Nuprl Lemma : coSet-equality
∀x,y:coSet{i:l}.
(x = y ∈ coSet{i:l} ⇐⇒ ∃R:coSet{i:l} ⟶ coSet{i:l} ⟶ ℙ'. (coSet-bisimulation{i:l}(x,y.R[x;y]) ∧ R[x;y]))
Proof
Definitions occuring in Statement :
coSet-bisimulation: coSet-bisimulation{i:l}(x,y.R[x; y]),
coSet: coSet{i:l},
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
uimplies: b supposing a,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strong-type-continuous: Continuous+(T.F[T]),
type-continuous: Continuous(T.F[T]),
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
cand: A c∧ B,
coSet-bisimulation: coSet-bisimulation{i:l}(x,y.R[x; y]),
istype: istype(T),
F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation
Lemmas referenced :
corec_wf,
istype-universe,
coinduction-principle,
subtype_rel_product,
subtype_rel_function,
subtype_rel_wf,
strong-continuous-depproduct,
strong-continuous-function,
continuous-id,
subtype_rel_weakening,
nat_wf,
istype-nat,
coSet-bisimulation_wf,
subtype_rel_self,
coSet_wf,
equal_wf,
coSet_subtype,
subtype_coSet,
subtype_rel_dep_function,
corec-subtype-coSet,
coSet-subtype-corec,
equal_functionality_wrt_subtype_rel2,
subtype_rel_transitivity
Rules used in proof :
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
productEquality,
universeEquality,
functionEquality,
cumulativity,
hypothesisEquality,
hypothesis,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
isect_memberFormation_alt,
because_Cache,
lambdaFormation_alt,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
isectEquality,
applyEquality,
functionIsType,
equalityIstype,
productIsType,
dependent_pairFormation_alt,
productElimination,
equalityIsType1,
hypothesis_subsumption,
functionExtensionality,
dependent_pairEquality_alt,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}x,y:coSet\{i:l\}.
(x = y \mLeftarrow{}{}\mRightarrow{} \mexists{}R:coSet\{i:l\} {}\mrightarrow{} coSet\{i:l\} {}\mrightarrow{} \mBbbP{}'. (coSet-bisimulation\{i:l\}(x,y.R[x;y]) \mwedge{} R[x;y]))
Date html generated:
2019_10_31-AM-06_32_51
Last ObjectModification:
2018_12_13-PM-02_29_34
Theory : constructive!set!theory
Home
Index