Nuprl Lemma : loop-class-state-classrel
∀[Info,B:Type]. ∀[X:EClass(B ⟶ B)]. ∀[init:Id ⟶ bag(B)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ loop-class-state(X;init)(e);↓∃b:B
(if first(e)
then b ↓∈ init loc(e)
else b ∈ loop-class-state(X;init)(pred(e))
fi
∧ if e ∈b X then ∃f:B ⟶ B. (f ∈ X(e) ∧ (v = (f b) ∈ B)) else v = b ∈ B fi ))
Proof
Definitions occuring in Statement :
loop-class-state: loop-class-state(X;init),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
loop-class-state: loop-class-state(X;init),
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
squash: ↓T,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
false: False,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,B:Type]. \mforall{}[X:EClass(B {}\mrightarrow{} B)]. \mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} loop-class-state(X;init)(e);\mdownarrow{}\mexists{}b:B
(if first(e)
then b \mdownarrow{}\mmember{} init loc(e)
else b \mmember{} loop-class-state(X;init)(pred(e))
fi
\mwedge{} if e \mmember{}\msubb{} X
then \mexists{}f:B {}\mrightarrow{} B. (f \mmember{} X(e) \mwedge{} (v = (f b)))
else v = b
fi ))
Date html generated:
2016_05_16-PM-11_35_32
Last ObjectModification:
2016_01_17-PM-07_09_35
Theory : event-ordering
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