Nuprl Lemma : cantor2baire-aux-pos
∀[a:ℕ ⟶ 𝔹]. ∀[n:ℕ].
cantor2baire-aux(a;n) ~ if a n then cantor2baire-aux(a;n-1) + 1 else cantor2baire-aux(a;n-1) fi supposing 0 < n
Proof
Definitions occuring in Statement :
cantor2baire-aux: cantor2baire-aux(a;n),
nat-pred: n-1,
nat: ℕ,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cantor2baire-aux: cantor2baire-aux(a;n),
nat: ℕ,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
primrec-unroll,
istype-void,
subtract-add-cancel,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_subtype_base,
equal_wf,
cantor2baire-aux_wf,
subtract_wf,
decidable__le,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
le_wf,
add_functionality_wrt_eq,
nat-pred_wf,
nat-pred-as-sub,
iff_weakening_equal,
nat_wf,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
Error :isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
hypothesisEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
independent_pairFormation,
Error :universeIsType,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
imageElimination,
universeEquality,
intEquality,
addEquality,
Error :dependent_set_memberEquality_alt,
Error :functionIsType,
imageMemberEquality,
baseClosed,
axiomSqEquality
Latex:
\mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbN{}].
cantor2baire-aux(a;n) \msim{} if a n then cantor2baire-aux(a;n-1) + 1 else cantor2baire-aux(a;n-1) fi
supposing 0 < n
Date html generated:
2019_06_20-PM-03_07_37
Last ObjectModification:
2018_10_03-PM-00_24_16
Theory : continuity
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