Nuprl Lemma : ireal-approx-1
∀[x:ℝ]. ∀[M:ℕ+]. 1-approx(x;M;x M)
Proof
Definitions occuring in Statement :
ireal-approx: j-approx(x;M;z),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rational-approx: (x within 1/n),
rge: x ≥ y
Lemmas referenced :
less_than'_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rabs_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
nat_plus_wf,
real_wf,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
nequal_wf,
subtype_rel_sets,
less_than_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_inversion,
int-rdiv-req,
rational-approx_wf,
rleq_functionality_wrt_implies,
rational-approx-property,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
multiplyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
lambdaFormation,
baseApply,
closedConclusion,
baseClosed,
setEquality,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[M:\mBbbN{}\msupplus{}]. 1-approx(x;M;x M)
Date html generated:
2018_05_22-PM-01_59_14
Last ObjectModification:
2017_10_25-AM-10_23_32
Theory : reals
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