Nuprl Lemma : fix_wf_corec_partial_nat
∀[F:Type ⟶ Type]
∀[f:⋂T:Type. ((T ⟶ partial(ℕ)) ⟶ F[T] ⟶ partial(ℕ))]. (fix(f) ∈ corec(T.F[T]) ⟶ partial(ℕ))
supposing ContinuousMonotone(T.F[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
partial: partial(T),
continuous-monotone: ContinuousMonotone(T.F[T]),
nat: ℕ,
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,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
fix_wf_corec-partial1,
nat_wf,
set-value-type,
le_wf,
int-value-type,
nat-mono,
partial_wf,
continuous-monotone_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
applyEquality,
cumulativity,
universeEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectEquality,
functionEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[f:\mcap{}T:Type. ((T {}\mrightarrow{} partial(\mBbbN{})) {}\mrightarrow{} F[T] {}\mrightarrow{} partial(\mBbbN{}))]. (fix(f) \mmember{} corec(T.F[T]) {}\mrightarrow{} partial(\mBbbN{}))
supposing ContinuousMonotone(T.F[T])
Date html generated:
2016_05_14-AM-06_25_05
Last ObjectModification:
2015_12_26-AM-11_58_22
Theory : co-recursion
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