Nuprl Lemma : corec-rel_wf
∀[F:Type ⟶ Type]
∀[G:⋂T:Type. ((T ⟶ T ⟶ ℙ) ⟶ F[T] ⟶ F[T] ⟶ ℙ)]. (corec-rel(G) ∈ corec(T.F[T]) ⟶ corec(T.F[T]) ⟶ ℙ)
supposing ContinuousMonotone(T.F[T])
Proof
Definitions occuring in Statement :
corec-rel: corec-rel(G),
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[s],
member: t ∈ T,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
top: Top,
lt_int: i <z j,
subtract: n - m,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
assert: ↑b,
bfalse: ff,
compose: f o g,
so_apply: x[s],
so_lambda: λ2x.t[x],
corec-rel: corec-rel(G),
corec: corec(T.F[T])
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp_unroll,
false_wf,
le_wf,
primrec-unroll,
true_wf,
top_wf,
decidable__le,
subtract_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_weakening2,
subtype_base_sq,
bool_subtype_base,
equal_wf,
eq_int_eq_false,
le_weakening,
subtype_rel_self,
iff_weakening_equal,
iff_imp_equal_bool,
lt_int_wf,
le-add-cancel2,
assert_of_lt_int,
assert_wf,
iff_wf,
primrec_wf,
not-le-2,
int_seg_wf,
nat_wf,
continuous-monotone_wf,
all_wf,
corec_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
voidEquality,
because_Cache,
unionElimination,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
instantiate,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
addLevel,
impliesFunctionality,
functionExtensionality,
cumulativity,
isectEquality,
functionEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[G:\mcap{}T:Type. ((T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}) {}\mrightarrow{} F[T] {}\mrightarrow{} F[T] {}\mrightarrow{} \mBbbP{})]
(corec-rel(G) \mmember{} corec(T.F[T]) {}\mrightarrow{} corec(T.F[T]) {}\mrightarrow{} \mBbbP{})
supposing ContinuousMonotone(T.F[T])
Date html generated:
2018_07_25-PM-01_30_31
Last ObjectModification:
2018_05_31-PM-02_00_17
Theory : co-recursion
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