Nuprl Lemma : fix-corec-family-partial1
∀[P:Type]. ∀[H:(P ⟶ Type) ⟶ P ⟶ Type]. ∀[A:Type].
(∀[F:(P ⟶ Type) ⟶ P ⟶ Type]. ∀[f:(i:P ⟶ (corec-family(H) i) ⟶ partial(A))
⟶ i:P
⟶ (corec-family(H) i)
⟶ partial(A)].
(fix(f) ∈ i:P ⟶ (corec-family(H) i) ⟶ partial(A))) supposing
(mono(A) and
value-type(A))
Proof
Definitions occuring in Statement :
corec-family: corec-family(H),
partial: partial(T),
mono: mono(T),
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
fix: fix(F),
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
top: Top,
so_lambda: λ2x.t[x],
prop: ℙ
Lemmas referenced :
void_wf,
fixpoint-induction-bottom2,
corec-family_wf,
partial_wf,
strictness-apply,
istype-void,
bottom_wf-partial,
mono_wf,
value-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
voidElimination,
thin,
instantiate,
extract_by_obid,
hypothesis,
Error :functionExtensionality_alt,
because_Cache,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
functionEquality,
applyEquality,
independent_isectElimination,
sqequalRule,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
Error :functionIsType,
Error :inhabitedIsType,
Error :universeIsType,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
Error :isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[H:(P {}\mrightarrow{} Type) {}\mrightarrow{} P {}\mrightarrow{} Type]. \mforall{}[A:Type].
(\mforall{}[F:(P {}\mrightarrow{} Type) {}\mrightarrow{} P {}\mrightarrow{} Type]. \mforall{}[f:(i:P {}\mrightarrow{} (corec-family(H) i) {}\mrightarrow{} partial(A))
{}\mrightarrow{} i:P
{}\mrightarrow{} (corec-family(H) i)
{}\mrightarrow{} partial(A)].
(fix(f) \mmember{} i:P {}\mrightarrow{} (corec-family(H) i) {}\mrightarrow{} partial(A))) supposing
(mono(A) and
value-type(A))
Date html generated:
2019_06_20-PM-00_35_29
Last ObjectModification:
2019_02_20-PM-02_32_07
Theory : co-recursion
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