Nuprl Lemma : continuous'-monotone-bunion
∀[F,G:Type ⟶ Type].
(continuous'-monotone{i:l}(T.F[T] ⋃ G[T])) supposing
(continuous'-monotone{i:l}(T.G[T]) and
continuous'-monotone{i:l}(T.F[T]))
Proof
Definitions occuring in Statement :
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
b-union: A ⋃ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
type-continuous': semi-continuous(λT.F[T]),
so_lambda: λ2x.t[x],
type-incr-chain: type-incr-chain{i:l}(),
prop: ℙ,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
subtype_rel_b-union,
subtype_rel_wf,
b-union_wf,
tunion_wf,
nat_wf,
type-incr-chain_wf,
and_wf,
type-monotone_wf,
type-continuous'_wf,
subtype_rel_tunion,
subtype_rel_b-union-left,
subtype_rel_transitivity,
subtype_rel_b-union-right
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
hypothesis,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
setElimination,
rename,
independent_pairEquality,
instantiate,
functionEquality,
imageElimination,
unionElimination,
equalityElimination,
lambdaFormation
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(continuous'-monotone\{i:l\}(T.F[T] \mcup{} G[T])) supposing
(continuous'-monotone\{i:l\}(T.G[T]) and
continuous'-monotone\{i:l\}(T.F[T]))
Date html generated:
2016_05_15-PM-06_54_10
Last ObjectModification:
2015_12_27-AM-11_43_38
Theory : general
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