Nuprl Lemma : fix_wf_corec_3parameter
∀[F,A,B,C:Type ⟶ Type].
(∀[G:Top ⟶ Top ⟶ Top ⟶ Top ⟶ Top ⋂ ⋂T:{T:Type| corec(T.F[T]) ⊆r T}
((A[T] ⟶ B[T] ⟶ C[T] ⟶ T) ⟶ A[F[T]] ⟶ B[F[T]] ⟶ C[F[T]] ⟶ F[T])].
∀[a:A[corec(T.F[T])]]. ∀[b:B[corec(T.F[T])]]. ∀[c:C[corec(T.F[T])]].
(fix(G) a b c ∈ corec(T.F[T]))) supposing
(Continuous+(T.C[T]) and
Continuous+(T.B[T]) and
Continuous+(T.A[T]))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
strong-type-continuous: Continuous+(T.F[T]),
isect2: T1 ⋂ T2,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
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,
uimplies: b supposing a,
corec: corec(T.F[T]),
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
prop: ℙ
Lemmas referenced :
fix_wf_corec2',
continuous-function,
continuous-id,
subtype_rel_self,
nat_wf,
isect2_subtype_rel3,
top_wf,
subtype_rel_wf,
corec_wf,
bool_wf,
primrec_wf,
int_seg_wf,
isect2_wf,
strong-type-continuous_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
isect_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
functionEquality,
applyEquality,
universeEquality,
independent_isectElimination,
hypothesis,
isectEquality,
cumulativity,
unionElimination,
equalityElimination,
instantiate,
setEquality,
setElimination,
rename,
because_Cache,
inrFormation,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
voidElimination,
voidEquality,
natural_numberEquality,
axiomEquality
Latex:
\mforall{}[F,A,B,C:Type {}\mrightarrow{} Type].
(\mforall{}[G:Top {}\mrightarrow{} Top {}\mrightarrow{} Top {}\mrightarrow{} Top {}\mrightarrow{} Top \mcap{} \mcap{}T:\{T:Type| corec(T.F[T]) \msubseteq{}r T\}
((A[T] {}\mrightarrow{} B[T] {}\mrightarrow{} C[T] {}\mrightarrow{} T)
{}\mrightarrow{} A[F[T]]
{}\mrightarrow{} B[F[T]]
{}\mrightarrow{} C[F[T]]
{}\mrightarrow{} F[T])]. \mforall{}[a:A[corec(T.F[T])]]. \mforall{}[b:B[corec(T.F[T])]].
\mforall{}[c:C[corec(T.F[T])]].
(fix(G) a b c \mmember{} corec(T.F[T]))) supposing
(Continuous+(T.C[T]) and
Continuous+(T.B[T]) and
Continuous+(T.A[T]))
Date html generated:
2019_06_20-PM-00_36_58
Last ObjectModification:
2018_08_03-PM-03_55_45
Theory : co-recursion
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