Nuprl Lemma : bag-from-member-function
∀[T:Type]
∀bs:bag(T). ∀P,Q:T ⟶ ℙ.
((∀i:T. Dec(Q[i]))
⇒ (∀i:T. (i ↓∈ bs ⇒ Q[i] ⇒ P[i]))
⇒ (∃b:bag(T). ((∀i:T. (i ↓∈ bs ⇒ Q[i] ⇒ i ↓∈ b)) ∧ (∀i:T. (i ↓∈ b ⇒ P[i])))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
decision: Decision,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
squash: ↓T
Lemmas referenced :
bag_wf,
decidable_wf,
all_wf,
and_wf,
bag-member_wf,
dec2bool_decidable,
bag-member-filter,
assert_wf,
subtype_rel_bag,
top_wf,
not_wf,
subtype_rel_union,
dec2bool_wf,
bag-filter_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
dependent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
setElimination,
introduction,
productElimination,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality
Latex:
\mforall{}[T:Type]
\mforall{}bs:bag(T). \mforall{}P,Q:T {}\mrightarrow{} \mBbbP{}.
((\mforall{}i:T. Dec(Q[i]))
{}\mRightarrow{} (\mforall{}i:T. (i \mdownarrow{}\mmember{} bs {}\mRightarrow{} Q[i] {}\mRightarrow{} P[i]))
{}\mRightarrow{} (\mexists{}b:bag(T). ((\mforall{}i:T. (i \mdownarrow{}\mmember{} bs {}\mRightarrow{} Q[i] {}\mRightarrow{} i \mdownarrow{}\mmember{} b)) \mwedge{} (\mforall{}i:T. (i \mdownarrow{}\mmember{} b {}\mRightarrow{} P[i])))))
Date html generated:
2016_05_15-PM-02_58_51
Last ObjectModification:
2016_01_16-AM-08_37_36
Theory : bags
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