Nuprl Lemma : coinduction-principle
∀[F:Type ⟶ Type]
∀[R:corec(T.F[T]) ⟶ corec(T.F[T]) ⟶ ℙ]
∀[x,y:corec(T.F[T])]. x = y ∈ corec(T.F[T]) supposing R[x;y] supposing x,y.R[x;y] is an T.F[T]-bisimulation
supposing ContinuousMonotone(T.F[T])
Proof
Definitions occuring in Statement :
F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nequal: a ≠ b ∈ T ,
F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation,
corec: corec(T.F[T]),
type-continuous: Continuous(T.F[T]),
type-monotone: Monotone(T.F[T]),
continuous-monotone: ContinuousMonotone(T.F[T])
Lemmas referenced :
F-bisimulation_wf,
corec_wf,
continuous-monotone_wf,
nat_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
primrec0_lemma,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_weakening,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
primrec-unroll,
int_seg_wf,
top_wf,
le_wf,
not-equal-2,
not-le-2,
primrec_wf,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
lambdaEquality,
applyEquality,
universeEquality,
independent_isectElimination,
functionEquality,
cumulativity,
minusEquality,
intEquality,
addEquality,
productElimination,
independent_pairFormation,
unionElimination,
voidEquality,
functionExtensionality,
dependent_functionElimination,
voidElimination,
independent_functionElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation,
instantiate,
promote_hyp,
dependent_pairFormation,
equalityElimination,
dependent_set_memberEquality,
isectEquality,
independent_pairEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[R:corec(T.F[T]) {}\mrightarrow{} corec(T.F[T]) {}\mrightarrow{} \mBbbP{}]
\mforall{}[x,y:corec(T.F[T])]. x = y supposing R[x;y] supposing x,y.R[x;y] is an T.F[T]-bisimulation
supposing ContinuousMonotone(T.F[T])
Date html generated:
2019_06_20-PM-00_37_14
Last ObjectModification:
2018_08_07-PM-05_54_06
Theory : co-recursion
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