Nuprl Lemma : monad-from_wf
∀[Mnd:Monad]. (monad-from(Mnd) ∈ Monad(TypeCat))
Proof
Definitions occuring in Statement :
monad-from: monad-from(Mnd),
cat-monad: Monad(C),
type-cat: TypeCat,
uall: ∀[x:A]. B[x],
member: t ∈ T,
monad: Monad
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
monad-from: monad-from(Mnd),
uimplies: b supposing a,
type-cat: TypeCat,
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],
compose: f o g,
cat-ob: cat-ob(C),
pi1: fst(t),
cat-arrow: cat-arrow(C),
pi2: snd(t),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
M-return: M-return(Mnd),
id_functor: 1,
nat-trans: nat-trans(C;D;F;G),
functor-comp: functor-comp(F;G)
Lemmas referenced :
M-map_wf,
M-bind_wf,
istype-universe,
mk-monad_wf,
cat_ob_pair_lemma,
istype-void,
cat_arrow_triple_lemma,
ob_mk_functor_lemma,
cat_comp_tuple_lemma,
arrow_mk_functor_lemma,
cat_id_tuple_lemma,
monad_wf,
type-cat_wf,
subtype_rel_self,
cat-ob_wf,
cat-arrow_wf,
mk-functor_wf,
M-return_wf,
equal_wf,
squash_wf,
true_wf,
trivial-equal,
M-associative,
iff_weakening_equal,
M-leftunit,
M-rightunit,
eta_conv,
compose_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaEquality_alt,
lambdaFormation_alt,
applyEquality,
hypothesisEquality,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
functionIsType,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
equalityIsType1,
independent_functionElimination,
sqequalRule,
instantiate,
universeEquality,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
functionExtensionality,
functionEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality_alt,
cumulativity,
lambdaFormation,
isectEquality,
lambdaEquality
Latex:
\mforall{}[Mnd:Monad]. (monad-from(Mnd) \mmember{} Monad(TypeCat))
Date html generated:
2019_10_31-AM-07_25_39
Last ObjectModification:
2018_11_08-PM-06_00_48
Theory : small!categories
Home
Index