Nuprl Lemma : iterated-classrel-invariant2
∀[Info,A,S:Type]. ∀[init:Id ⟶ bag(S)]. ∀[f:A ⟶ S ⟶ S].
∀X:EClass(A). ∀es:EO+(Info).
∀[P:E ⟶ S ⟶ ℙ]
∀e:E. ∀v:S.
((∀s:S. ∀e':E.
(e' ≤loc e
⇒ if first(e')
then s ↓∈ init loc(e')
else iterated-classrel(es;S;A;f;init;X;pred(e');s) ∧ P[pred(e');s]
fi
⇒ if e' ∈b X then ∀a:A. (a ∈ X(e') ⇒ P[e';f a s]) else P[e';s] fi ))
⇒ iterated-classrel(es;S;A;f;init;X;e;v)
⇒ P[e;v])
Proof
Definitions occuring in Statement :
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
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,
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,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
sq_stable: SqStable(P),
cand: A c∧ B
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[init:Id {}\mrightarrow{} bag(S)]. \mforall{}[f:A {}\mrightarrow{} S {}\mrightarrow{} S].
\mforall{}X:EClass(A). \mforall{}es:EO+(Info).
\mforall{}[P:E {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}]
\mforall{}e:E. \mforall{}v:S.
((\mforall{}s:S. \mforall{}e':E.
(e' \mleq{}loc e
{}\mRightarrow{} if first(e')
then s \mdownarrow{}\mmember{} init loc(e')
else iterated-classrel(es;S;A;f;init;X;pred(e');s) \mwedge{} P[pred(e');s]
fi
{}\mRightarrow{} if e' \mmember{}\msubb{} X then \mforall{}a:A. (a \mmember{} X(e') {}\mRightarrow{} P[e';f a s]) else P[e';s] fi ))
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e;v)
{}\mRightarrow{} P[e;v])
Date html generated:
2016_05_16-PM-01_57_30
Last ObjectModification:
2016_01_17-PM-07_44_22
Theory : event-ordering
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