Nuprl Lemma : poset-cat-dist-flip
∀I:Cname List. ∀x:cat-ob(poset-cat(I)). ∀a:nameset(I). (((x a) = 0 ∈ ℤ) ⇒ (1 ≤ poset-cat-dist(I;x;flip(x;a))))
Proof
Definitions occuring in Statement :
poset-cat-dist: poset-cat-dist(I;x;y),
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
nameset: nameset(L),
coordinate_name: Cname,
cat-ob: cat-ob(C),
list: T List,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
poset-cat-dist: poset-cat-dist(I;x;y),
nameset: nameset(L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
name-morph: name-morph(I;J),
bfalse: ff,
prop: ℙ,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
subtract: n - m,
le: A ≤ B,
true: True,
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
cand: A c∧ B,
name-morph-flip: flip(f;y),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
int_seg: {i..j-},
lelt: i ≤ j < k,
coordinate_name: Cname,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
l_member: (x ∈ l)
Lemmas referenced :
list-subtype,
coordinate_name_wf,
length-filter-pos,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
name-morph-flip_wf,
name-morph_wf,
nil_wf,
extd-nameset_subtype_int,
equal_wf,
nameset_wf,
list_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
less_than_wf,
length_wf,
equal-wf-T-base,
extd-nameset_wf,
all_wf,
assert_wf,
isname_wf,
cat-ob_wf,
poset-cat_wf,
l_exists_iff,
l_member_wf,
subtype_rel_self,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
subtype_base_sq,
bool_subtype_base,
eq-cname_wf,
assert-eq-cname,
nsub2-flip,
lelt_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
set_subtype_base,
le_wf,
int_subtype_base,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality,
hypothesisEquality,
sqequalRule,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
addEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
intEquality,
minusEquality,
setEquality,
functionEquality,
functionExtensionality,
dependent_pairFormation,
productEquality,
instantiate,
cumulativity,
dependent_set_memberEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}I:Cname List. \mforall{}x:cat-ob(poset-cat(I)). \mforall{}a:nameset(I).
(((x a) = 0) {}\mRightarrow{} (1 \mleq{} poset-cat-dist(I;x;flip(x;a))))
Date html generated:
2017_10_05-AM-10_28_53
Last ObjectModification:
2017_07_28-AM-11_23_54
Theory : cubical!sets
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