Nuprl Lemma : memory-classrel3-sv
∀[Info,A1,A2,A3,S:Type].
∀init:Id ⟶ S. ∀tr1:Id ⟶ A1 ⟶ S ⟶ S. ∀X1:EClass(A1). ∀tr2:Id ⟶ A2 ⟶ S ⟶ S. ∀X2:EClass(A2). ∀tr3:Id
⟶ A3
⟶ S
⟶ S.
∀X3:EClass(A3). ∀es:EO+(Info). ∀e:E. ∀v:S.
(disjoint-classrel(es;A2;X2;A3;X3)
⇒ disjoint-classrel(es;A3;X3;A1;X1)
⇒ disjoint-classrel(es;A2;X2;A1;X1)
⇒ single-valued-classrel(es;X3;A3)
⇒ single-valued-classrel(es;X2;A2)
⇒ single-valued-classrel(es;X1;A1)
⇒ (v ∈ memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
⇐⇒ prior-iterated-classrel(es;A1 + A2 + A3;S;v;X1 (+) X2 (+) X3;tr1 + tr2 + tr3 loc(e);λloc.{init loc};e)))
Proof
Definitions occuring in Statement :
disjoint-union-tr: tr1 + tr2,
disjoint-union-comb: X (+) Y,
memory-class3: memory-class3(init;tr1;X1;tr2;X2;tr3;X3),
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
single-valued-classrel: single-valued-classrel(es;X;T),
disjoint-classrel: disjoint-classrel(es;A;X;B;Y),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-loc: loc(e),
es-E: E,
Id: Id,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
single-bag: {x}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
Memory3: Memory3,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
squash: ↓T,
guard: {T}
Latex:
\mforall{}[Info,A1,A2,A3,S:Type].
\mforall{}init:Id {}\mrightarrow{} S. \mforall{}tr1:Id {}\mrightarrow{} A1 {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X1:EClass(A1). \mforall{}tr2:Id {}\mrightarrow{} A2 {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X2:EClass(A2).
\mforall{}tr3:Id {}\mrightarrow{} A3 {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X3:EClass(A3). \mforall{}es:EO+(Info). \mforall{}e:E. \mforall{}v:S.
(disjoint-classrel(es;A2;X2;A3;X3)
{}\mRightarrow{} disjoint-classrel(es;A3;X3;A1;X1)
{}\mRightarrow{} disjoint-classrel(es;A2;X2;A1;X1)
{}\mRightarrow{} single-valued-classrel(es;X3;A3)
{}\mRightarrow{} single-valued-classrel(es;X2;A2)
{}\mRightarrow{} single-valued-classrel(es;X1;A1)
{}\mRightarrow{} (v \mmember{} memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
\mLeftarrow{}{}\mRightarrow{} prior-iterated-classrel(es;A1 + A2 + A3;S;v;X1 (+) X2 (+) X3;tr1 + tr2 + tr3
loc(e);\mlambda{}loc.\{init loc\};e)))
Date html generated:
2016_05_17-AM-10_06_30
Last ObjectModification:
2016_01_17-PM-11_05_19
Theory : classrel!lemmas
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