Nuprl Lemma : Memory-loc-classrel1
∀[Info,B,A:Type]. ∀[f:Id ⟶ A ⟶ B ⟶ B]. ∀[init:Id ⟶ bag(B)]. ∀[X:EClass(A)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ Memory-loc-class(f;init;X)(e);↓((↑first(e)) ∧ v ↓∈ init loc(e))
∨ ((¬↑first(e))
∧ ((∃a:A
(a ∈ X(pred(e))
∧ (∃b:B
(b ∈ Memory-loc-class(f;init;X)(pred(e))
∧ (v = (f loc(e) a b) ∈ B)))))
∨ ((∀a:A. (¬a ∈ X(pred(e)))) ∧ v ∈ Memory-loc-class(f;init;X)(pred(e))))))
Proof
Definitions occuring in Statement :
Memory-loc-class: Memory-loc-class(f;init;X),
classrel: v ∈ X(e),
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,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
squash: ↓T,
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
Memory-loc-class: Memory-loc-class(f;init;X),
primed-class-opt: Prior(X)?b,
es-local-pred: last(P),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
Id: Id,
es-E: E,
es-base-E: es-base-E(es),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
true: True,
eclass: EClass(A[eo; e]),
nat: ℕ
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[f:Id {}\mrightarrow{} A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[X:EClass(A)]. \mforall{}[es:EO+(Info)].
\mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} Memory-loc-class(f;init;X)(e);\mdownarrow{}((\muparrow{}first(e)) \mwedge{} v \mdownarrow{}\mmember{} init loc(e))
\mvee{} ((\mneg{}\muparrow{}first(e))
\mwedge{} ((\mexists{}a:A
(a \mmember{} X(pred(e))
\mwedge{} (\mexists{}b:B
(b \mmember{} Memory-loc-class(f;init;X)(pred(e))
\mwedge{} (v = (f loc(e) a b))))))
\mvee{} ((\mforall{}a:A. (\mneg{}a \mmember{} X(pred(e))))
\mwedge{} v \mmember{} Memory-loc-class(f;init;X)(pred(e))))))
Date html generated:
2016_05_17-AM-09_22_22
Last ObjectModification:
2016_01_17-PM-11_14_54
Theory : classrel!lemmas
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