Nuprl Lemma : strong-continuity2_biject_retract-ext
∀[T,S,U:Type].
∀r:ℕ ⟶ U
((U ⊆r ℕ)
⇒ (∀x:U. ((r x) = x ∈ U))
⇒ (∀g:S ⟶ U
(Bij(S;U;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,
subtype_rel: A ⊆r B,
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 :
member: t ∈ T,
bij_inv: bij_inv(bi),
pi1: fst(t),
pi2: snd(t),
strong-continuity2_biject_retract,
biject-inverse,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
strong-continuity2_biject_retract,
lifting-strict-spread,
istype-void,
strict4-apply,
strict4-spread,
biject-inverse
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}[T,S,U:Type].
\mforall{}r:\mBbbN{} {}\mrightarrow{} U
((U \msubseteq{}r \mBbbN{})
{}\mRightarrow{} (\mforall{}x:U. ((r x) = x))
{}\mRightarrow{} (\mforall{}g:S {}\mrightarrow{} U
(Bij(S;U;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:
2019_06_20-PM-02_50_34
Last ObjectModification:
2019_03_26-AM-07_44_57
Theory : continuity
Home
Index