Nuprl Lemma : fix_wf_corec-partial1
∀[A:Type]
(∀[F:Type ⟶ Type]
∀[f:⋂T:Type. ((T ⟶ partial(A)) ⟶ F[T] ⟶ partial(A))]. (fix(f) ∈ corec(T.F[T]) ⟶ partial(A))
supposing ContinuousMonotone(T.F[T])) supposing
(mono(A) and
value-type(A))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
partial: partial(T),
mono: mono(T),
continuous-monotone: ContinuousMonotone(T.F[T]),
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
fix-corec-partial1,
corec_wf,
partial_wf,
subtype_rel_dep_function,
corec_subtype,
subtype_rel_self,
equal_wf,
continuous-monotone_wf,
mono_wf,
value-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
universeEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaFormation,
productElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
isectEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]
(\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[f:\mcap{}T:Type. ((T {}\mrightarrow{} partial(A)) {}\mrightarrow{} F[T] {}\mrightarrow{} partial(A))]
(fix(f) \mmember{} corec(T.F[T]) {}\mrightarrow{} partial(A))
supposing ContinuousMonotone(T.F[T])) supposing
(mono(A) and
value-type(A))
Date html generated:
2017_04_14-AM-07_48_35
Last ObjectModification:
2017_02_27-PM-03_18_06
Theory : co-recursion
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