Nuprl Lemma : State-comb-classrel-mem2
∀[Info,B,A:Type]. ∀[f:A ⟶ B ⟶ B]. ∀[init:Id ⟶ bag(B)].
∀X:EClass(A). ∀es:EO+(Info). ∀e:E.
∀[v:B]
(v ∈ State-comb(init;f;X)(e)
⇐⇒ if e ∈b X
then ↓∃w:B. ∃a:A. (w ∈ Memory-class(f;init;X)(e) ∧ (v = (f a w) ∈ B) ∧ a ∈ X(e))
else v ∈ Memory-class(f;init;X)(e)
fi )
Proof
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
Memory-class: Memory-class(f;init;X),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
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],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
iterated_classrel: iterated_classrel(es;S;A;f;init;X;e;v),
sq_stable: SqStable(P),
cand: A c∧ B,
not: ¬A
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[init:Id {}\mrightarrow{} bag(B)].
\mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e:E.
\mforall{}[v:B]
(v \mmember{} State-comb(init;f;X)(e)
\mLeftarrow{}{}\mRightarrow{} if e \mmember{}\msubb{} X
then \mdownarrow{}\mexists{}w:B. \mexists{}a:A. (w \mmember{} Memory-class(f;init;X)(e) \mwedge{} (v = (f a w)) \mwedge{} a \mmember{} X(e))
else v \mmember{} Memory-class(f;init;X)(e)
fi )
Date html generated:
2016_05_17-AM-09_57_42
Last ObjectModification:
2016_01_17-PM-11_08_55
Theory : classrel!lemmas
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