Nuprl Lemma : baire2cantor2baire
∀a:ℕ ⟶ ℕ. (init0(a) ⇒ increasing-sequence(a) ⇒ (cantor2baire(baire2cantor(a)) = a ∈ (ℕ ⟶ ℕ)))
Proof
Definitions occuring in Statement :
init0: init0(a),
cantor2baire: cantor2baire(a),
baire2cantor: baire2cantor(a),
increasing-sequence: increasing-sequence(a),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
increasing-sequence: increasing-sequence(a),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
label: ...$L... t,
nequal: a ≠ b ∈ T ,
baire2cantor: baire2cantor(a),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
guard: {T},
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
subtype_rel: A ⊆r B,
init0: init0(a),
less_than': less_than'(a;b),
le: A ≤ B,
cantor2baire-aux: cantor2baire-aux(a;n),
cantor2baire: cantor2baire(a),
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
nat-pred-as-sub,
iff_weakening_equal,
ifthenelse_wf,
true_wf,
squash_wf,
add_nat_wf,
int_term_value_add_lemma,
itermAdd_wf,
btrue_wf,
bfalse_wf,
nat-pred_wf,
subtract-add-cancel,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
primrec-unroll,
le_wf,
false_wf,
primrec0_lemma,
init0_wf,
increasing-sequence_wf,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
addEquality,
cumulativity,
instantiate,
promote_hyp,
applyLambdaEquality,
because_Cache,
productElimination,
equalityTransitivity,
equalityElimination,
equalitySymmetry,
levelHypothesis,
addLevel,
dependent_set_memberEquality,
functionEquality,
applyEquality,
unionElimination,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
functionExtensionality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (init0(a) {}\mRightarrow{} increasing-sequence(a) {}\mRightarrow{} (cantor2baire(baire2cantor(a)) = a))
Date html generated:
2017_04_21-AM-11_22_42
Last ObjectModification:
2017_04_20-PM-03_57_46
Theory : continuity
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