Nuprl Lemma : corec-subtype-corec
∀[F,G:Type ⟶ Type]. (corec(T.F[T]) ⊆r corec(T.G[T])) supposing (Monotone(T.G[T]) and (∀T:Type. (F[T] ⊆r G[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],
all: ∀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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
type-monotone: Monotone(T.F[T]),
prop: ℙ
Lemmas referenced :
corec-subtype-corec2,
subtype_rel_wf,
type-monotone_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
hypothesis,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
functionEquality
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(corec(T.F[T]) \msubseteq{}r corec(T.G[T])) supposing (Monotone(T.G[T]) and (\mforall{}T:Type. (F[T] \msubseteq{}r G[T])))
Date html generated:
2016_05_14-AM-06_21_57
Last ObjectModification:
2015_12_26-PM-00_00_03
Theory : co-recursion
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