Nuprl Lemma : monad-equations
∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x:cat-ob(C)].
((((cat-comp(C) M(x) M(M(x)) M(x) monad-unit(M;M(x)) monad-op(M;x)) = (cat-id(C) M(x)) ∈ (cat-arrow(C) M(x) M(x)))
∧ ((cat-comp(C) M(x) M(M(x)) M(x) (arrow(monad-functor(M)) x M(x) monad-unit(M;x)) monad-op(M;x))
= (cat-id(C) M(x))
∈ (cat-arrow(C) M(x) M(x))))
∧ ((cat-comp(C) M(M(M(x))) M(M(x)) M(x) monad-op(M;M(x)) monad-op(M;x))
= (cat-comp(C) M(M(M(x))) M(M(x)) M(x) (arrow(monad-functor(M)) M(M(x)) M(x) monad-op(M;x)) monad-op(M;x))
∈ (cat-arrow(C) M(M(M(x))) M(x))))
Proof
Definitions occuring in Statement :
monad-op: monad-op(M;x),
monad-unit: monad-unit(M;x),
monad-fun: M(x),
monad-functor: monad-functor(M),
cat-monad: Monad(C),
functor-arrow: arrow(F),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
monad-functor: monad-functor(M),
pi1: fst(t),
pi2: snd(t),
monad-unit: monad-unit(M;x),
monad-op: monad-op(M;x),
monad-fun: M(x),
spreadn: spread3,
cat-monad: Monad(C),
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-monad_wf,
cat-ob_wf
Rules used in proof :
isect_memberEquality,
isectElimination,
extract_by_obid,
axiomEquality,
independent_pairEquality,
because_Cache,
independent_pairFormation,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. \mforall{}[x:cat-ob(C)].
((((cat-comp(C) M(x) M(M(x)) M(x) monad-unit(M;M(x)) monad-op(M;x)) = (cat-id(C) M(x)))
\mwedge{} ((cat-comp(C) M(x) M(M(x)) M(x) (arrow(monad-functor(M)) x M(x) monad-unit(M;x)) monad-op(M;x))
= (cat-id(C) M(x))))
\mwedge{} ((cat-comp(C) M(M(M(x))) M(M(x)) M(x) monad-op(M;M(x)) monad-op(M;x))
= (cat-comp(C) M(M(M(x))) M(M(x)) M(x) (arrow(monad-functor(M)) M(M(x)) M(x) monad-op(M;x))
monad-op(M;x))))
Date html generated:
2017_01_19-PM-02_58_26
Last ObjectModification:
2017_01_17-PM-05_26_26
Theory : small!categories
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