Nuprl Lemma : cantor-interval_wf
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. (cantor-interval(a;b;f;n) ∈ ℝ × ℝ)
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cantor-interval: cantor-interval(a;b;f;n),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
primrec_wf,
real_wf,
int_seg_wf,
bool_wf,
eqtt_to_assert,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
radd_wf,
int-rmul_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesis,
because_Cache,
hypothesisEquality,
independent_pairEquality,
lambdaEquality,
productElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
baseClosed,
dependent_pairFormation,
promote_hyp,
axiomEquality,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}]. (cantor-interval(a;b;f;n) \mmember{} \mBbbR{} \mtimes{} \mBbbR{})
Date html generated:
2017_10_03-AM-09_48_44
Last ObjectModification:
2017_07_28-AM-08_00_45
Theory : reals
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