Nuprl Lemma : poset-cat-arrow-filter-nil
∀I:Cname List. ∀J:nameset(I) List. ∀c1,c2:cat-ob(poset-cat(I)). ∀f:cat-arrow(poset-cat(I)) c1 c2.
((filter(λz.(c1 z =z c2 z);J) = [] ∈ ({x:nameset(J)| ↑(c1 x =z c2 x)} List))
⇒ (∀j∈J.((c1 j) = 0 ∈ ℕ2) ∧ ((c2 j) = 1 ∈ ℕ2)))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
eq_int: (i =z j),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
squash: ↓T,
sq_stable: SqStable(P),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
name-morph: name-morph(I;J),
respects-equality: respects-equality(S;T),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
uiff: uiff(P;Q)
Lemmas referenced :
list_wf,
nameset_wf,
coordinate_name_wf,
subtype_rel_list,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
select_wf,
sq_stable__le,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
assert_wf,
eq_int_wf,
filter_wf5,
extd-nameset_wf,
nil_wf,
istype-assert,
isname_wf,
l_member_wf,
respects-equality-list,
respects-equality-set,
respects-equality-set-trivial2,
cat-arrow_wf,
poset-cat_wf,
cat-ob_wf,
int_seg_wf,
length_wf,
poset-cat-arrow-not-equal,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
equal_functionality_wrt_subtype_rel2,
filter_is_nil_implies,
list-subtype,
neg_assert_of_eq_int,
extd-nameset_subtype_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
applyEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
sqequalRule,
productElimination,
instantiate,
cumulativity,
intEquality,
natural_numberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
setEquality,
setIsType,
functionIsType,
sqequalBase,
inhabitedIsType
Latex:
\mforall{}I:Cname List. \mforall{}J:nameset(I) List. \mforall{}c1,c2:cat-ob(poset-cat(I)). \mforall{}f:cat-arrow(poset-cat(I)) c1 c2.
((filter(\mlambda{}z.(c1 z =\msubz{} c2 z);J) = []) {}\mRightarrow{} (\mforall{}j\mmember{}J.((c1 j) = 0) \mwedge{} ((c2 j) = 1)))
Date html generated:
2019_11_05-PM-00_31_50
Last ObjectModification:
2018_12_10-AM-09_22_27
Theory : cubical!sets
Home
Index