Nuprl Lemma : strong-continuity2-no-inner-squash-cantor2
∀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 :
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 :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
all_wf,
strong-continuity2-half-squash-surject-biject,
trivial-biject-exists,
surject-nat-bool,
subtype_rel_self,
bool_wf,
nat_wf
Rules used in proof :
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
functionEquality,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}
\00D9(\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_09_29-PM-06_06_53
Last ObjectModification:
2017_09_04-AM-10_59_13
Theory : continuity
Home
Index