Nuprl Lemma : Memory-class-invariant-sv
∀[Info,B,A:Type].
∀P:B ⟶ ℙ. ∀f:A ⟶ B ⟶ B. ∀init:Id ⟶ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e:E. ∀v:B.
(single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e);B)
⇒ (∀a:A. ∀e':E. ((e' <loc e) ⇒ a ∈ X(e') ⇒ (∀s:B. (P[s] ⇒ P[f a s]))))
⇒ (∀v:B. (v ↓∈ init loc(e) ⇒ P[v]))
⇒ v ∈ Memory-class(f;init;X)(e)
⇒ P[v])
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-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
sq_stable: SqStable(P),
not: ¬A,
false: False,
uimplies: b supposing a,
prop: ℙ,
guard: {T},
so_apply: x[s],
es-E: E,
es-base-E: es-base-E(es),
true: True,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,B,A:Type].
\mforall{}P: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{}e:E. \mforall{}v:B.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e);B)
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e':E. ((e' <loc e) {}\mRightarrow{} a \mmember{} X(e') {}\mRightarrow{} (\mforall{}s:B. (P[s] {}\mRightarrow{} P[f a s]))))
{}\mRightarrow{} (\mforall{}v:B. (v \mdownarrow{}\mmember{} init loc(e) {}\mRightarrow{} P[v]))
{}\mRightarrow{} v \mmember{} Memory-class(f;init;X)(e)
{}\mRightarrow{} P[v])
Date html generated:
2016_05_17-AM-09_24_04
Last ObjectModification:
2016_01_17-PM-11_10_38
Theory : classrel!lemmas
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