Nuprl Lemma : strong-continuity2-no-inner-squash-unique-bool
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ?)
∀f:ℕ ⟶ 𝔹. ∃n:ℕ. (((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ (m = n ∈ ℕ)))))
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,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
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 :
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
strong-continuity3: strong-continuity3(T;F),
all: ∀x:A. B[x]
Lemmas referenced :
nat_wf,
surject-nat-bool,
bool_wf,
strong-continuity3-half-squash-surject
Rules used in proof :
functionEquality,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
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{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} (m = n)))))
Date html generated:
2017_09_29-PM-06_05_39
Last ObjectModification:
2017_09_04-PM-00_13_19
Theory : continuity
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