Nuprl Lemma : qlog-lemma
∀e:{e:ℚ| 0 < e} . ∀q:{q:ℚ| (e ≤ q) ∧ q < 1} . {nr:ℕ × ℚ| let n,r = nr in (r = q ↑ n ∈ ℚ) ∧ (e ≤ r) ∧ r * r < e}
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
sq_type: SQType(T),
less_than': less_than'(a;b),
subtract: n - m,
sq_stable: SqStable(P),
pi1: fst(t)
Lemmas referenced :
rationals_wf,
qle_wf,
qless_wf,
int-subtype-rationals,
uniform-comp-nat-induction,
nat_plus_wf,
equal_wf,
qexp_wf,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
qmul_wf,
set_wf,
nat_wf,
istype-nat,
set-value-type,
int-value-type,
rationals-value-type,
decidable__qless,
nat_plus_subtype_nat,
int_seg_wf,
int_seg_properties,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
subtract_wf,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-less_than,
qless_complement_qorder,
qless_transitivity_2_qorder,
qmul_preserves_qle,
trivial-int-eq1,
qexp-add,
qexp-positive,
qle-minus,
qmul_com,
qmul_ac_1_qrng,
qminus-minus,
qexp-one,
le_wf,
istype-void,
qle_weakening_lt_qorder,
qexp_preserves_qle,
qle_weakening_eq_qorder,
qless_irreflexivity,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
subtype_base_sq,
int_subtype_base,
mul-commutes,
istype-universe,
qexp-mul,
exp_unroll_q,
add-commutes,
qexp1,
sq_stable_from_decidable,
not_wf,
qmul_assoc,
subtract-add-cancel,
decidable__equal_int,
qlog-bound,
pi1_wf_top,
subtract_nat_wf,
subtract-is-int-iff,
false_wf,
decidable__qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
productIsType,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
functionEquality,
setEquality,
productEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
because_Cache,
spreadEquality,
inhabitedIsType,
isect_memberFormation_alt,
intEquality,
multiplyEquality,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
isectIsType,
functionIsType,
productElimination,
imageElimination,
independent_pairEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
hyp_replacement,
applyLambdaEquality,
minusEquality,
isect_memberEquality_alt,
promote_hyp,
cumulativity,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}e:\{e:\mBbbQ{}| 0 < e\} . \mforall{}q:\{q:\mBbbQ{}| (e \mleq{} q) \mwedge{} q < 1\} .
\{nr:\mBbbN{} \mtimes{} \mBbbQ{}| let n,r = nr in (r = q \muparrow{} n) \mwedge{} (e \mleq{} r) \mwedge{} r * r < e\}
Date html generated:
2020_05_20-AM-09_26_50
Last ObjectModification:
2020_01_01-AM-11_45_02
Theory : rationals
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