Nuprl Lemma : equiv-on-corec
∀F:Type ⟶ Type
(ContinuousMonotone(T.F[T])
⇒ (∀G:⋂T:Type. ((T ⟶ T ⟶ ℙ) ⟶ F[T] ⟶ F[T] ⟶ ℙ)
((∀T:Type. ∀E:T ⟶ T ⟶ ℙ. (EquivRel(T;x,y.E x y) ⇒ EquivRel(F[T];x,y.G E x y)))
⇒ EquivRel(corec(T.F[T]);x,y.corec-rel(G) x y))))
Proof
Definitions occuring in Statement :
corec-rel: corec-rel(G),
corec: corec(T.F[T]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
continuous-monotone: ContinuousMonotone(T.F[T]),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
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 y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
corec-rel: corec-rel(G),
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
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,
equiv_rel_true,
bool_wf,
bfalse_wf,
equiv_rel_wf,
set_wf,
primrec-wf2,
all_wf,
continuous-monotone_wf,
corec_wf,
corec-rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
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
(ContinuousMonotone(T.F[T])
{}\mRightarrow{} (\mforall{}G:\mcap{}T:Type. ((T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}) {}\mrightarrow{} F[T] {}\mrightarrow{} F[T] {}\mrightarrow{} \mBbbP{})
((\mforall{}T:Type. \mforall{}E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (EquivRel(T;x,y.E x y) {}\mRightarrow{} EquivRel(F[T];x,y.G E x y)))
{}\mRightarrow{} EquivRel(corec(T.F[T]);x,y.corec-rel(G) x y))))
Date html generated:
2018_07_25-PM-01_30_39
Last ObjectModification:
2018_05_31-AM-10_55_18
Theory : co-recursion
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