Nuprl Lemma : ratbound_wf
∀[x:ℤ × ℕ+]. (ratbound(x) ∈ {m:ℕ+| |ratreal(x)| ≤ r(m)} )
Proof
Definitions occuring in Statement :
ratbound: ratbound(x),
ratreal: ratreal(r),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ratbound: ratbound(x),
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
decidable: Dec(P),
le: A ≤ B,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
ge: i ≥ j
Lemmas referenced :
div_bounds_1,
absval_wf,
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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_subtype_base,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nequal-le-implies,
decidable__lt,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
itermAdd_wf,
int_term_value_add_lemma,
rleq_wf,
rabs_wf,
ratreal_wf,
int-to-real_wf,
nat_plus_wf,
rdiv_wf,
rless-int,
rless_wf,
rleq_functionality,
rabs_functionality,
ratreal-req,
req_weakening,
rabs-of-nonneg,
rleq-int,
decidable__le,
rneq_wf,
squash_wf,
true_wf,
real_wf,
rabs-int,
iff_weakening_equal,
absval_pos,
nat_plus_subtype_nat,
rneq_functionality,
rabs-rdiv,
subtype_rel_self,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
rdiv_functionality,
req_transitivity,
rmul_functionality,
rmul-rinv,
rmul-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nat_properties,
div_rem_sum,
nat_plus_inc_int_nzero,
istype-le,
rem_bounds_1,
int_term_value_mul_lemma,
set_subtype_base,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
productElimination,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
callbyvalueReduce,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
inhabitedIsType,
because_Cache,
remainderEquality,
setElimination,
rename,
lambdaFormation_alt,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityIstype,
applyEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
divideEquality,
equalityTransitivity,
closedConclusion,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
dependent_set_memberEquality_alt,
imageMemberEquality,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
addEquality,
independent_pairEquality,
productIsType,
inrFormation_alt,
applyLambdaEquality,
imageElimination,
universeEquality,
multiplyEquality
Latex:
\mforall{}[x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. (ratbound(x) \mmember{} \{m:\mBbbN{}\msupplus{}| |ratreal(x)| \mleq{} r(m)\} )
Date html generated:
2019_10_30-AM-09_33_22
Last ObjectModification:
2019_01_11-PM-01_23_12
Theory : reals
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