Nuprl Lemma : iterated-classrel-as-prior
∀[Info,A,S:Type]. ∀[init:Id ⟶ bag(S)]. ∀[f:A ⟶ S ⟶ S]. ∀[X:EClass(A)].
∀es:EO+(Info). ∀e:E.
∀[v:S]
(iterated-classrel(es;S;A;f;init;X;e;v)
⇐⇒ ∃z:S
(prior-iterated-classrel(es;A;S;z;X;f;init;e)
∧ ((∃a:A. (a ∈ X(e) ∧ (v = (f a z) ∈ S))) ∨ ((∀a:A. (¬a ∈ X(e))) ∧ (v = z ∈ S)))))
Proof
Definitions occuring in Statement :
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
cand: A c∧ B,
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
false: False,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[init:Id {}\mrightarrow{} bag(S)]. \mforall{}[f:A {}\mrightarrow{} S {}\mrightarrow{} S]. \mforall{}[X:EClass(A)].
\mforall{}es:EO+(Info). \mforall{}e:E.
\mforall{}[v:S]
(iterated-classrel(es;S;A;f;init;X;e;v)
\mLeftarrow{}{}\mRightarrow{} \mexists{}z:S
(prior-iterated-classrel(es;A;S;z;X;f;init;e)
\mwedge{} ((\mexists{}a:A. (a \mmember{} X(e) \mwedge{} (v = (f a z)))) \mvee{} ((\mforall{}a:A. (\mneg{}a \mmember{} X(e))) \mwedge{} (v = z)))))
Date html generated:
2016_05_16-PM-01_56_58
Last ObjectModification:
2015_12_29-PM-02_19_37
Theory : event-ordering
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