Nuprl Lemma : square-between-lemma3
∀a:ℕ. ∀b,n:ℕ+. (∃q:ℚ [((a/b) - (1/n) < q * q ∧ q * q < (a/b) + (1/n) ∧ (0 ≤ q))])
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
qadd: r + s,
rationals: ℚ,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
qsub: r - s,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
qge: a ≥ b
Lemmas referenced :
nat_wf,
divide_wf,
mul_bounds_1a,
nat_plus_subtype_nat,
le_wf,
set-value-type,
equal_wf,
istype-int,
int-value-type,
square-between-lemma2,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
qless_wf,
qsub_wf,
qdiv_wf,
subtype_rel_set,
rationals_wf,
int-subtype-rationals,
less_than_wf,
int_nzero-rational,
nat_plus_inc_int_nzero,
qmul_wf,
qadd_wf,
qle_wf,
nat_plus_wf,
div_rem_sum,
div_bounds_1,
rem_bounds_1,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
qmul_preserves_qle2,
qle-int,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
qle_witness,
squash_wf,
true_wf,
qmul-qdiv-cancel,
subtype_rel_self,
iff_weakening_equal,
qmul_preserves_qless,
qless-int,
decidable__lt,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_comm_qrng,
qadd_comm_q,
qmul-qdiv-cancel3,
qless_functionality_wrt_implies_1,
qle_weakening_eq_qorder,
qmul_functionality_wrt_qle,
qmul-mul,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
qadd-add,
qless_witness,
qmul_preserves_qle,
qmul_one_qrng,
qless_transitivity_2_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
sqequalRule,
universeIsType,
natural_numberEquality,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIsType1,
inhabitedIsType,
lambdaEquality_alt,
independent_isectElimination,
intEquality,
dependent_functionElimination,
instantiate,
cumulativity,
productElimination,
because_Cache,
independent_functionElimination,
productIsType,
remainderEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
divideEquality,
addEquality,
unionElimination,
isect_memberFormation_alt,
imageElimination,
imageMemberEquality,
universeEquality,
minusEquality
Latex:
\mforall{}a:\mBbbN{}. \mforall{}b,n:\mBbbN{}\msupplus{}. (\mexists{}q:\mBbbQ{} [((a/b) - (1/n) < q * q \mwedge{} q * q < (a/b) + (1/n) \mwedge{} (0 \mleq{} q))])
Date html generated:
2019_10_16-PM-00_38_04
Last ObjectModification:
2018_10_10-AM-11_04_39
Theory : rationals
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