Nuprl Lemma : cantor-to-int-bounded
∀F:(ℕ ⟶ 𝔹) ⟶ ℤ. ∃B:ℕ. ∀f:ℕ ⟶ 𝔹. (|F f| ≤ B)
Proof
Definitions occuring in Statement :
absval: |i|,
nat: ℕ,
bool: 𝔹,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_member: (x ∈ l),
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
bfalse: ff,
subtype_rel: A ⊆r B,
listp: A List+,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
true: True,
guard: {T},
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
cantor-to-int-uniform-continuity,
istype-nat,
bool_wf,
istype-int,
finite-function,
int_seg_wf,
nsub_finite,
finite-bool,
finite-iff-listable,
bfalse_wf,
nat_properties,
decidable__lt,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-le,
absval_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__le,
istype-less_than,
imax-list-nat,
map-length,
map_wf,
nat_wf,
imax-list-ub,
length-map,
select-map,
subtype_rel_list,
top_wf,
le_weakening,
squash_wf,
true_wf,
equal_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_seg_properties,
select_wf,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
le_wf,
ifthenelse_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
functionIsType,
universeIsType,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality_alt,
independent_functionElimination,
functionEquality,
unionElimination,
imageElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyEquality,
inhabitedIsType,
equalityElimination,
dependent_set_memberEquality_alt,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
intEquality,
functionExtensionality,
functionExtensionality_alt,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
cumulativity,
hyp_replacement
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbZ{}. \mexists{}B:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (|F f| \mleq{} B)
Date html generated:
2019_10_15-AM-10_26_19
Last ObjectModification:
2019_06_26-PM-02_45_48
Theory : continuity
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