Nuprl Lemma : assert-bag-all
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. (∀x:T. (x ↓∈ bs ⇒ (↑p[x])) ⇐⇒ ↑bag-all(x.p[x];bs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-all: bag-all(x.p[x];bs),
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
bag-all: bag-all(x.p[x];bs),
bag-map: bag-map(f;bs),
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
bl-all: (∀x∈L.P[x])_b,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
false: False,
or: P ∨ Q,
true: True,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
bfalse: ff
Lemmas referenced :
eqtt_to_assert,
bag-all_wf,
all_wf,
bag-member_wf,
assert_wf,
assert_witness,
bag_wf,
bool_wf,
deq_wf,
equal-wf-T-base,
list_wf,
list-subtype-bag,
permutation_wf,
equal_wf,
equal-wf-base,
bl-all_wf,
map_wf,
l_member_wf,
assert-bl-all,
l_all_map,
l_all_iff,
bag-member-list,
decidable-equal-deq,
list_induction,
reduce_wf,
band_wf,
btrue_wf,
reduce_nil_lemma,
reduce_cons_lemma,
null_nil_lemma,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
true_wf,
iff_imp_equal_bool,
assert_elim,
subtype_base_sq,
bool_subtype_base,
iff_weakening_equal,
or_wf,
cons_member,
cons_wf,
assert_of_band,
map_nil_lemma,
map_cons_lemma,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
productElimination,
independent_isectElimination,
functionEquality,
independent_pairEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
because_Cache,
universeEquality,
pointwiseFunctionalityForEquality,
baseClosed,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
rename,
axiomEquality,
productEquality,
setElimination,
setEquality,
voidElimination,
voidEquality,
unionElimination,
natural_numberEquality,
addLevel,
levelHypothesis,
instantiate,
imageElimination,
imageMemberEquality,
impliesFunctionality,
allFunctionality,
equalityElimination,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)].
(\mforall{}x:T. (x \mdownarrow{}\mmember{} bs {}\mRightarrow{} (\muparrow{}p[x])) \mLeftarrow{}{}\mRightarrow{} \muparrow{}bag-all(x.p[x];bs))
Date html generated:
2018_05_21-PM-09_47_11
Last ObjectModification:
2017_07_26-PM-06_30_08
Theory : bags_2
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