Nuprl Lemma : iterated-classrel-invariant
∀[Info,A,S:Type]. ∀[init:Id ⟶ bag(S)]. ∀[f:A ⟶ S ⟶ S]. ∀[X:EClass(A)]. ∀[P:S ⟶ ℙ].
∀es:EO+(Info). ∀e:E. ∀v:S.
((∀s:S. (s ↓∈ init loc(e) ⇒ P[s]))
⇒ (∀a:A. ∀e':E. ∀s:S.
(e' ≤loc e ⇒ a ∈ X(e') ⇒ prior-iterated-classrel(es;A;S;s;X;f;init;e') ⇒ P[s] ⇒ P[f a s]))
⇒ iterated-classrel(es;S;A;f;init;X;e;v)
⇒ P[v])
Proof
Definitions occuring in Statement :
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: 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,
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],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P)
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[init:Id {}\mrightarrow{} bag(S)]. \mforall{}[f:A {}\mrightarrow{} S {}\mrightarrow{} S]. \mforall{}[X:EClass(A)]. \mforall{}[P:S {}\mrightarrow{} \mBbbP{}].
\mforall{}es:EO+(Info). \mforall{}e:E. \mforall{}v:S.
((\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{} prior-iterated-classrel(es;A;S;s;X;f;init;e')
{}\mRightarrow{} P[s]
{}\mRightarrow{} P[f a s]))
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e;v)
{}\mRightarrow{} P[v])
Date html generated:
2016_05_16-PM-01_57_14
Last ObjectModification:
2016_01_17-PM-07_42_10
Theory : event-ordering
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