Nuprl Lemma : weak-continuity-implies-strong-cantor
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ
∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ?)
∀f:ℕ ⟶ 𝔹. ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ?) supposing ↑isl(M n f)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ext2Cantor: ext2Cantor(n;f;d),
int_seg: {i..j-},
lelt: i ≤ j < k,
isl: isl(x)
Lemmas referenced :
strong-continuity2-implies-uniform-continuity2-nat,
nat_wf,
bool_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
ext2Cantor_wf,
int_seg_wf,
btrue_wf,
unit_wf2,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
all_wf,
exists_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
isect_wf,
assert_wf,
isl_wf,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
functionEquality,
hypothesis,
dependent_pairFormation,
lambdaEquality,
isectElimination,
setElimination,
rename,
because_Cache,
unionElimination,
equalityElimination,
sqequalRule,
independent_isectElimination,
inlEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
inrEquality,
axiomEquality,
independent_pairFormation,
productEquality,
unionEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
isect_memberFormation
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}
\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} (\mBbbN{}?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}
((\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f)))) \mwedge{} (\mforall{}n:\mBbbN{}. (M n f) = (inl (F f)) supposing \muparrow{}isl(M n f)))
Date html generated:
2017_04_17-AM-10_00_05
Last ObjectModification:
2017_02_27-PM-05_53_14
Theory : continuity
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