Nuprl Lemma : fix-corec-partial1
∀[A:Type]
(∀[F:Type ⟶ Type]. ∀[f:(corec(T.F[T]) ⟶ partial(A)) ⟶ corec(T.F[T]) ⟶ partial(A)].
(fix(f) ∈ corec(T.F[T]) ⟶ partial(A))) supposing
(mono(A) and
value-type(A))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
partial: partial(T),
mono: mono(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),
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],
prop: ℙ,
all: ∀x:A. B[x]
Lemmas referenced :
bottom_wf_function,
fixpoint-induction-bottom2,
value-type_wf,
mono_wf,
partial_wf,
corec_wf,
void_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
voidElimination,
thin,
instantiate,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
because_Cache,
independent_isectElimination,
cumulativity,
lambdaFormation
Latex:
\mforall{}[A:Type]
(\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}[f:(corec(T.F[T]) {}\mrightarrow{} partial(A)) {}\mrightarrow{} corec(T.F[T]) {}\mrightarrow{} partial(A)].
(fix(f) \mmember{} corec(T.F[T]) {}\mrightarrow{} partial(A))) supposing
(mono(A) and
value-type(A))
Date html generated:
2016_05_14-AM-06_24_58
Last ObjectModification:
2016_01_12-PM-03_19_28
Theory : co-recursion
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