Nuprl Lemma : approximate-qsqrt
∀a:{a:ℚ| 0 ≤ a} . ∀n:ℕ+. (∃q:ℚ [((0 ≤ q) ∧ |(q * q) - a| < (1/n))])
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
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,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
true: True,
guard: {T},
nat: ℕ,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
qsub: r - s
Lemmas referenced :
sq_stable_from_decidable,
qle_wf,
int-subtype-rationals,
decidable__qle,
better-q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
equal-wf-base,
rationals_wf,
int_subtype_base,
assert-qeq,
istype-assert,
sq_exists_wf,
qless_wf,
qabs_wf,
qsub_wf,
qmul_wf,
qdiv_wf,
subtype_rel_set,
less_than_wf,
int_nzero-rational,
nat_plus_inc_int_nzero,
nat_plus_wf,
qmul_preserves_qle2,
qle-int,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
qle_witness,
squash_wf,
true_wf,
qmul_zero_qrng,
subtype_rel_self,
qmul-qdiv-cancel,
iff_weakening_equal,
square-between-lemma3,
istype-le,
decidable__lt,
istype-less_than,
qabs-as-qmax,
qmax_strict_lb,
qadd_preserves_qless,
qadd_wf,
qadd_com,
qadd_comm_q,
qadd_inv_assoc_q,
qmul_over_plus_qrng,
qinv_inv_q,
mon_assoc_q,
qadd_ac_1_q,
qinverse_q,
mon_ident_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
applyEquality,
hypothesis,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
universeIsType,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
functionEquality,
lambdaEquality_alt,
productEquality,
closedConclusion,
natural_numberEquality,
intEquality,
inhabitedIsType,
independent_isectElimination,
setIsType,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
isect_memberFormation_alt,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
productIsType,
minusEquality
Latex:
\mforall{}a:\{a:\mBbbQ{}| 0 \mleq{} a\} . \mforall{}n:\mBbbN{}\msupplus{}. (\mexists{}q:\mBbbQ{} [((0 \mleq{} q) \mwedge{} |(q * q) - a| < (1/n))])
Date html generated:
2020_05_20-AM-09_30_52
Last ObjectModification:
2019_12_31-PM-04_59_54
Theory : rationals
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