Nuprl Lemma : continuous-function
∀[A,B:Type ⟶ Type]. (Continuous(T.A[T] ⟶ B[T])) supposing (Continuous(T.B[T]) and Continuous+(T.A[T]))
Proof
Definitions occuring in Statement :
strong-type-continuous: Continuous+(T.F[T]),
type-continuous: Continuous(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
type-continuous: Continuous(T.F[T]),
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B
Lemmas referenced :
false_wf,
le_wf,
nat_wf,
type-continuous_wf,
strong-type-continuous_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
thin,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
applyEquality,
universeEquality,
isectEquality,
functionEquality,
axiomEquality,
cumulativity,
isect_memberEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination
Latex:
\mforall{}[A,B:Type {}\mrightarrow{} Type].
(Continuous(T.A[T] {}\mrightarrow{} B[T])) supposing (Continuous(T.B[T]) and Continuous+(T.A[T]))
Date html generated:
2017_04_14-AM-07_36_18
Last ObjectModification:
2017_02_27-PM-03_08_40
Theory : subtype_1
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