Nuprl Lemma : iterated-classrel-trans
∀[Info,A,S:Type]. ∀[init:Id ⟶ bag(S)]. ∀[f:A ⟶ S ⟶ S]. ∀[R:S ⟶ S ⟶ ℙ].
∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:S.
(single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);S)
⇒ Trans(S;x,y.R[x;y])
⇒ (∀a:A. ∀e:E.
((e1 <loc e)
⇒ e ≤loc e2
⇒ a ∈ X(e)
⇒ (∀s:S. (iterated-classrel(es;S;A;f;init;X;pred(e);s) ⇒ R[s;f a s]))))
⇒ (e1 <loc e2)
⇒ iterated-classrel(es;S;A;f;init;X;e1;v1)
⇒ iterated-classrel(es;S;A;f;init;X;e2;v2)
⇒ (((∃e:E. ((e1 <loc e) ∧ e ≤loc e2 ∧ (∃a:A. a ∈ X(e)))) ⇒ R[v1;v2])
∧ ((∀e:E. ((e1 <loc e) ⇒ e ≤loc e2 ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ (v1 = v2 ∈ S))))
Proof
Definitions occuring in Statement :
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
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-locl: (e <loc e'),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
single-valued-bag: single-valued-bag(b;T),
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},
nat: ℕ,
prop: ℙ,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
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,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
es-locl: (e <loc e'),
so_apply: x[s1;s2],
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
es-le: e ≤loc e' ,
trans: Trans(T;x,y.E[x; y]),
true: True,
label: ...$L... t
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[init:Id {}\mrightarrow{} bag(S)]. \mforall{}[f:A {}\mrightarrow{} S {}\mrightarrow{} S]. \mforall{}[R:S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}].
\mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e1,e2:E. \mforall{}v1,v2:S.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} Trans(S;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:S. (iterated-classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} R[s;f a s]))))
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e1;v1)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e2;v2)
{}\mRightarrow{} (((\mexists{}e:E. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))) {}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e))))) {}\mRightarrow{} (v1 = v2))))
Date html generated:
2016_05_16-PM-01_57_53
Last ObjectModification:
2016_01_17-PM-07_45_49
Theory : event-ordering
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