Nuprl Lemma : qlog-bound
∀e:{e:ℚ| 0 < e} . ∀q:{q:ℚ| (e ≤ q) ∧ q < 1} . ∃N:ℕ+. q ↑ N < e
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
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],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
false: False,
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
qsub: r - s,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
qeq: qeq(r;s),
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
gt: i > j,
nat: ℕ
Lemmas referenced :
set_wf,
rationals_wf,
qle_wf,
qless_wf,
int-subtype-rationals,
decidable__qless,
sq_stable_from_decidable,
small-reciprocal,
squash_wf,
decidable__cand,
decidable__qle,
sq_stable__and,
qless_witness,
qsub_wf,
qdiv_wf,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
equal-wf-T-base,
qadd_preserves_qless,
qadd_wf,
true_wf,
qmul_one_qrng,
qadd_comm_q,
qadd_ac_1_q,
mon_ident_q,
qadd_assoc,
iff_weakening_equal,
qmul_preserves_qless,
qmul_wf,
qmul-qdiv-cancel,
subtype_rel_set,
less_than_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int-equal-in-rationals,
not_wf,
decidable__lt,
decidable__equal_int,
intformnot_wf,
int_formula_prop_not_lemma,
subtype_base_sq,
int_subtype_base,
equal_wf,
qdiv-self,
assert-qeq,
equal-wf-base,
qless_transitivity,
qless-int,
mul_nat_plus,
subtract_wf,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
nat_plus_wf,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-swap,
le-add-cancel,
qexp_wf,
nat_plus_subtype_nat,
mul_bounds_1b,
qexp-greater-one,
qinv-positive,
decidable__le,
intformle_wf,
itermMultiply_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
le_wf,
qadd-add,
itermAdd_wf,
int_term_value_add_lemma,
qmul-mul,
qmul_assoc_qrng,
qmul_ac_1_qrng,
qexp-non-zero,
qexp-qdiv,
qexp-one,
qexp-positive-iff,
assert_wf,
isEven_wf,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
applyEquality,
because_Cache,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination,
isect_memberEquality,
independent_isectElimination,
productElimination,
minusEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_pairFormation,
intEquality,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
impliesFunctionality,
instantiate,
cumulativity,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
isect_memberFormation,
inrFormation,
inlFormation
Latex:
\mforall{}e:\{e:\mBbbQ{}| 0 < e\} . \mforall{}q:\{q:\mBbbQ{}| (e \mleq{} q) \mwedge{} q < 1\} . \mexists{}N:\mBbbN{}\msupplus{}. q \muparrow{} N < e
Date html generated:
2018_05_22-AM-00_08_57
Last ObjectModification:
2017_07_26-PM-06_52_18
Theory : rationals
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