Nuprl Lemma : canonical-bound-property
∀x:ℝ. ((r(-canonical-bound(x)) ≤ x) ∧ (x ≤ r(canonical-bound(x))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
canonical-bound: canonical-bound(r),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
minus: -n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
int-to-real: r(n),
rleq: x ≤ y,
rsub: x - y,
rnonneg: rnonneg(x),
rminus: -(x),
radd: a + b,
accelerate: accelerate(k;f),
and: P ∧ Q,
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
squash: ↓T,
prop: ℙ,
real: ℝ,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
nat: ℕ,
absval: |i|,
sq_stable: SqStable(P),
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
uiff: uiff(P;Q),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
canonical-bound_wf,
value-type-has-value,
int-value-type,
le_wf,
squash_wf,
true_wf,
istype-int,
reg-seq-list-add-as-l_sum,
cons_wf,
nat_plus_wf,
nil_wf,
nat_plus_properties,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
subtype_base_sq,
int_subtype_base,
subtype_rel_self,
iff_weakening_equal,
map_cons_lemma,
map_nil_lemma,
l_sum_cons_lemma,
l_sum_nil_lemma,
mul_cancel_in_le,
div_rem_sum2,
nequal_wf,
rem_bounds_absval,
absval_wf,
real_wf,
sq_stable__le,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
less_than_wf,
absval_pos,
istype-le,
absval_ifthenelse,
lt_int_wf,
sq_stable__all,
le_witness_for_triv,
itermAdd_wf,
int_term_value_add_lemma,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
false_wf,
add-associates,
mul-associates,
minus-one-mul,
mul-commutes,
mul-swap,
zero-add,
add-commutes,
add-swap,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
callbyvalueReduce,
sqleReflexivity,
independent_pairFormation,
intEquality,
independent_isectElimination,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
minusEquality,
divideEquality,
functionEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
instantiate,
cumulativity,
equalityIstype,
baseClosed,
sqequalBase,
imageMemberEquality,
universeEquality,
productElimination,
because_Cache,
addEquality,
closedConclusion,
remainderEquality,
applyLambdaEquality,
functionIsTypeImplies,
functionIsType,
pointwiseFunctionality,
promote_hyp,
baseApply
Latex:
\mforall{}x:\mBbbR{}. ((r(-canonical-bound(x)) \mleq{} x) \mwedge{} (x \mleq{} r(canonical-bound(x))))
Date html generated:
2019_10_29-AM-09_35_19
Last ObjectModification:
2019_01_18-AM-11_03_33
Theory : reals
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