Nuprl Lemma : corec-rel-wf2
∀[F:𝕌' ⟶ 𝕌']
∀[G:⋂T:𝕌'. ((T ⟶ T ⟶ ℙ) ⟶ F[T] ⟶ F[T] ⟶ ℙ)]. (corec-rel(G) ∈ corec(T.F[T]) ⟶ corec(T.F[T]) ⟶ ℙ)
supposing continuous-monotone{i':l}(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,
squash: ↓T,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
assert: ↑b,
bfalse: ff,
uiff: uiff(P;Q),
compose: f o g,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
corec-rel: corec-rel(G),
istype: istype(T),
corec: corec(T.F[T])
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp_unroll,
istype-void,
le_wf,
primrec-unroll,
true_wf,
istype-top,
subtract-1-ge-0,
le_weakening2,
subtype_base_sq,
bool_subtype_base,
equal_wf,
eq_int_eq_false,
le_weakening,
int_subtype_base,
subtype_rel_self,
iff_weakening_equal,
iff_imp_equal_bool,
lt_int_wf,
iff_functionality_wrt_iff,
assert_wf,
iff_weakening_uiff,
assert_of_lt_int,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
le-add-cancel2,
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,
top_wf,
int_seg_wf,
nat_wf,
continuous-monotone_wf,
corec_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
Error :isect_memberEquality_alt,
because_Cache,
Error :inhabitedIsType,
instantiate,
applyEquality,
imageElimination,
universeEquality,
Error :equalityIsType4,
baseClosed,
imageMemberEquality,
productElimination,
addEquality,
unionElimination,
minusEquality,
cumulativity,
functionExtensionality,
Error :isectIsType,
Error :functionIsType
Latex:
\mforall{}[F:\mBbbU{}' {}\mrightarrow{} \mBbbU{}']
\mforall{}[G:\mcap{}T:\mBbbU{}'. ((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 continuous-monotone\{i':l\}(T.F[T])
Date html generated:
2019_06_20-PM-00_37_25
Last ObjectModification:
2018_10_01-AM-10_08_50
Theory : co-recursion
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