Nuprl Lemma : State-loc-comb-invariant-sv2
∀[Info,A,S:Type].
∀es:EO+(Info). ∀P:E ⟶ S ⟶ ℙ. ∀init:Id ⟶ bag(S). ∀f:Id ⟶ A ⟶ S ⟶ S. ∀X:EClass(A). ∀e:E. ∀v:S.
(single-valued-bag(init loc(e);S)
⇒ single-valued-classrel(es;X;A)
⇒ (∀s:S. ∀e':E.
(e' ≤loc e
⇒ if first(e') then s ↓∈ init loc(e') else s ∈ State-loc-comb(init;f;X)(pred(e')) ∧ P[pred(e');s] fi
⇒ if e' ∈b X then ∀a:A. (a ∈ X(e') ⇒ P[e';f loc(e') a s]) else P[e';s] fi ))
⇒ v ∈ State-loc-comb(init;f;X)(e)
⇒ P[e;v])
Proof
Definitions occuring in Statement :
State-loc-comb: State-loc-comb(init;f;X),
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
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 ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: 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,
subtype_rel: A ⊆r B,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
not: ¬A,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
sq_stable: SqStable(P)
Latex:
\mforall{}[Info,A,S:Type].
\mforall{}es:EO+(Info). \mforall{}P:E {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}init:Id {}\mrightarrow{} bag(S). \mforall{}f:Id {}\mrightarrow{} A {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X:EClass(A). \mforall{}e:E. \mforall{}v:S.
(single-valued-bag(init loc(e);S)
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}s:S. \mforall{}e':E.
(e' \mleq{}loc e
{}\mRightarrow{} if first(e')
then s \mdownarrow{}\mmember{} init loc(e')
else s \mmember{} State-loc-comb(init;f;X)(pred(e')) \mwedge{} P[pred(e');s]
fi
{}\mRightarrow{} if e' \mmember{}\msubb{} X then \mforall{}a:A. (a \mmember{} X(e') {}\mRightarrow{} P[e';f loc(e') a s]) else P[e';s] fi ))
{}\mRightarrow{} v \mmember{} State-loc-comb(init;f;X)(e)
{}\mRightarrow{} P[e;v])
Date html generated:
2016_05_17-AM-10_03_30
Last ObjectModification:
2016_01_17-PM-11_05_05
Theory : classrel!lemmas
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