Nuprl Lemma : State-loc-comb-invariant-sv1
∀[Info,A,S:Type]. ∀[P:S ⟶ ℙ].
∀init:Id ⟶ bag(S). ∀f:Id ⟶ A ⟶ S ⟶ S. ∀X:EClass(A). ∀es:EO+(Info). ∀e:E. ∀v:S.
(single-valued-bag(init loc(e);S)
⇒ single-valued-classrel(es;X;A)
⇒ (∀s:S. (s ↓∈ init loc(e) ⇒ P[s]))
⇒ (∀a:A. ∀e':E. ∀s:S.
(e' ≤loc e
⇒ a ∈ X(e')
⇒ if first(e') then s ↓∈ init loc(e') else s ∈ State-loc-comb(init;f;X)(pred(e')) fi
⇒ P[s]
⇒ P[f loc(e') a s]))
⇒ v ∈ State-loc-comb(init;f;X)(e)
⇒ P[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),
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[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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
or: P ∨ Q,
and: P ∧ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
so_apply: x[s],
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[P:S {}\mrightarrow{} \mBbbP{}].
\mforall{}init:Id {}\mrightarrow{} bag(S). \mforall{}f:Id {}\mrightarrow{} A {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e:E. \mforall{}v:S.
(single-valued-bag(init loc(e);S)
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}s:S. (s \mdownarrow{}\mmember{} init loc(e) {}\mRightarrow{} P[s]))
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e':E. \mforall{}s:S.
(e' \mleq{}loc e
{}\mRightarrow{} a \mmember{} X(e')
{}\mRightarrow{} if first(e') then s \mdownarrow{}\mmember{} init loc(e') else s \mmember{} State-loc-comb(init;f;X)(pred(e')) fi
{}\mRightarrow{} P[s]
{}\mRightarrow{} P[f loc(e') a s]))
{}\mRightarrow{} v \mmember{} State-loc-comb(init;f;X)(e)
{}\mRightarrow{} P[v])
Date html generated:
2016_05_17-AM-10_03_21
Last ObjectModification:
2016_01_17-PM-11_04_16
Theory : classrel!lemmas
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