Nuprl Lemma : fix_wf_corec_parameter
∀[F:Type ⟶ Type]. ∀[A:Type]. ∀[G:Top ⟶ Top ⟶ Top ⋂ ⋂T:Type. ((A ⟶ T) ⟶ A ⟶ F[T])]. ∀[a:A].
(fix(G) a ∈ corec(T.F[T]))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
isect2: T1 ⋂ T2,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
member: t ∈ T,
apply: f a,
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,
corec: corec(T.F[T]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strong-type-continuous: Continuous+(T.F[T]),
type-continuous: Continuous(T.F[T]),
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
bfalse: ff,
top: Top,
nat: ℕ
Lemmas referenced :
fix_wf_corec2,
continuous-function,
continuous-constant,
continuous-id,
subtype_rel_self,
nat_wf,
isect2_subtype_rel3,
top_wf,
subtype_rel_wf,
bool_wf,
primrec_wf,
int_seg_wf,
isect2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
isect_memberEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
functionEquality,
universeEquality,
independent_isectElimination,
because_Cache,
hypothesis,
isectEquality,
applyEquality,
cumulativity,
unionElimination,
equalityElimination,
instantiate,
inrFormation,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
voidElimination,
voidEquality,
natural_numberEquality,
setElimination,
rename,
axiomEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}[A:Type]. \mforall{}[G:Top {}\mrightarrow{} Top {}\mrightarrow{} Top \mcap{} \mcap{}T:Type. ((A {}\mrightarrow{} T) {}\mrightarrow{} A {}\mrightarrow{} F[T])]. \mforall{}[a:A].
(fix(G) a \mmember{} corec(T.F[T]))
Date html generated:
2016_05_14-AM-06_19_18
Last ObjectModification:
2015_12_26-PM-00_02_37
Theory : co-recursion
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