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
Lemmas :
loop-class-state-fun-eq,
single-bag_wf,
parallel-class_wf,
eclass1_wf,
bag_size_single_lemma,
assert_of_le_int,
parallel-class-single-val,
eclass1-disjoint-classrel,
disjoint-classrel-symm,
parallel-class-disjoint-classrel,
eclass1-single-val,
single-valued-bag-single,
member-eclass_wf,
bool_wf,
eqtt_to_assert,
es-first_wf2,
es-loc_wf,
classfun-res_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
classfun_wf,
state-class3_wf,
loop-class-state-functional,
false_wf,
es-pred_wf,
iff_weakening_equal,
sv_bag_only_single_lemma,
disjoint-classrel_wf,
single-valued-classrel_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
eclass_wf,
Id_wf,
member-parallel-class-bool,
member-eclass-eclass1,
classfun-res-parallel-class-left,
classfun-res-eclass1,
classfun-res-parallel-class-right,
assert_of_bor,
assert_wf
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:
2015_07_23-AM-11_30_27
Last ObjectModification:
2015_02_04-PM-04_45_59
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