Nuprl Lemma : cantor2baire2cantor
∀a:ℕ ⟶ 𝔹. (initF(a) ⇒ (baire2cantor(cantor2baire(a)) = a ∈ (ℕ ⟶ 𝔹)))
Proof
Definitions occuring in Statement :
initF: initF(a),
cantor2baire: cantor2baire(a),
baire2cantor: baire2cantor(a),
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
bnot: ¬bb,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
initF: initF(a),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
assert: ↑b,
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
top: Top,
cantor2baire-aux: cantor2baire-aux(a;n),
nat-pred: n-1,
guard: {T},
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
member: t ∈ T,
cantor2baire: cantor2baire(a),
baire2cantor: baire2cantor(a),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
btrue_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_term_value_add_lemma,
itermAdd_wf,
assert_of_eq_int,
nat-pred_wf,
cantor2baire-aux_wf,
eq_int_wf,
eqtt_to_assert,
equal-wf-base,
int_formula_prop_wf,
decidable__equal_int,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
cantor2baire-aux-pos,
assert_wf,
btrue_neq_bfalse,
assert_elim,
bfalse_wf,
iff_imp_equal_bool,
primrec0_lemma,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
bool_wf,
initF_wf,
nat_wf,
le_wf,
false_wf,
decidable__equal_nat
Rules used in proof :
promote_hyp,
addEquality,
productElimination,
equalityElimination,
baseClosed,
computeAll,
int_eqEquality,
dependent_pairFormation,
rename,
setElimination,
levelHypothesis,
because_Cache,
addLevel,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
intEquality,
independent_isectElimination,
cumulativity,
instantiate,
functionEquality,
applyEquality,
unionElimination,
isectElimination,
hypothesis,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
functionExtensionality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (initF(a) {}\mRightarrow{} (baire2cantor(cantor2baire(a)) = a))
Date html generated:
2017_04_21-AM-11_22_34
Last ObjectModification:
2017_04_20-PM-03_53_37
Theory : continuity
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