Nuprl Lemma : memory-class3-classrel
∀[Info,A1,A2,A3,B:Type]. ∀[init:Id ⟶ B]. ∀[tr1:Id ⟶ A1 ⟶ B ⟶ B]. ∀[X1:EClass(A1)]. ∀[tr2:Id ⟶ A2 ⟶ B ⟶ B].
∀[X2:EClass(A2)]. ∀[tr3:Id ⟶ A3 ⟶ B ⟶ B]. ∀[X3:EClass(A3)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e);↓if first(e)
then v = (init loc(e)) ∈ B
else ∃b:B
(b ∈ memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
∧ if pred(e) ∈b X1 ∨bpred(e) ∈b X2 ∨bpred(e) ∈b X3
then (∃a1:A1
(a1 ∈ X1(pred(e)) ∧ (v = (tr1 loc(e) a1 b) ∈ B)))
∨ (∃a2:A2
(a2 ∈ X2(pred(e))
∧ (v = (tr2 loc(e) a2 b) ∈ B)))
∨ (∃a3:A3
(a3 ∈ X3(pred(e))
∧ (v = (tr3 loc(e) a3 b) ∈ B)))
else v = b ∈ B
fi )
fi )
Proof
Definitions occuring in Statement :
memory-class3: memory-class3(init;tr1;X1;tr2;X2;tr3;X3),
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,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_apply: x[s],
bor: p ∨bq,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
memory-class3: memory-class3(init;tr1;X1;tr2;X2;tr3;X3),
not: ¬A,
sq_or: a ↓∨ b,
cand: A c∧ B,
es-E: E,
es-base-E: es-base-E(es),
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[Info,A1,A2,A3,B:Type]. \mforall{}[init:Id {}\mrightarrow{} B]. \mforall{}[tr1:Id {}\mrightarrow{} A1 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X1:EClass(A1)]. \mforall{}[tr2:Id
{}\mrightarrow{} A2
{}\mrightarrow{} B
{}\mrightarrow{} B].
\mforall{}[X2:EClass(A2)]. \mforall{}[tr3:Id {}\mrightarrow{} A3 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X3:EClass(A3)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e);\mdownarrow{}if first(e)
then v = (init loc(e))
else \mexists{}b:B
(b \mmember{} memory-class3(init;tr1;X1;tr2;X2;
tr3;X3)(pred(e))
\mwedge{} if pred(e) \mmember{}\msubb{} X1
\mvee{}\msubb{}pred(e) \mmember{}\msubb{} X2
\mvee{}\msubb{}pred(e) \mmember{}\msubb{} X3
then (\mexists{}a1:A1
(a1 \mmember{} X1(pred(e))
\mwedge{} (v = (tr1 loc(e) a1 b))))
\mvee{} (\mexists{}a2:A2
(a2 \mmember{} X2(pred(e))
\mwedge{} (v = (tr2 loc(e) a2 b))))
\mvee{} (\mexists{}a3:A3
(a3 \mmember{} X3(pred(e))
\mwedge{} (v = (tr3 loc(e) a3 b))))
else v = b
fi )
fi )
Date html generated:
2016_05_17-AM-11_19_01
Last ObjectModification:
2016_01_18-AM-00_24_18
Theory : process-model
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