Nuprl Lemma : fix_wf_corec_system
∀[F:Type ⟶ Type]
∀[I:Type]. ∀[G:⋂T:{T:Type| (F[T] ⊆r T) ∧ (corec(T.F[T]) ⊆r T)} . ((I ⟶ T) ⟶ I ⟶ F[T])].
(fix(G) ∈ I ⟶ corec(T.F[T]))
supposing Monotone(T.F[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
type-monotone: Monotone(T.F[T]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
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],
and: P ∧ Q,
cand: A c∧ B,
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 ,
prop: ℙ,
subtype_rel: A ⊆r B,
bfalse: ff,
top: Top
Lemmas referenced :
fix_wf_corec1,
continuous-function,
continuous-constant,
continuous-id,
subtype_rel_self,
nat_wf,
subtype_rel_wf,
corec_wf,
set_wf,
top_wf,
bool_wf,
type-monotone_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
independent_isectElimination,
hypothesis,
isectEquality,
applyEquality,
cumulativity,
independent_pairFormation,
isect_memberEquality,
unionElimination,
equalityElimination,
setElimination,
rename,
dependent_set_memberEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
functionExtensionality,
because_Cache,
voidElimination,
voidEquality,
axiomEquality,
setEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[I:Type]. \mforall{}[G:\mcap{}T:\{T:Type| (F[T] \msubseteq{}r T) \mwedge{} (corec(T.F[T]) \msubseteq{}r T)\} . ((I {}\mrightarrow{} T) {}\mrightarrow{} I {}\mrightarrow{} F[T])].
(fix(G) \mmember{} I {}\mrightarrow{} corec(T.F[T]))
supposing Monotone(T.F[T])
Date html generated:
2019_06_20-PM-00_36_53
Last ObjectModification:
2018_08_07-PM-05_28_48
Theory : co-recursion
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