Nuprl Lemma : poset-cat-dist-non-zero
∀[I:Cname List]. ∀[x,y:cat-ob(poset-cat(I))].
(null(filter(λx1.((x x1 =z 0) ∧b (y x1 =z 1));I)) ~ ff) supposing
((¬(x = y ∈ cat-ob(poset-cat(I)))) and
(cat-arrow(poset-cat(I)) x y))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
null: null(as),
filter: filter(P;l),
list: T List,
band: p ∧b q,
eq_int: (i =z j),
bfalse: ff,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
poset-cat-dist: poset-cat-dist(I;x;y),
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
name-morph: name-morph(I;J),
so_lambda: λ2x.t[x],
so_apply: x[s],
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
cons: [a / b]
Lemmas referenced :
poset-cat-dist-zero,
le_wf,
poset-cat-dist_wf,
nat_wf,
not_wf,
equal_wf,
cat-ob_wf,
poset-cat_wf,
cat-arrow_wf,
list_wf,
coordinate_name_wf,
filter_wf5,
l_member_wf,
eq_int_wf,
nameset_wf,
extd-nameset_wf,
nil_wf,
all_wf,
assert_wf,
isname_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
extd-nameset_subtype_int,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
productElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
promote_hyp,
sqequalAxiom,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
functionEquality,
because_Cache,
functionExtensionality,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
dependent_functionElimination,
dependent_pairFormation,
intEquality,
voidElimination,
voidEquality,
computeAll,
hypothesis_subsumption,
addEquality
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y:cat-ob(poset-cat(I))].
(null(filter(\mlambda{}x1.((x x1 =\msubz{} 0) \mwedge{}\msubb{} (y x1 =\msubz{} 1));I)) \msim{} ff) supposing
((\mneg{}(x = y)) and
(cat-arrow(poset-cat(I)) x y))
Date html generated:
2017_10_05-AM-10_28_47
Last ObjectModification:
2017_07_28-AM-11_23_50
Theory : cubical!sets
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