Nuprl Lemma : fix-mutual-corec-partial1
∀[A:Type]
(∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type]. ∀[f:(i:ℕk ⟶ m-corec(T.F[T];i) ⟶ partial(A))
⟶ i:ℕk
⟶ m-corec(T.F[T];i)
⟶ partial(A)].
(fix(f) ∈ i:ℕk ⟶ m-corec(T.F[T];i) ⟶ partial(A))) supposing
(mono(A) and
value-type(A))
Proof
Definitions occuring in Statement :
m-corec: m-corec(T.F[T];i),
partial: partial(T),
mono: mono(T),
int_seg: {i..j-},
nat: ℕ,
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],
natural_number: $n,
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],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
top: Top
Lemmas referenced :
void_wf,
m-corec_wf,
int_seg_wf,
partial_wf,
nat_wf,
mono_wf,
value-type_wf,
fixpoint-induction-bottom2,
strictness-apply,
bottom_wf-partial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
voidElimination,
thin,
instantiate,
extract_by_obid,
hypothesis,
because_Cache,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
natural_numberEquality,
setElimination,
rename,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
independent_isectElimination,
voidEquality
Latex:
\mforall{}[A:Type]
(\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} Type]. \mforall{}[f:(i:\mBbbN{}k {}\mrightarrow{} m-corec(T.F[T];i) {}\mrightarrow{} partial(A))
{}\mrightarrow{} i:\mBbbN{}k
{}\mrightarrow{} m-corec(T.F[T];i)
{}\mrightarrow{} partial(A)].
(fix(f) \mmember{} i:\mBbbN{}k {}\mrightarrow{} m-corec(T.F[T];i) {}\mrightarrow{} partial(A))) supposing
(mono(A) and
value-type(A))
Date html generated:
2018_05_21-PM-00_17_50
Last ObjectModification:
2017_10_18-PM-02_46_28
Theory : co-recursion
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