Nuprl Lemma : assert-bag_all
∀[T:Type]. ∀[f:T ⟶ 𝔹]. ∀[b:bag(T)]. (∀x:T. (x ↓∈ b ⇒ (↑f[x])) ⇐⇒ ↑bag_all(b;f))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag_all: bag_all(b;f),
bag: bag(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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bag_all: bag_all(b;f),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
nil: [],
it: ⋅,
btrue: tt,
true: True,
cons-bag: x.b,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
sq_or: a ↓∨ b,
or: P ∨ Q,
guard: {T},
empty-bag: {},
top: Top,
false: False,
band: p ∧b q,
bfalse: ff,
sq_type: SQType(T)
Lemmas referenced :
all_wf,
bag-member_wf,
assert_wf,
bag_all_wf,
assert_witness,
bag_wf,
bool_wf,
bag_to_squash_list,
sq_stable__all,
sq_stable_from_decidable,
decidable__assert,
squash_wf,
list_induction,
list-subtype-bag,
list_wf,
nil_wf,
bag_all-cons,
assert_of_band,
bag-member-cons,
equal_wf,
cons_wf,
bag_all-empty,
bag-member-empty-iff,
empty-bag_wf,
true_wf,
bool_cases_sqequal,
cons-bag_wf,
band_wf,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base
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,
functionEquality,
hypothesis,
applyEquality,
functionExtensionality,
productElimination,
independent_pairEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
because_Cache,
universeEquality,
imageElimination,
promote_hyp,
rename,
independent_isectElimination,
natural_numberEquality,
voidEquality,
voidElimination,
inlFormation,
imageMemberEquality,
baseClosed,
inrFormation,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
unionElimination,
comment,
dependent_set_memberEquality,
setElimination,
setEquality,
equalityTransitivity,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:bag(T)]. (\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}f[x])) \mLeftarrow{}{}\mRightarrow{} \muparrow{}bag\_all(b;f))
Date html generated:
2016_10_25-AM-10_28_45
Last ObjectModification:
2016_07_12-AM-06_45_14
Theory : bags
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