Nuprl Lemma : corec_subtype
∀[F:Type ⟶ Type]. corec(T.F[T]) ⊆r F[corec(T.F[T])] supposing Continuous(T.F[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
type-continuous: Continuous(T.F[T]),
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],
member: t ∈ T,
uimplies: b supposing a,
type-continuous: Continuous(T.F[T]),
so_apply: x[s],
nat: ℕ,
corec: corec(T.F[T]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
subtype_rel_self,
primrec1_lemma,
primrec_add,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
type-continuous_wf,
corec_wf,
subtype_rel_transitivity,
nat_wf,
int_seg_wf,
top_wf,
primrec_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
instantiate,
lemma_by_obid,
universeEquality,
hypothesisEquality,
hypothesis,
applyEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
isectEquality,
independent_isectElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
voidEquality,
intEquality,
minusEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. corec(T.F[T]) \msubseteq{}r F[corec(T.F[T])] supposing Continuous(T.F[T])
Date html generated:
2016_05_14-AM-06_21_36
Last ObjectModification:
2016_01_14-PM-08_03_28
Theory : co-recursion
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