Nuprl Lemma : square-between-lemma1
∀n:ℕ+. ∀k:ℕn - 1. (∃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,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
uiff: uiff(P;Q),
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T)
Lemmas referenced :
qdiv_wf,
isqrt_wf,
mul_bounds_1a,
int_seg_properties,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
le_wf,
int_seg_subtype_nat,
false_wf,
int_nzero-rational,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
nequal_wf,
isqrt-property,
nat_wf,
qmul_preserves_qle,
qmul_wf,
qless-int,
nat_properties,
decidable__lt,
qmul_preserves_qless,
qdiv-non-neg1,
qle-int,
itermAdd_wf,
int_term_value_add_lemma,
less_than_wf,
equal_wf,
qle_wf,
subtype_rel_set,
rationals_wf,
lelt_wf,
int-subtype-rationals,
subtype_rel_sets,
qless_wf,
int_seg_wf,
nat_plus_wf,
qmul-mul,
equal-wf-T-base,
int-equal-in-rationals,
not_wf,
squash_wf,
true_wf,
qmul-qdiv-cancel3,
qmul_assoc_qrng,
qmul-qdiv-cancel,
iff_weakening_equal,
qmul_ac_1_qrng,
qmul_comm_qrng,
qmul_assoc,
subtype_base_sq,
decidable__equal_int,
mul_preserves_lt,
itermMinus_wf,
int_term_value_minus_lemma,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_set_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
applyLambdaEquality,
addLevel,
impliesFunctionality,
imageElimination,
imageMemberEquality,
universeEquality,
instantiate,
cumulativity,
minusEquality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}k:\mBbbN{}n - 1. (\mexists{}q:\mBbbQ{} [(((k/n) \mleq{} (q * q)) \mwedge{} q * q < (k + 1/n) \mwedge{} (0 \mleq{} q))])
Date html generated:
2018_05_22-AM-00_29_14
Last ObjectModification:
2017_07_26-PM-06_57_32
Theory : rationals
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