Nuprl Lemma : strong-continuity2-no-inner-squash-unique
∀F:(ℕ ⟶ ℕ) ⟶ ℕ
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?)
∀f:ℕ ⟶ ℕ. ∃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),
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 :
squash: ↓T,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
strong-continuity3: strong-continuity3(T;F)
Lemmas referenced :
subtype_rel_self,
nat_wf,
strong-continuity3-half-squash
Rules used in proof :
functionEquality,
dependent_functionElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \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_38
Last ObjectModification:
2017_09_03-PM-09_28_17
Theory : continuity
Home
Index