Nuprl Lemma : state-class3-fun-eq
∀[Info,B,A1,A2,A3:Type]. ∀[init:Id ⟶ B]. ∀[tr1:Id ⟶ A1 ⟶ B ⟶ B]. ∀[tr2:Id ⟶ A2 ⟶ B ⟶ B].
∀[tr3:Id ⟶ A3 ⟶ B ⟶ B]. ∀[X1:EClass(A1)]. ∀[X2:EClass(A2)]. ∀[X3:EClass(A3)]. ∀[es:EO+(Info)]. ∀[e:E].
(state-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
= if e ∈b X1
then if first(e)
then tr1 loc(e) X1@e (init loc(e))
else tr1 loc(e) X1@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if e ∈b X2
then if first(e)
then tr2 loc(e) X2@e (init loc(e))
else tr2 loc(e) X2@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if e ∈b X3
then if first(e)
then tr3 loc(e) X3@e (init loc(e))
else tr3 loc(e) X3@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if first(e) then init loc(e)
else state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
∈ B) supposing
(disjoint-classrel(es;A1;X1;A2;X2) and
disjoint-classrel(es;A1;X1;A3;X3) and
disjoint-classrel(es;A2;X2;A3;X3) and
single-valued-classrel(es;X1;A1) and
single-valued-classrel(es;X2;A2) and
single-valued-classrel(es;X3;A3))
Proof
Definitions occuring in Statement :
state-class3: state-class3(init;tr1;X1;tr2;X2;tr3;X3),
classfun-res: X@e,
classfun: X(e),
single-valued-classrel: single-valued-classrel(es;X;T),
disjoint-classrel: disjoint-classrel(es;A;X;B;Y),
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 ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
state-class3: state-class3(init;tr1;X1;tr2;X2;tr3;X3),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
btrue: tt,
true: True,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bor: p ∨bq
Latex:
\mforall{}[Info,B,A1,A2,A3:Type]. \mforall{}[init:Id {}\mrightarrow{} B]. \mforall{}[tr1:Id {}\mrightarrow{} A1 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[tr2:Id {}\mrightarrow{} A2 {}\mrightarrow{} B {}\mrightarrow{} B].
\mforall{}[tr3:Id {}\mrightarrow{} A3 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X1:EClass(A1)]. \mforall{}[X2:EClass(A2)]. \mforall{}[X3:EClass(A3)]. \mforall{}[es:EO+(Info)].
\mforall{}[e:E].
(state-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
= if e \mmember{}\msubb{} X1
then if first(e)
then tr1 loc(e) X1@e (init loc(e))
else tr1 loc(e) X1@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if e \mmember{}\msubb{} X2
then if first(e)
then tr2 loc(e) X2@e (init loc(e))
else tr2 loc(e) X2@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if e \mmember{}\msubb{} X3
then if first(e)
then tr3 loc(e) X3@e (init loc(e))
else tr3 loc(e) X3@e state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
if first(e) then init loc(e)
else state-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi ) supposing
(disjoint-classrel(es;A1;X1;A2;X2) and
disjoint-classrel(es;A1;X1;A3;X3) and
disjoint-classrel(es;A2;X2;A3;X3) and
single-valued-classrel(es;X1;A1) and
single-valued-classrel(es;X2;A2) and
single-valued-classrel(es;X3;A3))
Date html generated:
2016_05_17-AM-11_18_24
Last ObjectModification:
2015_12_29-PM-05_23_00
Theory : process-model
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