Nuprl Lemma : strong-continuity2-no-inner-squash-cantor3
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ2
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ2?)
∀f:ℕ ⟶ 𝔹. ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ2?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ2?) supposing ↑isl(M n f))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
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,
true: True,
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 :
true: True,
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
strong-continuity2-half-squash-surject-biject,
trivial-biject-exists,
surject-nat-bool,
less_than_wf,
retraction-nat-nsub,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
bool_wf,
nat_wf
Rules used in proof :
independent_functionElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_pairFormation,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
functionEquality,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}2
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} (\mBbbN{}2?)
\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_09_29-PM-06_06_55
Last ObjectModification:
2017_09_04-AM-10_57_18
Theory : continuity
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