_rel_sets,
nequal_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
qless_witness,
qsqrt_wf,
set_wf,
sq_stable_from_decidable,
decidable__qless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
functionEquality,
setEquality,
natural_numberEquality,
because_Cache,
hypothesisEquality,
lambdaFormation,
setElimination,
rename,
lambdaEquality,
productEquality,
dependent_functionElimination,
intEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[r:\{r:\mBbbQ{}| 0 \mleq{} r\} ]. \mforall{}[n:\mBbbN{}\msupplus{}]. |(qsqrt(r;n) * qsqrt(r;n)) - r| < (1/n)
Date html generated:
2018_05_22-AM-00_30_29
Last ObjectModification:
2018_05_19-PM-04_10_06
Theory : rationals
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