Nuprl Lemma : concat-lifting-loc-member
∀[B:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type]. ∀[bags:k:ℕn ⟶ bag(A k)]. ∀[f:Id ⟶ funtype(n;A;bag(B))]. ∀[b:B]. ∀[l:Id].
(b ↓∈ concat-lifting-loc(n;bags;l;f)
⇐⇒ ↓∃lst:k:ℕn ⟶ (A k). ((∀[k:ℕn]. lst k ↓∈ bags k) ∧ b ↓∈ uncurry-rev(n;f l) lst))
Proof
Definitions occuring in Statement :
concat-lifting-loc: concat-lifting-loc(n;bags;loc;f),
Id: Id,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
uncurry-rev: uncurry-rev(n;f),
bag-member: x ↓∈ bs,
bag: bag(T),
funtype: funtype(n;A;T)
Definitions unfolded in proof :
concat-lifting-loc: concat-lifting-loc(n;bags;loc;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs
Latex:
\mforall{}[B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[bags:k:\mBbbN{}n {}\mrightarrow{} bag(A k)]. \mforall{}[f:Id {}\mrightarrow{} funtype(n;A;bag(B))].
\mforall{}[b:B]. \mforall{}[l:Id].
(b \mdownarrow{}\mmember{} concat-lifting-loc(n;bags;l;f)
\mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}lst:k:\mBbbN{}n {}\mrightarrow{} (A k). ((\mforall{}[k:\mBbbN{}n]. lst k \mdownarrow{}\mmember{} bags k) \mwedge{} b \mdownarrow{}\mmember{} uncurry-rev(n;f l) lst))
Date html generated:
2016_05_17-AM-09_15_28
Last ObjectModification:
2016_01_17-PM-11_14_26
Theory : classrel!lemmas
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