Nuprl Lemma : unique-corec-solution
∀[F:Type ⟶ Type]
∀[I:Type]
∀G:⋂T:{T:Type| (F[T] ⊆r T) ∧ (corec(T.F[T]) ⊆r T)} . ((I ⟶ T) ⟶ I ⟶ F[T])
∃!s:I ⟶ corec(T.F[T]). (s = (G s) ∈ (I ⟶ corec(T.F[T])))
supposing ContinuousMonotone(T.F[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
continuous-monotone: ContinuousMonotone(T.F[T]),
exists!: ∃!x:T. P[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
type-continuous: Continuous(T.F[T]),
all: ∀x:A. B[x],
ext-eq: A ≡ B,
cand: A c∧ B,
istype: istype(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
exists!: ∃!x:T. P[x],
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s1;s2],
indexed-F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation (indexed I)
Lemmas referenced :
subtype_rel_wf,
nat_wf,
corec-ext,
corec_wf,
subtype_rel_dep_function,
continuous-monotone_wf,
fix_wf_corec_system,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
indexed-coinduction-principle,
all_wf,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
Error :isect_memberEquality_alt,
isectElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :universeIsType,
extract_by_obid,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
universeEquality,
Error :functionIsType,
rename,
Error :lambdaFormation_alt,
independent_isectElimination,
independent_pairFormation,
Error :lambdaEquality_alt,
applyEquality,
Error :dependent_set_memberEquality_alt,
because_Cache,
Error :productIsType,
functionEquality,
cumulativity,
functionExtensionality,
Error :isectIsType,
Error :setIsType,
setElimination,
Error :dependent_pairFormation_alt,
Error :functionExtensionality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
Error :equalityIsType1,
productEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[I:Type]
\mforall{}G:\mcap{}T:\{T:Type| (F[T] \msubseteq{}r T) \mwedge{} (corec(T.F[T]) \msubseteq{}r T)\} . ((I {}\mrightarrow{} T) {}\mrightarrow{} I {}\mrightarrow{} F[T])
\mexists{}!s:I {}\mrightarrow{} corec(T.F[T]). (s = (G s))
supposing ContinuousMonotone(T.F[T])
Date html generated:
2019_06_20-PM-00_37_32
Last ObjectModification:
2018_10_01-AM-10_05_11
Theory : co-recursion
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