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
Lemmas :
bool_wf,
eqtt_to_assert,
bag-member-single,
es-loc_wf,
squash_wf,
bag-member_wf,
single-bag_wf,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
classrel_wf,
loop-class-memory_wf,
parallel-class_wf,
eclass1_wf,
Id_wf,
es-first_wf2,
event-ordering+_subtype,
exists_wf,
es-pred_wf,
member-eclass_wf,
loop-class-memory-classrel,
bor_wf,
iff_transitivity,
assert_wf,
or_wf,
assert_of_bor,
uiff_wf,
bool_cases,
member-parallel-class-bool,
assert_of_bnot,
bnot_wf,
not_wf,
member-eclass-eclass1,
parallel-classrel,
eclass1-classrel,
iff_weakening_equal,
es-pred-loc-base,
es-E_wf,
not_assert_elim,
assert_elim,
and_wf,
btrue_neq_bfalse,
false_wf
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:
2015_07_23-AM-11_30_49
Last ObjectModification:
2015_02_04-PM-04_49_51
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