Nuprl Lemma : subtype-corec2
∀[A,B:Type ⟶ Type].
(let P = corec(T.A[T] ⟶ B[T]) in P ⊆r (A[P] ⟶ B[P])) supposing (Continuous+(T.B[T]) and Continuous+(T.A[T]))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
strong-type-continuous: Continuous+(T.F[T]),
let: let,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
let: let,
so_lambda: λ2x.t[x],
so_apply: x[s],
type-continuous: Continuous(T.F[T]),
guard: {T},
prop: ℙ
Lemmas referenced :
nat_wf,
corec_subtype,
continuous-function,
subtype_rel_weakening,
corec_wf,
strong-type-continuous_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
functionEquality,
cumulativity,
lemma_by_obid,
universeEquality,
rename,
lambdaEquality,
applyEquality,
independent_isectElimination,
isectEquality
Latex:
\mforall{}[A,B:Type {}\mrightarrow{} Type].
(let P = corec(T.A[T] {}\mrightarrow{} B[T]) in
P \msubseteq{}r (A[P] {}\mrightarrow{} B[P])) supposing
(Continuous+(T.B[T]) and
Continuous+(T.A[T]))
Date html generated:
2016_05_14-AM-06_21_50
Last ObjectModification:
2015_12_26-PM-00_00_09
Theory : co-recursion
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