Nuprl Lemma : poset-cat-dist-zero
∀[I:Cname List]. ∀[x,y:cat-ob(poset-cat(I))].
uiff(x = y ∈ cat-ob(poset-cat(I));poset-cat-dist(I;x;y) ≤ 0) supposing cat-arrow(poset-cat(I)) x y
Proof
Definitions occuring in Statement :
poset-cat-dist: poset-cat-dist(I;x;y),
poset-cat: poset-cat(J),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
pi1: fst(t),
name-morph: name-morph(I;J),
cat-arrow: cat-arrow(C),
pi2: snd(t),
poset-cat-dist: poset-cat-dist(I;x;y),
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
nameset: nameset(L),
less_than': less_than'(a;b),
l_all: (∀x∈L.P[x]),
squash: ↓T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
sq_type: SQType(T),
eq_int: (i =z j),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
l_member: (x ∈ l),
coordinate_name: Cname,
int_upper: {i...},
nat: ℕ,
cons: [a / b],
ge: i ≥ j ,
true: True
Lemmas referenced :
extd-nameset-nil,
less_than'_wf,
poset-cat-dist_wf,
equal_wf,
cat-ob_wf,
poset-cat_wf,
nameset_wf,
all_wf,
assert_wf,
isname_wf,
nil_wf,
coordinate_name_wf,
extd-nameset_wf,
le_wf,
cat-arrow_wf,
list_wf,
list-subtype,
filter_is_nil,
band_wf,
eq_int_wf,
extd-nameset_subtype_int,
length_of_nil_lemma,
false_wf,
int_seg_wf,
length_wf,
subtype_base_sq,
extd-nameset_subtype_base,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
int_subtype_base,
int_seg_subtype,
int_seg_cases,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
assert_of_le_int,
equal-wf-T-base,
decidable__cand,
member_filter,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
sq_stable__l_member,
decidable__equal-coordinate_name,
set_subtype_base,
select_member,
lelt_wf,
filter_wf5,
l_member_wf,
list-cases,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
product_subtype_list,
length_of_cons_lemma,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
not_wf,
equal-wf-base,
nsub2_subtype_extd-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
rename,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
isectElimination,
natural_numberEquality,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
functionExtensionality,
functionEquality,
isect_memberEquality,
voidElimination,
independent_isectElimination,
lambdaFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
independent_functionElimination,
hypothesis_subsumption,
addEquality,
equalityElimination,
productEquality,
setEquality,
promote_hyp
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y:cat-ob(poset-cat(I))].
uiff(x = y;poset-cat-dist(I;x;y) \mleq{} 0) supposing cat-arrow(poset-cat(I)) x y
Date html generated:
2017_10_05-AM-10_28_41
Last ObjectModification:
2017_07_28-AM-11_23_45
Theory : cubical!sets
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