Nuprl Lemma : qroot
∀k:{2...}. ∀a:{a:ℚ| (0 ≤ a) ∨ (↑isOdd(k))} . ∀n:ℕ+. (∃q:ℚ [((0 ≤ a ⇐⇒ 0 ≤ q) ∧ |q ↑ k - a| < (1/n))])
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
rationals: ℚ,
isOdd: isOdd(n),
int_upper: {i...},
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
or: P ∨ Q,
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,
prop: ℙ,
or: P ∨ Q,
subtype_rel: A ⊆r B,
int_upper: {i...},
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,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat: ℕ,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
assert: ↑b,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
subtract: n - m,
less_than: a < b,
true: True,
ge: i ≥ j ,
exp: i^n,
primrec: primrec(n;b;c),
rev_uimplies: rev_uimplies(P;Q),
isOdd: isOdd(n),
qsub: r - s,
qmul: r * s,
qabs: |r|,
has-value: (a)↓,
has-valueall: has-valueall(a)
Lemmas referenced :
sq_stable_from_decidable,
qle_wf,
assert_wf,
isOdd_wf,
decidable__or,
decidable__qle,
decidable__assert,
better-q-elim,
nat_plus_properties,
iff_weakening_uiff,
qeq_wf2,
equal-wf-base,
rationals_wf,
int_subtype_base,
assert-qeq,
int-subtype-rationals,
istype-assert,
qdiv_wf,
or_wf,
sq_exists_wf,
iff_wf,
qless_wf,
qabs_wf,
qsub_wf,
qexp_wf,
upper_subtype_nat,
istype-false,
subtype_rel_set,
less_than_wf,
int_nzero-rational,
nat_plus_inc_int_nzero,
nat_plus_wf,
istype-int_upper,
bool_wf,
eq_int_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_eq_int,
bfalse_wf,
set-value-type,
equal_wf,
union-value-type,
unit_wf2,
ifthenelse_wf,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
mul_nat_plus,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int-value-type,
exp-fastexp,
subtract_wf,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
istype-le,
exp_wf_nat_plus,
itermMultiply_wf,
int_term_value_mul_lemma,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
set_subtype_base,
neg_assert_of_eq_int,
btrue_neq_bfalse,
intformeq_wf,
int_formula_prop_eq_lemma,
mul_preserves_le,
nat_plus_subtype_nat,
le_wf,
add-associates,
add-swap,
add-commutes,
squash_wf,
true_wf,
nat_wf,
nat_properties,
ge_wf,
le_witness_for_triv,
exp_wf2,
subtract-1-ge-0,
exp_step,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_weakening2,
subtract-add-cancel,
add-subtract-cancel,
exp_preserves_le,
itermAdd_wf,
int_term_value_add_lemma,
minus-minus,
iroot-lemma2,
absval_wf,
lt_int_wf,
assert_of_lt_int,
qdiv-non-neg1,
qless-int,
qle-int,
qmul_preserves_qle2,
qle_witness,
qmul_wf,
qmul_zero_qrng,
subtype_rel_self,
qmul-qdiv-cancel,
iff_weakening_equal,
bnot_wf,
not_wf,
iff_transitivity,
assert_of_bnot,
decidable__equal_int,
itermMinus_wf,
int_term_value_minus_lemma,
zero-mul,
istype-universe,
exp-zero,
multiply-is-int-iff,
add-is-int-iff,
mul_bounds_1a,
exp_wf4,
false_wf,
absval_unfold,
istype-top,
mul_preserves_lt,
uiff_transitivity,
exp_wf3,
qmul-preserves-eq,
qmul-mul,
qmul_ac_1_qrng,
qmul_comm_qrng,
int-equal-in-rationals,
assert_of_band,
not_assert_elim,
qmul_preserves_qless,
equal-wf-T-base,
qmul_one_qrng,
qmul-qdiv-cancel3,
int_upper_wf,
exp-of-mul,
le_int_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
qexp-non-zero,
modulus_wf_int_mod,
int-subtype-int_mod,
int_mod_wf,
qexp-qdiv,
qexp-exp,
exp-minusone,
qmul_preserves_qle,
qadd_wf,
qmul_over_plus_qrng,
qadd-add,
qabs-qminus,
qinv_inv_q,
qadd_comm_q,
qmul_over_minus_qrng,
qmul_assoc,
qadd_preserves_qless,
qless_transitivity_1_qorder,
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
qpositive_wf,
assert-qpositive,
qadd_inv_assoc_q,
mon_ident_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
unionEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
hypothesisEquality,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
unionIsType,
universeIsType,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
functionEquality,
lambdaEquality_alt,
productEquality,
independent_pairFormation,
intEquality,
inhabitedIsType,
setIsType,
unionElimination,
instantiate,
cumulativity,
equalityTransitivity,
cutEval,
dependent_set_memberEquality_alt,
equalityIstype,
approximateComputation,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
int_eqEquality,
multiplyEquality,
equalityElimination,
closedConclusion,
promote_hyp,
equalityIsType4,
baseApply,
equalityIsType2,
equalityIsType1,
addEquality,
minusEquality,
intWeakElimination,
functionIsTypeImplies,
dependent_set_memberFormation_alt,
productIsType,
functionIsType,
isect_memberFormation_alt,
universeEquality,
pointwiseFunctionality,
lessCases,
axiomSqEquality,
sqequalBase,
callbyvalueReduce
Latex:
\mforall{}k:\{2...\}. \mforall{}a:\{a:\mBbbQ{}| (0 \mleq{} a) \mvee{} (\muparrow{}isOdd(k))\} . \mforall{}n:\mBbbN{}\msupplus{}.
(\mexists{}q:\mBbbQ{} [((0 \mleq{} a \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} q) \mwedge{} |q \muparrow{} k - a| < (1/n))])
Date html generated:
2019_10_16-PM-00_37_36
Last ObjectModification:
2019_06_25-PM-00_20_44
Theory : rationals
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