Nuprl Lemma : Memory-class-trans2
∀[Info,B,A:Type].
∀R:B ⟶ B ⟶ ℙ. ∀f:A ⟶ B ⟶ B. ∀init:Id ⟶ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B.
(Trans(B;x,y.R[x;y])
⇒ (∀a:A. ∀e:E. (e1 ≤loc e ⇒ (e <loc e2) ⇒ a ∈ X(e) ⇒ (∀s:B. (s ∈ Memory-class(f;init;X)(e) ⇒ R[s;f a s]))))
⇒ single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);B)
⇒ v1 ∈ Memory-class(f;init;X)(e1)
⇒ v2 ∈ Memory-class(f;init;X)(e2)
⇒ (e1 <loc e2)
⇒ (∀e:E. (e1 ≤loc e ⇒ (e <loc e2) ⇒ (∀a:A. (¬a ∈ X(e)))))
⇒ (v1 = v2 ∈ B))
Proof
Definitions occuring in Statement :
Memory-class: Memory-class(f;init;X),
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
squash: ↓T,
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
or: P ∨ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
exists: ∃x:A. B[x],
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
es-locl: (e <loc e'),
true: True,
rev_implies: P ⇐ Q,
es-E: E,
es-base-E: es-base-E(es)
Latex:
\mforall{}[Info,B,A:Type].
\mforall{}R:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. \mforall{}f:A {}\mrightarrow{} B {}\mrightarrow{} B. \mforall{}init:Id {}\mrightarrow{} bag(B). \mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e1,e2:E.
\mforall{}v1,v2:B.
(Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e2)
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:B. (s \mmember{} Memory-class(f;init;X)(e) {}\mRightarrow{} R[s;f a s]))))
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);B)
{}\mRightarrow{} v1 \mmember{} Memory-class(f;init;X)(e1)
{}\mRightarrow{} v2 \mmember{} Memory-class(f;init;X)(e2)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} (\mforall{}e:E. (e1 \mleq{}loc e {}\mRightarrow{} (e <loc e2) {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e)))))
{}\mRightarrow{} (v1 = v2))
Date html generated:
2016_05_17-AM-09_24_43
Last ObjectModification:
2016_01_17-PM-11_12_08
Theory : classrel!lemmas
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