Nuprl Lemma : continuous-monotone-depfunction
∀[A:Type]. ∀[F:Type ⟶ A ⟶ Type]. ContinuousMonotone(T.a:A ⟶ F[T;a]) supposing ∀a:A. ContinuousMonotone(T.F[T;a])
Proof
Definitions occuring in Statement :
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
type-continuous: Continuous(T.F[T]),
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_dep_function,
subtype_rel_wf,
nat_wf,
all_wf,
continuous-monotone_wf,
false_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
independent_isectElimination,
hypothesis,
lambdaFormation,
dependent_functionElimination,
productElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
isectEquality,
functionEquality,
cumulativity,
independent_pairEquality,
instantiate,
dependent_set_memberEquality,
natural_numberEquality,
functionExtensionality
Latex:
\mforall{}[A:Type]. \mforall{}[F:Type {}\mrightarrow{} A {}\mrightarrow{} Type].
ContinuousMonotone(T.a:A {}\mrightarrow{} F[T;a]) supposing \mforall{}a:A. ContinuousMonotone(T.F[T;a])
Date html generated:
2016_05_13-PM-04_09_42
Last ObjectModification:
2015_12_26-AM-11_22_45
Theory : subtype_1
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