Nuprl Lemma : continuous-monotone-set
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[F:Type ⟶ Type].
(ContinuousMonotone(T.{x:F[T]| P[x]} )) supposing ((∀T:Type. (F[T] ⊆r A)) and ContinuousMonotone(T.F[T]))
Proof
Definitions occuring in Statement :
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
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,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
type-continuous: Continuous(T.F[T]),
guard: {T},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
isect_subtype_rel_trivial,
set_wf,
le_wf,
false_wf,
continuous-monotone_wf,
all_wf,
nat_wf,
subtype_rel_wf,
subtype_rel_set,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
lambdaEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
because_Cache,
lambdaFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
isectEquality,
setEquality,
functionEquality,
cumulativity,
independent_pairEquality,
instantiate,
dependent_set_memberEquality,
natural_numberEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[F:Type {}\mrightarrow{} Type].
(ContinuousMonotone(T.\{x:F[T]| P[x]\} )) supposing ((\mforall{}T:Type. (F[T] \msubseteq{}r A)) and ContinuousMonotone(T\000C.F[T]))
Date html generated:
2016_05_13-PM-04_10_02
Last ObjectModification:
2016_01_14-PM-07_29_48
Theory : subtype_1
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