Nuprl Lemma : fix_wf_corec2
∀[F,H:Type ⟶ Type].
∀[G:⋂T:Type. (H[T] ⟶ H[F[T]]) ⋂ Top ⟶ H[Top]]. (fix(G) ∈ H[corec(T.F[T])]) supposing Continuous(T.H[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
type-continuous: Continuous(T.F[T]),
isect2: T1 ⋂ T2,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
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,
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
so_apply: x[s],
or: P ∨ Q,
so_lambda: λ2x.t[x],
bfalse: ff,
prop: ℙ
Lemmas referenced :
fix_wf_corec2',
isect2_subtype_rel3,
top_wf,
subtype_rel_wf,
corec_wf,
isect2_subtype_rel2,
bool_wf,
isect2_wf,
type-continuous_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
introduction,
independent_isectElimination,
isect_memberEquality,
unionElimination,
equalityElimination,
sqequalRule,
applyEquality,
instantiate,
isectEquality,
universeEquality,
cumulativity,
functionEquality,
setElimination,
rename,
inlFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
because_Cache,
axiomEquality
Latex:
\mforall{}[F,H:Type {}\mrightarrow{} Type].
\mforall{}[G:\mcap{}T:Type. (H[T] {}\mrightarrow{} H[F[T]]) \mcap{} Top {}\mrightarrow{} H[Top]]. (fix(G) \mmember{} H[corec(T.F[T])])
supposing Continuous(T.H[T])
Date html generated:
2016_05_14-AM-06_19_08
Last ObjectModification:
2015_12_26-PM-00_02_39
Theory : co-recursion
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