Nuprl Lemma : cantor-to-general-cantor
∀B:ℕ ⟶ ℕ+
∃f:(ℕ ⟶ 𝔹) ⟶ n:ℕ ⟶ ℕB[n]
(Surj(ℕ ⟶ 𝔹;n:ℕ ⟶ ℕB[n];f)
∧ (∀k:ℕ. ∃j:ℕ. ∀p,q:ℕ ⟶ 𝔹. ((p = q ∈ (ℕj ⟶ 𝔹)) ⇒ ((f p) = (f q) ∈ (n:ℕk ⟶ ℕB[n])))))
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
inject: Inj(A;B;f),
compose: f o g,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
equipollent: A ~ B,
pi1: fst(t),
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
so_apply: x[s],
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
cand: A c∧ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
le: A ≤ B,
guard: {T},
false: False,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
int_seg_subtype,
equal_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
subtype_rel_function,
istype-universe,
equipollent_wf,
equipollent-int_seg-shift,
decidable__equal_int_seg,
compose_wf,
surject_wf,
lelt_wf,
ifthenelse_wf,
equipollent_transitivity,
equipollent-two,
indep-function_functionality_wrt_equipollent,
equipollent-exp,
equal-wf-base,
primrec-wf2,
le_weakening2,
le_weakening,
le_functionality,
imax_ub,
exp_functionality_wrt_le_1,
imax_nat,
add-zero,
zero-mul,
add-mul-special,
add-swap,
minus-one-mul,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
primrec-wf-nat-plus,
general_add_assoc,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
int_subtype_base,
le_wf,
set_subtype_base,
istype-false,
decidable__equal_int,
add-subtract-cancel,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
primrec-unroll,
primrec0_lemma,
int_seg_wf,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add_nat_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_seg_properties,
nat_wf,
primrec_wf,
nat_plus_subtype_nat,
exp_wf2,
istype-le,
log-property,
istype-nat,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformand_wf,
nat_plus_properties,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
nat_plus_wf,
imax_nat_plus,
istype-less_than,
log_wf,
imax_wf
Rules used in proof :
functionExtensionality,
productEquality,
setIsType,
baseApply,
inlFormation_alt,
functionEquality,
universeEquality,
sqequalBase,
productIsType,
intEquality,
cumulativity,
instantiate,
equalityElimination,
promote_hyp,
pointwiseFunctionality,
imageElimination,
addEquality,
productElimination,
functionIsType,
equalityIstype,
int_eqEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
voidElimination,
isect_memberEquality_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
universeIsType,
rename,
setElimination,
closedConclusion,
applyEquality,
because_Cache,
hypothesis,
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
dependent_pairFormation_alt,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}
\mexists{}f:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} n:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[n]
(Surj(\mBbbN{} {}\mrightarrow{} \mBbbB{};n:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[n];f) \mwedge{} (\mforall{}k:\mBbbN{}. \mexists{}j:\mBbbN{}. \mforall{}p,q:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((p = q) {}\mRightarrow{} ((f p) = (f q)))))
Date html generated:
2019_10_15-AM-10_26_32
Last ObjectModification:
2019_10_03-PM-06_54_31
Theory : continuity
Home
Index