Nuprl Lemma : bag-maximal?-append
∀[T:Type]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[b1,b2:bag(T)]. ∀[x:T].
uiff(↑bag-maximal?(b1 + b2;x;R);(↑bag-maximal?(b1;x;R)) ∧ (↑bag-maximal?(b2;x;R)))
Proof
Definitions occuring in Statement :
bag-maximal?: bag-maximal?(bg;x;R),
bag-append: as + bs,
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-maximal?: bag-maximal?(bg;x;R),
bag-append: as + bs,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
true: True,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
bfalse: ff,
not: ¬A,
false: False,
so_apply: x[s],
band: p ∧b q,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x]
Lemmas referenced :
assert_wf,
bag-maximal?_wf,
bag-append_wf,
squash_wf,
list-subtype-bag,
list_accum_append,
subtype_rel_list,
top_wf,
decidable__assert,
list_accum_wf,
bool_wf,
btrue_wf,
band_wf,
iff_imp_equal_bool,
true_wf,
subtype_base_sq,
bool_subtype_base,
assert_functionality_wrt_uiff,
assert_witness,
bfalse_wf,
false_wf,
list_accum_invariant,
not_wf,
assert_elim,
not_assert_elim,
and_wf,
equal_wf,
btrue_neq_bfalse,
uiff_wf,
bag_wf,
bag_to_squash_list,
sq_stable__uiff,
sq_stable_from_decidable,
sq_stable__and
Rules used in proof :
because_Cache,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesis,
functionExtensionality,
applyEquality,
productEquality,
isect_memberEquality,
independent_isectElimination,
lambdaEquality,
sqequalRule,
voidElimination,
voidEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
natural_numberEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isect_memberFormation,
productElimination,
independent_pairEquality,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
functionEquality,
universeEquality,
imageElimination,
promote_hyp,
hyp_replacement,
Error :applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[b1,b2:bag(T)]. \mforall{}[x:T].
uiff(\muparrow{}bag-maximal?(b1 + b2;x;R);(\muparrow{}bag-maximal?(b1;x;R)) \mwedge{} (\muparrow{}bag-maximal?(b2;x;R)))
Date html generated:
2016_10_25-AM-10_24_33
Last ObjectModification:
2016_07_12-AM-06_41_10
Theory : bags
Home
Index