Nuprl Lemma : indexed-coinduction-principle
∀[I:Type]. ∀[F:Type ⟶ Type].
∀[R:(I ⟶ corec(T.F[T])) ⟶ (I ⟶ corec(T.F[T])) ⟶ ℙ]
∀[x,y:I ⟶ corec(T.F[T])]. x = y ∈ (I ⟶ corec(T.F[T])) supposing R[x;y]
supposing x,y.R[x;y] is an T.F[T]-bisimulation (indexed I)
supposing ContinuousMonotone(T.F[T])
Proof
Definitions occuring in Statement :
indexed-F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation (indexed I),
corec: corec(T.F[T]),
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
continuous-monotone: ContinuousMonotone(T.F[T]),
type-monotone: Monotone(T.F[T]),
type-continuous: Continuous(T.F[T]),
corec: corec(T.F[T]),
indexed-F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation (indexed I),
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
decidable: Dec(P),
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
top: Top,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ
Lemmas referenced :
nat_wf,
equal_wf,
and_wf,
le_weakening2,
primrec_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-swap,
le-add-cancel,
istype-le,
top_wf,
int_seg_wf,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
less-iff-le,
add_functionality_wrt_le,
add-associates,
istype-int,
add-zero,
add-commutes,
le-add-cancel2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
primrec0_lemma,
istype-void,
subtype_rel_self,
subtract-1-ge-0,
istype-nat,
indexed-F-bisimulation_wf,
corec_wf,
continuous-monotone_wf,
istype-universe
Rules used in proof :
lambdaEquality,
functionEquality,
dependent_set_memberEquality,
isect_memberEquality,
functionExtensionality,
isect_memberFormation,
independent_pairEquality,
Error :isectIsType,
closedConclusion,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
imageElimination,
minusEquality,
Error :productIsType,
applyLambdaEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
addEquality,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
cumulativity,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
Error :functionIsTypeImplies,
Error :functionExtensionality_alt,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
Error :isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :universeIsType,
extract_by_obid,
Error :lambdaEquality_alt,
applyEquality,
because_Cache,
Error :functionIsType,
independent_isectElimination,
universeEquality,
instantiate
Latex:
\mforall{}[I:Type]. \mforall{}[F:Type {}\mrightarrow{} Type].
\mforall{}[R:(I {}\mrightarrow{} corec(T.F[T])) {}\mrightarrow{} (I {}\mrightarrow{} corec(T.F[T])) {}\mrightarrow{} \mBbbP{}]
\mforall{}[x,y:I {}\mrightarrow{} corec(T.F[T])]. x = y supposing R[x;y]
supposing x,y.R[x;y] is an T.F[T]-bisimulation (indexed I)
supposing ContinuousMonotone(T.F[T])
Date html generated:
2019_06_20-PM-01_05_24
Last ObjectModification:
2019_06_20-PM-00_59_24
Theory : co-recursion
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