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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