Nuprl Lemma : poset-cat-arrow-flip
∀I:Cname List. ∀x:cat-ob(poset-cat(I)). ∀a:nameset(I). (((x a) = 0 ∈ ℤ) ⇒ (cat-arrow(poset-cat(I)) x flip(x;a)))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
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,
name-morph-flip: flip(f;y),
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
sq_stable: SqStable(P),
squash: ↓T,
cat-ob: cat-ob(C),
name-morph: name-morph(I;J),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
assert_of_le_int,
eq-cname_wf,
bool_wf,
eqtt_to_assert,
assert-eq-cname,
subtract_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
coordinate_name_wf,
set_subtype_base,
le_wf,
int_subtype_base,
false_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
nameset_wf,
extd-nameset_wf,
nil_wf,
all_wf,
assert_wf,
isname_wf,
l_member_wf,
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,
equal-wf-T-base,
cat-ob_wf,
poset-cat_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
lambdaEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
functionEquality,
functionExtensionality,
dependent_set_memberEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}I:Cname List. \mforall{}x:cat-ob(poset-cat(I)). \mforall{}a:nameset(I).
(((x a) = 0) {}\mRightarrow{} (cat-arrow(poset-cat(I)) x flip(x;a)))
Date html generated:
2017_10_05-AM-10_27_48
Last ObjectModification:
2017_07_28-AM-11_23_20
Theory : cubical!sets
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