Nuprl Lemma : strong-continuity2_biject
∀[T,S:Type].
∀g:S ⟶ ℕ
(Bij(S;ℕ;g)
⇒ (∀F:(ℕ ⟶ T) ⟶ S
(strong-continuity2(T;g o F)
⇒ (∃M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (S?)
∀f:ℕ ⟶ T
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (S?)))
∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (S?) supposing ↑isl(M n f)))))))
Proof
Definitions occuring in Statement :
strong-continuity2: strong-continuity2(T;F),
biject: Bij(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_wf,
zero-le-nat,
subtype_rel_self,
nat_wf,
strong-continuity2_biject_retract
Rules used in proof :
universeEquality,
rename,
setElimination,
sqequalRule,
dependent_set_memberEquality,
applyEquality,
lambdaFormation,
because_Cache,
independent_functionElimination,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[T,S:Type].
\mforall{}g:S {}\mrightarrow{} \mBbbN{}
(Bij(S;\mBbbN{};g)
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} S
(strong-continuity2(T;g o F)
{}\mRightarrow{} (\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (S?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} T
((\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_05_19
Last ObjectModification:
2017_09_04-AM-10_36_13
Theory : continuity
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