Nuprl Lemma : union-continuous-type-monotone
∀[F:Type ⟶ Type]. (Monotone(T.F[T])) supposing (union-continuous{i:l}(T.F[T]) and (∀A,B:Type. (A ≡ B ⇒ F[A] ≡ F[B])))
Proof
Definitions occuring in Statement :
union-continuous: union-continuous{i:l}(T.F[T]),
type-monotone: Monotone(T.F[T]),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
union-continuous: union-continuous{i:l}(T.F[T]),
tunion: ⋃x:A.B[x],
ifthenelse: if b then t else f fi ,
btrue: tt,
pi2: snd(t),
all: ∀x:A. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
bool: 𝔹,
bfalse: ff,
guard: {T}
Lemmas referenced :
subtype_rel_transitivity,
bfalse_wf,
tunion_wf,
btrue_wf,
ifthenelse_wf,
bool_wf,
ext-eq_wf,
all_wf,
union-continuous_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
sqequalRule,
axiomEquality,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
cumulativity,
functionEquality,
imageMemberEquality,
dependent_pairEquality,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
imageElimination,
unionElimination,
independent_isectElimination
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
(Monotone(T.F[T])) supposing
(union-continuous\{i:l\}(T.F[T]) and
(\mforall{}A,B:Type. (A \mequiv{} B {}\mRightarrow{} F[A] \mequiv{} F[B])))
Date html generated:
2016_05_13-PM-04_10_24
Last ObjectModification:
2016_01_14-PM-07_29_50
Theory : subtype_1
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