Nuprl Lemma : real-interval-m-TB
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . m-TB({x:ℝ| x ∈ [a, b]} ;rmetric())
Proof
Definitions occuring in Statement :
m-TB: m-TB(X;d),
rmetric: rmetric(),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
totally-bounded: totally-bounded(A),
nat: ℕ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
nat_plus: ℕ+,
rset-member: x ∈ A,
rmetric: rmetric(),
mdist: mdist(d;x;y),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
interval-totally-bounded,
m-TB-iff,
real_wf,
i-member_wf,
rccint_wf,
rmetric_wf,
metric-on-subtype,
member_rccint_lemma,
istype-void,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
rless-int-fractions2,
istype-less_than,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_wf,
int_seg_wf,
rabs_wf,
rsub_wf,
int_seg_properties,
nat_plus_properties,
mdist_wf,
istype-nat,
sq_stable__rleq,
rleq_weakening_rless,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
le_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setEquality,
hypothesis,
setElimination,
rename,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
setIsType,
universeIsType,
productElimination,
independent_functionElimination,
closedConclusion,
natural_numberEquality,
addEquality,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
multiplyEquality,
productIsType,
functionIsType,
imageElimination,
inhabitedIsType,
functionExtensionality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseApply,
intEquality,
sqequalBase
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . m-TB(\{x:\mBbbR{}| x \mmember{} [a, b]\} ;rmetric())
Date html generated:
2019_10_30-AM-06_51_15
Last ObjectModification:
2019_10_10-PM-05_33_47
Theory : reals
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