Nuprl Lemma : nearby-cases
∀n:ℕ+. ∀x,y:ℝ. ((x < y) ∨ (y < x) ∨ (|x - y| ≤ (r1/r(n))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
or: 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,
so_lambda: λ2x.t[x],
so_apply: x[s],
real: ℝ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
nat: ℕ,
cand: A c∧ B,
less_than: a < b,
less_than': less_than'(a;b),
req_int_terms: t1 ≡ t2,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_stable: SqStable(P),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rdiv: (x/y),
rge: x ≥ y,
le: A ≤ B
Lemmas referenced :
nat_plus_wf,
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,
set-value-type,
equal_wf,
less_than_wf,
int-value-type,
real_wf,
rless_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
intformeq_wf,
int_formula_prop_eq_lemma,
itermAdd_wf,
int_term_value_add_lemma,
rational-approx-property,
int-rdiv_wf,
int_subtype_base,
nequal_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
int-rdiv-req,
req_weakening,
rsub-rdiv,
rabs-of-nonneg,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
squash_wf,
true_wf,
rabs-int,
subtype_rel_self,
iff_weakening_equal,
absval_pos,
istype-le,
req_transitivity,
rabs-rdiv,
rdiv_functionality,
rmul-is-positive,
rmul_wf,
req_inversion,
rmul-int,
absval_wf,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermMinus_wf,
rminus-int,
radd-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
subtype_base_sq,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
sq_stable__less_than,
int_term_value_minus_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
zero-rleq-rabs,
rmul-rdiv,
rmul-one,
req_functionality,
rmul_preserves_rleq,
rinv_wf2,
rmul-rinv3,
real_term_value_mul_lemma,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
rleq_weakening_equal,
istype-false,
rmul-rinv,
rmul_functionality,
uimplies_transitivity,
rational-approx_wf,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
uiff_transitivity,
radd_functionality,
radd-rdiv,
rleq-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
isectElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inhabitedIsType,
intEquality,
applyEquality,
because_Cache,
addEquality,
inlFormation_alt,
unionIsType,
closedConclusion,
inrFormation_alt,
productElimination,
applyLambdaEquality,
dependent_set_memberFormation_alt,
baseApply,
baseClosed,
sqequalBase,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality,
productIsType,
minusEquality,
cumulativity,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x,y:\mBbbR{}. ((x < y) \mvee{} (y < x) \mvee{} (|x - y| \mleq{} (r1/r(n))))
Date html generated:
2019_10_29-AM-10_06_14
Last ObjectModification:
2019_04_01-PM-11_00_37
Theory : reals
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