Nuprl Lemma : atan-size-bound_wf
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. (atan-size-bound(x) ∈ {a:{2...}| |x| ≤ (r1/r(a))} )
Proof
Definitions occuring in Statement :
atan-size-bound: atan-size-bound(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
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,
and: P ∧ Q,
rational-upper-approx: above x within 1/n,
atan-size-bound: atan-size-bound(x),
subtype_rel: A ⊆r B,
real: ℝ,
sq_type: SQType(T),
guard: {T},
rabs: |x|,
has-value: (a)↓,
int_upper: {i...},
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
rneq: x ≠ y,
true: True,
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
nat: ℕ
Lemmas referenced :
rational-upper-approx-property,
rabs_wf,
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,
absval-non-neg,
value-type-has-value,
int-value-type,
imax_wf,
divide_wfa,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
nequal_wf,
imax_ub,
istype-false,
istype-le,
rleq_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
int_upper_properties,
rless_wf,
int-rdiv_wf,
real_wf,
imax_unfold,
iff_weakening_equal,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
rleq_functionality,
req_weakening,
int-rdiv-req,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions,
div_rem_sum2,
rem_bounds_1,
decidable__le,
subtract_wf,
remainder_wfa,
itermSubtract_wf,
itermMultiply_wf,
int_term_value_subtract_lemma,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
productElimination,
callbyvalueReduce,
sqleReflexivity,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
because_Cache,
addEquality,
int_eqEquality,
independent_pairFormation,
inrFormation_alt,
equalityIstype,
baseClosed,
sqequalBase,
axiomEquality,
setIsType,
closedConclusion,
imageMemberEquality,
baseApply,
sqequalIntensionalEquality,
equalityElimination,
promote_hyp,
imageElimination,
multiplyEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(2))\} ]. (atan-size-bound(x) \mmember{} \{a:\{2...\}| |x| \mleq{} (r1/r(a))\} )
Date html generated:
2019_10_31-AM-06_08_08
Last ObjectModification:
2019_04_03-PM-04_54_46
Theory : reals_2
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