Nuprl Lemma : poset-functor_wf
∀[J,K:Cname List]. ∀[f:name-morph(J;K)]. (poset-functor(J;K;f) ∈ Functor(poset-cat(K);poset-cat(J)))
Proof
Definitions occuring in Statement :
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
name-morph: name-morph(I;J),
coordinate_name: Cname,
cat-functor: Functor(C1;C2),
list: T List,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cat-functor: Functor(C1;C2),
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cand: A c∧ B,
name-comp: (f o g),
compose: f o g,
uext: uext(g),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
nameset: nameset(L),
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
ge: i ≥ j
Lemmas referenced :
name-comp_wf,
nil_wf,
coordinate_name_wf,
name-morph_wf,
assert_witness,
le_int_wf,
assert_of_le_int,
nameset_wf,
all_wf,
assert_wf,
extd-nameset_subtype_int,
le_reflexive,
cat-ob_wf,
poset-cat_wf,
equal_wf,
cat-arrow_wf,
cat-id_wf,
cat-comp_wf,
list_wf,
isname_wf,
bool_wf,
eqtt_to_assert,
assert-isname,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not-assert-isname,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_functionality,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalRule,
dependent_pairEquality,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
productElimination,
independent_isectElimination,
functionEquality,
functionExtensionality,
lambdaFormation,
dependent_functionElimination,
independent_pairFormation,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
natural_numberEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqEquality,
intEquality,
voidEquality,
computeAll
Latex:
\mforall{}[J,K:Cname List]. \mforall{}[f:name-morph(J;K)].
(poset-functor(J;K;f) \mmember{} Functor(poset-cat(K);poset-cat(J)))
Date html generated:
2017_10_05-AM-10_29_00
Last ObjectModification:
2017_07_28-AM-11_23_58
Theory : cubical!sets
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