Nuprl Lemma : square-between-lemma2
∀n:ℕ+. ∀k:ℕ. (∃q:ℚ [(((k/n) ≤ (q * q)) ∧ q * q < (k + 1/n) ∧ (0 ≤ q))])
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__lt,
subtract_wf,
nat_wf,
nat_plus_wf,
square-between-lemma1,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
add_nat_plus,
isqrt_wf,
add_nat_wf,
divide_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-false,
div_rem_sum,
nat_plus_inc_int_nzero,
rem_bounds_1,
div_bounds_1,
isqrt-property,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
int_subtype_base,
mul_preserves_lt,
itermSubtract_wf,
itermMultiply_wf,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
set-value-type,
equal_wf,
int-value-type,
subtype_base_sq,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add-associates,
mul-associates,
zero-add,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
qmul_wf,
int-subtype-rationals,
qmul_preserves_qle,
qdiv_wf,
qless-int,
qmul_preserves_qless,
qmul_preserves_qle2,
qle-int,
qle_witness,
qle_wf,
qmul_zero_qrng,
iff_weakening_equal,
int_nzero-rational,
qless_wf,
subtype_rel_set,
qmul-mul,
set_subtype_base,
iff_weakening_uiff,
equal-wf-base,
rationals_wf,
int-equal-in-rationals,
qless_witness,
qmul_assoc_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
natural_numberEquality,
unionElimination,
universeIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
productIsType,
dependent_set_memberFormation_alt,
addEquality,
because_Cache,
imageMemberEquality,
baseClosed,
applyEquality,
productElimination,
divideEquality,
imageElimination,
equalityIsType4,
inhabitedIsType,
multiplyEquality,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
intEquality,
cutEval,
promote_hyp,
instantiate,
cumulativity,
minusEquality,
isect_memberFormation_alt
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}k:\mBbbN{}. (\mexists{}q:\mBbbQ{} [(((k/n) \mleq{} (q * q)) \mwedge{} q * q < (k + 1/n) \mwedge{} (0 \mleq{} q))])
Date html generated:
2019_10_16-PM-00_37_55
Last ObjectModification:
2018_10_10-AM-11_04_45
Theory : rationals
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