Nuprl Lemma : bag-member-lifting-2
∀[C,B,A:Type]. ∀[f:A ⟶ B ⟶ C]. ∀[as:bag(A)]. ∀[bs:bag(B)]. ∀[c:C].
uiff(c ↓∈ lifting-2(f) as bs;↓∃a:A. ∃b:B. (a ↓∈ as ∧ b ↓∈ bs ∧ (c = (f a b) ∈ C)))
Proof
Definitions occuring in Statement :
lifting-2: lifting-2(f),
bag-member: x ↓∈ bs,
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
uall: ∀[x:A]. B[x],
bag-member: x ↓∈ bs,
so_lambda: λ2x.t[x],
so_apply: x[s],
lifting-2: lifting-2(f),
lifting2: lifting2(f;abag;bbag),
lifting-gen-rev: lifting-gen-rev(n;f;bags),
lifting-gen-list-rev: lifting-gen-list-rev(n;bags),
eq_int: (i =z j),
select: L[n],
cons: [a / b],
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
btrue: tt,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__bag-member,
bag-member-single,
single-bag_wf,
bag-combine_wf,
bag-member-combine,
bag_wf,
equal_wf,
and_wf,
exists_wf,
squash_wf,
lifting-2_wf,
bag-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
lemma_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
because_Cache,
functionEquality,
universeEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_isectElimination,
dependent_pairFormation,
independent_functionElimination
Latex:
\mforall{}[C,B,A:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} C]. \mforall{}[as:bag(A)]. \mforall{}[bs:bag(B)]. \mforall{}[c:C].
uiff(c \mdownarrow{}\mmember{} lifting-2(f) as bs;\mdownarrow{}\mexists{}a:A. \mexists{}b:B. (a \mdownarrow{}\mmember{} as \mwedge{} b \mdownarrow{}\mmember{} bs \mwedge{} (c = (f a b))))
Date html generated:
2016_05_15-PM-03_05_19
Last ObjectModification:
2016_01_16-AM-08_35_14
Theory : bags
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