Nuprl Lemma : infinitesmal-difference
∀[x,y:ℝ]. uiff(x = y;∀[k:ℕ+]. (|x - y| ≤ (r1/r(k))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
iff_weakening_uiff,
uall_wf,
nat_plus_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
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,
req_wf,
infinitesmal-zero,
less_than'_wf,
req_witness,
uiff_wf,
real_wf,
req-implies-req,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0
Rules used in proof :
cut,
addLevel,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
hypothesis,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairEquality,
applyEquality,
minusEquality,
cumulativity,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[x,y:\mBbbR{}]. uiff(x = y;\mforall{}[k:\mBbbN{}\msupplus{}]. (|x - y| \mleq{} (r1/r(k))))
Date html generated:
2017_10_03-AM-08_52_14
Last ObjectModification:
2017_07_28-AM-07_35_04
Theory : reals
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