Nuprl Lemma : Kleisli-adjunction_wf
∀[C:SmallCategory]. ∀[M:Monad(C)]. (Kl(C;M) ∈ KlF(C;M) -| KlG(C;M))
Proof
Definitions occuring in Statement :
Kleisli-adjunction: Kl(C;M),
Kleisli-right: KlG(C;M),
Kleisli-left: KlF(C;M),
Kleisli-cat: Kl(C;M),
cat-monad: Monad(C),
counit-unit-adjunction: F -| G,
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Kleisli-adjunction: Kl(C;M),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
Kleisli-right: KlG(C;M),
Kleisli-left: KlF(C;M),
Kleisli-cat: Kl(C;M),
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_apply: x[s],
mk-cat: mk-cat,
uimplies: b supposing a,
cat_comp: g o f,
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cat-ob: cat-ob(C),
pi1: fst(t),
cat-arrow: cat-arrow(C),
pi2: snd(t)
Lemmas referenced :
mk-adjunction_wf,
Kleisli-cat_wf,
Kleisli-left_wf,
Kleisli-right_wf,
ob_mk_functor_lemma,
cat_arrow_triple_lemma,
cat-ob_wf,
monad-unit_wf,
cat_comp_tuple_lemma,
arrow_mk_functor_lemma,
cat_id_tuple_lemma,
cat_ob_pair_lemma,
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
monad-fun_wf,
cat-comp-assoc,
monad-extend_wf,
cat-id_wf,
iff_weakening_equal,
cat-comp_wf,
monad-unit-extend,
cat-comp-ident,
cat-monad_wf,
small-category_wf,
subtype_rel-equal,
cat-comp-ident1,
cat-comp-ident2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
hypothesis,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
lambdaFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
independent_pairFormation,
axiomEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. (Kl(C;M) \mmember{} KlF(C;M) -| KlG(C;M))
Date html generated:
2017_10_05-AM-00_52_54
Last ObjectModification:
2017_07_28-AM-09_21_10
Theory : small!categories
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