Nuprl Lemma : simple-comb-concat-classrel
∀[Info,B:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type]. ∀[Xs:k:ℕn ⟶ EClass(A k)]. ∀[f:(k:ℕn ⟶ (A k)) ⟶ bag(B)].
∀[F:(k:ℕn ⟶ bag(A k)) ⟶ bag(B)].
∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ simple-comb(F;Xs)(e);↓∃vs:k:ℕn ⟶ (A k). ((∀k:ℕn. vs[k] ∈ Xs[k](e)) ∧ v ↓∈ f vs))
supposing ∀v:B. ∀bs:k:ℕn ⟶ bag(A k). (v ↓∈ F bs ⇐⇒ ↓∃vs:k:ℕn ⟶ (A k). ((∀k:ℕn. vs k ↓∈ bs k) ∧ v ↓∈ f vs))
Proof
Definitions occuring in Statement :
simple-comb: simple-comb(F;Xs),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
int_seg: {i..j-},
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀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,
bag-member: x ↓∈ bs,
bag: bag(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],
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
simple-comb: simple-comb(F;Xs),
eclass: EClass(A[eo; e]),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
guard: {T},
sq_stable: SqStable(P),
rev_implies: P ⇐ Q
Latex:
\mforall{}[Info,B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[Xs:k:\mBbbN{}n {}\mrightarrow{} EClass(A k)]. \mforall{}[f:(k:\mBbbN{}n {}\mrightarrow{} (A k)) {}\mrightarrow{} bag(B)].
\mforall{}[F:(k:\mBbbN{}n {}\mrightarrow{} bag(A k)) {}\mrightarrow{} bag(B)].
\mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} simple-comb(F;Xs)(e);\mdownarrow{}\mexists{}vs:k:\mBbbN{}n {}\mrightarrow{} (A k). ((\mforall{}k:\mBbbN{}n. vs[k] \mmember{} Xs[k](e)) \mwedge{} v \mdownarrow{}\mmember{} f vs))
supposing \mforall{}v:B. \mforall{}bs:k:\mBbbN{}n {}\mrightarrow{} bag(A k).
(v \mdownarrow{}\mmember{} F bs \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}vs:k:\mBbbN{}n {}\mrightarrow{} (A k). ((\mforall{}k:\mBbbN{}n. vs k \mdownarrow{}\mmember{} bs k) \mwedge{} v \mdownarrow{}\mmember{} f vs))
Date html generated:
2016_05_17-AM-09_19_02
Last ObjectModification:
2016_01_17-PM-11_13_30
Theory : classrel!lemmas
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