Nuprl Lemma : strong-continuous-b-union
∀[F,G:Type ⟶ Type].
(Continuous+(T.F[T] ⋃ G[T])) supposing ((∀T,S:Type. (¬F[T] ⋂ G[S])) and Continuous+(T.G[T]) and Continuous+(T.F[T]))
Proof
Definitions occuring in Statement :
strong-type-continuous: Continuous+(T.F[T]),
isect2: T1 ⋂ T2,
b-union: A ⋃ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
btrue: tt,
bfalse: ff,
isect2: T1 ⋂ T2,
it: ⋅,
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
bfalse_wf,
btrue_wf,
isect2_subtype_rel,
b-union-void,
isect2_subtype_rel2,
isect2-b-union-subtype,
strong-subtype-implies,
strong-subtype-b-union,
bool_wf,
subtype_rel_b-union,
ifthenelse_wf,
le_wf,
false_wf,
strong-type-continuous_wf,
isect2_wf,
not_wf,
all_wf,
b-union_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
isectEquality,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
isect_memberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
cumulativity,
universeEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
natural_numberEquality,
lambdaFormation,
imageElimination,
unionElimination,
equalityElimination,
imageMemberEquality,
dependent_pairEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
rename,
voidElimination,
baseClosed
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(Continuous+(T.F[T] \mcup{} G[T])) supposing
((\mforall{}T,S:Type. (\mneg{}F[T] \mcap{} G[S])) and
Continuous+(T.G[T]) and
Continuous+(T.F[T]))
Date html generated:
2016_05_13-PM-04_12_14
Last ObjectModification:
2016_01_14-PM-07_30_01
Theory : subtype_1
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