Nuprl Lemma : canon-bnd_wf
∀[x:ℝ]. (canon-bnd(x) ∈ {k:{3...}| ∀n:ℕ+. (|x n| ≤ (n * k))} )
Proof
Definitions occuring in Statement :
canon-bnd: canon-bnd(x),
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 :
and: P ∧ Q,
ge: i ≥ j ,
guard: {T},
nat: ℕ,
subtype_rel: A ⊆r B,
false: False,
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat_plus: ℕ+,
int_upper: {i...},
canon-bnd: canon-bnd(x),
real: ℝ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
regular-int-seq: k-regular-seq(f),
subtract: n - m,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True
Lemmas referenced :
real_wf,
nat_plus_wf,
istype-le,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermVar_wf,
intformle_wf,
intformand_wf,
nat_properties,
decidable__le,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
absval_wf,
nat_plus_properties,
add-zero,
zero-mul,
add-mul-special,
add-swap,
add-commutes,
one-mul,
mul-commutes,
minus-one-mul,
mul-distributes,
subtract_wf,
int-triangle-inequality,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
int_term_value_mul_lemma,
itermMultiply_wf,
iff_weakening_equal,
subtype_rel_self,
absval_mul,
add_functionality_wrt_eq,
true_wf,
squash_wf,
add_functionality_wrt_le,
le_weakening,
le_functionality,
nat_plus_subtype_nat,
absval_pos,
absval-non-neg,
le_wf,
set_subtype_base,
nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf
Rules used in proof :
axiomEquality,
multiplyEquality,
functionIsType,
lambdaFormation_alt,
independent_pairFormation,
int_eqEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
universeIsType,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
applyEquality,
isectElimination,
extract_by_obid,
addEquality,
dependent_set_memberEquality_alt,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inhabitedIsType,
minusEquality,
intEquality,
cumulativity,
instantiate,
productElimination,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[x:\mBbbR{}]. (canon-bnd(x) \mmember{} \{k:\{3...\}| \mforall{}n:\mBbbN{}\msupplus{}. (|x n| \mleq{} (n * k))\} )
Date html generated:
2019_10_16-PM-03_06_37
Last ObjectModification:
2019_10_10-PM-03_16_17
Theory : reals
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