Nuprl Lemma : canonical-bound_wf
∀[r:ℝ]. (canonical-bound(r) ∈ {k:{2...}| ∀n:ℕ+. (|r n| ≤ ((2 * n) * k))} )
Proof
Definitions occuring in Statement :
canonical-bound: canonical-bound(r),
real: ℝ,
absval: |i|,
int_upper: {i...},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
real: ℝ,
nat_plus: ℕ+,
guard: {T},
int_upper: {i...},
all: ∀x:A. B[x],
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],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
ge: i ≥ j ,
uiff: uiff(P;Q),
and: P ∧ Q,
canonical-bound: canonical-bound(r),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
canon-bnd: canon-bnd(x)
Lemmas referenced :
canon-bnd_wf,
div_rem_sum,
absval_wf,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rem_bounds_1,
add_nat_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
nat_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_preserves_le,
nat_plus_subtype_nat,
nat_plus_wf,
real_wf,
nat_plus_properties,
le_functionality,
le_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
applyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityIstype,
sqequalBase,
because_Cache,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productElimination,
int_eqEquality,
independent_pairFormation,
divideEquality,
multiplyEquality,
functionIsType
Latex:
\mforall{}[r:\mBbbR{}]. (canonical-bound(r) \mmember{} \{k:\{2...\}| \mforall{}n:\mBbbN{}\msupplus{}. (|r n| \mleq{} ((2 * n) * k))\} )
Date html generated:
2019_10_16-PM-03_06_42
Last ObjectModification:
2019_01_31-PM-04_38_30
Theory : reals
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