Nuprl Lemma : rational-lower-approx-property
∀x:ℝ. ∀n:ℕ+. (((below x within 1/n) ≤ x) ∧ (x ≤ ((below x within 1/n) + (r1/r(n)))))
Proof
Definitions occuring in Statement :
rational-lower-approx: (below x within 1/n),
rdiv: (x/y),
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[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],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
rational-approx: (x within 1/n),
rational-lower-approx: (below x within 1/n),
real: ℝ,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rat_term_to_real: rat_term_to_real(f;t),
rtermSubtract: left "-" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
cand: A c∧ B,
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
rational-approx-property,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
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,
nat_plus_wf,
real_wf,
value-type-has-value,
int-value-type,
subtract_wf,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
nequal_wf,
int-to-real_wf,
rsub_wf,
rdiv_wf,
rless-int,
rless_wf,
req_functionality,
int-rdiv_functionality,
req_inversion,
rsub-int,
req_weakening,
int-rdiv-req,
rsub_functionality,
req-int-fractions,
nat_plus_inc_int_nzero,
decidable__equal_int,
assert-rat-term-eq2,
rtermDivide_wf,
rtermSubtract_wf,
rtermVar_wf,
rtermConstant_wf,
rleq_wf,
radd_wf,
rabs_wf,
iff_weakening_uiff,
rleq_functionality,
radd_functionality,
rabs-difference-bound-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
itermAdd_wf,
int_term_value_add_lemma,
req_transitivity,
radd-rdiv,
rdiv_functionality,
radd-int,
req_wf,
rleq_weakening,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
intEquality,
applyEquality,
because_Cache,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
inrFormation_alt,
productElimination,
applyLambdaEquality,
promote_hyp,
addEquality
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (((below x within 1/n) \mleq{} x) \mwedge{} (x \mleq{} ((below x within 1/n) + (r1/r(n)))))
Date html generated:
2019_10_29-AM-10_01_11
Last ObjectModification:
2019_04_02-AM-10_01_41
Theory : reals
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