Nuprl Lemma : mk-monad_wf
∀[C:SmallCategory]. ∀[T:Functor(C;C)]. ∀[u:nat-trans(C;C;1;T)]. ∀[m:nat-trans(C;C;functor-comp(T;T);T)].
(mk-monad(T;
u;
m) ∈ Monad(C)) supposing
((∀X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (u (T X)) (m X)) = (cat-id(C) (T X)) ∈ (cat-arrow(C) (T X) (T X)))) and
(∀X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (T X (T X) (u X)) (m X))
= (cat-id(C) (T X))
∈ (cat-arrow(C) (T X) (T X)))) and
(∀X:cat-ob(C)
((cat-comp(C) (T (T (T X))) (T (T X)) (T X) (m (T X)) (m X))
= (cat-comp(C) (T (T (T X))) (T (T X)) (T X) (T (T (T X)) (T X) (m X)) (m X))
∈ (cat-arrow(C) (T (T (T X))) (T X)))))
Proof
Definitions occuring in Statement :
mk-monad: mk-monad,
cat-monad: Monad(C),
id_functor: 1,
functor-comp: functor-comp(F;G),
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
functor-comp: functor-comp(F;G),
nat-trans: nat-trans(C;D;F;G),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
id_functor: 1,
uimplies: b supposing a,
cat-monad: Monad(C),
mk-monad: mk-monad,
prop: ℙ,
spreadn: spread3,
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
all_wf,
equal_wf,
cat-comp_wf,
functor-arrow_wf,
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
cat-id_wf,
nat-trans_wf,
functor-comp_wf,
id_functor_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_set_memberEquality,
dependent_pairEquality,
hypothesisEquality,
independent_pairEquality,
setElimination,
rename,
because_Cache,
isectElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
productEquality,
setEquality,
functionEquality,
independent_pairFormation,
productElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:Functor(C;C)]. \mforall{}[u:nat-trans(C;C;1;T)].
\mforall{}[m:nat-trans(C;C;functor-comp(T;T);T)].
(mk-monad(T;
u;
m) \mmember{} Monad(C)) supposing
((\mforall{}X:cat-ob(C). ((cat-comp(C) (T X) (T (T X)) (T X) (u (T X)) (m X)) = (cat-id(C) (T X)))) and
(\mforall{}X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (T X (T X) (u X)) (m X)) = (cat-id(C) (T X)))) and
(\mforall{}X:cat-ob(C)
((cat-comp(C) (T (T (T X))) (T (T X)) (T X) (m (T X)) (m X))
= (cat-comp(C) (T (T (T X))) (T (T X)) (T X) (T (T (T X)) (T X) (m X)) (m X)))))
Date html generated:
2020_05_20-AM-07_58_33
Last ObjectModification:
2017_01_16-PM-08_02_54
Theory : small!categories
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