Nuprl Lemma : b_all-squash-exists-bag2
∀[A,B:Type]. ∀[as:bag(A)]. ∀[P:A ⟶ B ⟶ ℙ].
  ↓∃bs:bag(A × B). ((bag-map(λx.(fst(x));bs) = as ∈ bag(A)) ∧ b_all(A × B;bs;x.↓P[fst(x);snd(x)])) 
  supposing b_all(A;as;x.↓∃y:B. P[x;y])
Proof
Definitions occuring in Statement : 
b_all: b_all(T;b;x.P[x])
, 
bag-map: bag-map(f;bs)
, 
bag: bag(T)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
exists: ∃x:A. B[x]
, 
squash: ↓T
, 
and: P ∧ Q
, 
lambda: λx.A[x]
, 
function: x:A ⟶ B[x]
, 
product: x:A × B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
squash: ↓T
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
b_all: b_all(T;b;x.P[x])
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
exists_wf, 
pi2_wf, 
squash_wf, 
b_all_wf, 
pi1_wf, 
bag-map_wf, 
bag_wf, 
equal_wf, 
and_wf, 
bag-member_wf, 
b_all-squash-exists-bag
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
independent_isectElimination, 
hypothesis, 
imageElimination, 
productElimination, 
dependent_pairFormation, 
independent_pairFormation, 
lambdaFormation, 
dependent_functionElimination, 
independent_functionElimination, 
productEquality, 
imageMemberEquality, 
baseClosed, 
isect_memberEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
cumulativity, 
universeEquality
Latex:
\mforall{}[A,B:Type].  \mforall{}[as:bag(A)].  \mforall{}[P:A  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].
    \mdownarrow{}\mexists{}bs:bag(A  \mtimes{}  B).  ((bag-map(\mlambda{}x.(fst(x));bs)  =  as)  \mwedge{}  b\_all(A  \mtimes{}  B;bs;x.\mdownarrow{}P[fst(x);snd(x)])) 
    supposing  b\_all(A;as;x.\mdownarrow{}\mexists{}y:B.  P[x;y])
Date html generated:
2016_05_15-PM-02_41_42
Last ObjectModification:
2016_01_16-AM-08_46_42
Theory : bags
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