Nuprl Lemma : monad-extend-comp
∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x,y,z:cat-ob(C)]. ∀[g:cat-arrow(C) x M(y)]. ∀[h:cat-arrow(C) y M(z)].
(monad-extend(C;M;x;z;cat-comp(C) x M(y) M(z) g monad-extend(C;M;y;z;h))
= (cat-comp(C) M(x) M(y) M(z) monad-extend(C;M;x;y;g) monad-extend(C;M;y;z;h))
∈ (cat-arrow(C) M(x) M(z)))
Proof
Definitions occuring in Statement :
monad-extend: monad-extend(C;M;x;y;f),
monad-fun: M(x),
cat-monad: Monad(C),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cat-monad: Monad(C),
nat-trans: nat-trans(C;D;F;G),
monad-fun: M(x),
functor-comp: functor-comp(F;G),
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
id_functor: 1,
monad-extend: monad-extend(C;M;x;y;f),
monad-functor: monad-functor(M),
pi1: fst(t),
monad-op: monad-op(M;x),
pi2: snd(t),
spreadn: spread3,
and: P ∧ Q,
true: True,
functor_arrow: F(f),
cat_comp: g o f,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
cat-arrow_wf,
monad-fun_wf,
cat-ob_wf,
cat-monad_wf,
small-category_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
equal_wf,
squash_wf,
true_wf,
cat-comp-assoc,
functor-arrow-comp,
iff_weakening_equal,
cat_comp_wf,
functor_arrow_wf,
cat_comp_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
because_Cache,
functionExtensionality,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. \mforall{}[x,y,z:cat-ob(C)]. \mforall{}[g:cat-arrow(C) x M(y)]. \mforall{}[h:cat-arrow(C) y
M(z)].
(monad-extend(C;M;x;z;cat-comp(C) x M(y) M(z) g monad-extend(C;M;y;z;h))
= (cat-comp(C) M(x) M(y) M(z) monad-extend(C;M;x;y;g) monad-extend(C;M;y;z;h)))
Date html generated:
2017_10_05-AM-00_52_24
Last ObjectModification:
2017_07_28-AM-09_20_53
Theory : small!categories
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