Nuprl Lemma : poset-id-functor
∀[I:Cname List]. (poset-functor(I;I;1) = 1 ∈ Functor(poset-cat(I);poset-cat(I)))
Proof
Definitions occuring in Statement :
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
id-morph: 1,
coordinate_name: Cname,
id_functor: 1,
cat-functor: Functor(C1;C2),
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
id_functor: 1,
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
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],
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
pi1: fst(t),
name-morph: name-morph(I;J),
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
assert: ↑b,
false: False,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
bfalse: ff,
true: True,
rev_uimplies: rev_uimplies(P;Q),
id-morph: 1,
name-comp: (f o g),
compose: f o g,
uext: uext(g)
Lemmas referenced :
list_wf,
coordinate_name_wf,
poset-cat_wf,
poset-functor_wf,
id-morph_wf,
id_functor_wf,
cat_ob_pair_lemma,
ob_pair_lemma,
ob_mk_functor_lemma,
name-comp-id-left,
nil_wf,
subtype_rel_self,
name-morph_wf,
cat-ob_wf,
cat_arrow_triple_lemma,
arrow_pair_lemma,
arrow_mk_functor_lemma,
cat-arrow_wf,
equal-functors,
nameset_wf,
assert_wf,
le_int_wf,
extd-nameset_subtype_int,
equal_wf,
bnot_wf,
not_wf,
le_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_witness,
name-comp_wf,
isname-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
because_Cache,
independent_isectElimination,
functionExtensionality,
rename,
setElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
productElimination,
independent_pairFormation,
impliesFunctionality,
equalityElimination
Latex:
\mforall{}[I:Cname List]. (poset-functor(I;I;1) = 1)
Date html generated:
2017_10_05-AM-10_29_13
Last ObjectModification:
2017_07_28-AM-11_24_07
Theory : cubical!sets
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