Nuprl Lemma : least-upper-bound
∀[A:Set(ℝ)]
((∃x:ℝ. (x ∈ A))
⇒ bounded-above(A)
⇒ (∃b:ℝ. sup(A) = b ⇐⇒ ∀x,y:ℝ. ((x < y) ⇒ ((∃a:ℝ. ((a ∈ A) ∧ (x < a))) ∨ A ≤ y))))
Proof
Definitions occuring in Statement :
sup: sup(A) = b,
bounded-above: bounded-above(A),
upper-bound: A ≤ b,
rset-member: x ∈ A,
rset: Set(ℝ),
rless: x < y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
sup: sup(A) = b,
uimplies: b supposing a,
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
upper-bound: A ≤ b,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rset: Set(ℝ),
subtype_rel: A ⊆r B,
rset-member: x ∈ A,
strict-upper-bounds: strict-upper-bounds(A),
strict-upper-bound: A < b,
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermSubtract: left "-" right,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
rtermAdd: left "+" right,
pi2: snd(t),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rgt: x > y,
closed-rset: closed-rset(A),
upper-bounds: upper-bounds(A),
member-closure: y ∈ closure(A),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rless_wf,
real_wf,
sup_wf,
rset-member_wf,
upper-bound_wf,
bounded-above_wf,
rset_wf,
rless-cases,
rsub_wf,
rless-implies-rless,
int-to-real_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_transitivity2,
rleq_weakening_rless,
le_witness_for_triv,
sup-iff-closure,
closures-meet,
strict-upper-bounds_wf,
bounded-above-strict,
subtype_rel_self,
rleq_wf,
rdiv_wf,
rless-int,
rleq-int-fractions2,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
rless-int-fractions3,
strict-upper-bound_wf,
rmul_wf,
radd_wf,
rmul_preserves_rless,
trivial-rsub-rless,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermVar_wf,
rtermAdd_wf,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
req-implies-req,
req_wf,
rinv_wf2,
itermMultiply_wf,
itermAdd_wf,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rless_functionality,
req_transitivity,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
req_weakening,
real_term_value_mul_lemma,
real_term_value_add_lemma,
rleq_weakening_equal,
rleq_weakening,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
member-closure_wf,
upper-bounds-closed,
istype-nat,
converges-to_wf,
upper-bounds_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
productIsType,
functionIsType,
because_Cache,
unionIsType,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
inlFormation_alt,
inrFormation_alt,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
rename,
setElimination,
applyEquality,
instantiate,
universeEquality,
closedConclusion,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
minusEquality,
cumulativity,
intEquality,
equalityIstype,
sqequalBase,
promote_hyp
Latex:
\mforall{}[A:Set(\mBbbR{})]
((\mexists{}x:\mBbbR{}. (x \mmember{} A))
{}\mRightarrow{} bounded-above(A)
{}\mRightarrow{} (\mexists{}b:\mBbbR{}. sup(A) = b \mLeftarrow{}{}\mRightarrow{} \mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} ((\mexists{}a:\mBbbR{}. ((a \mmember{} A) \mwedge{} (x < a))) \mvee{} A \mleq{} y))))
Date html generated:
2019_10_29-AM-10_43_13
Last ObjectModification:
2019_04_19-PM-06_29_38
Theory : reals
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