Nuprl Lemma : iterated_classrel_mem
∀[Info,A,S:Type]. ∀[init:Id ─→ bag(S)]. ∀[f:A ─→ S ─→ S]. ∀[X:EClass(A)].
∀es:EO+(Info). ∀R:A ─→ S ─→ S ─→ ℙ. ∀e1,e2:E. ∀v1,v2:S. ∀a:A.
(single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);S)
⇒ (∀s1,s2:S. ∀a:A. SqStable(R[a;s1;s2]))
⇒ (∀a1,a2:A. ∀s:S. ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ e' ≤loc e2
⇒ a1 ∈ X(e)
⇒ iterated_classrel(es;S;A;f;init;X;e;s)
⇒ a2 ∈ X(e')
⇒ R[a1;s;f a2 s]))
⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e1,e2:E.
((e1 <loc e2)
⇒ a1 ∈ X(e1)
⇒ iterated_classrel(es;S;A;f;init;X;e1;s1)
⇒ a2 ∈ X(e2)
⇒ iterated_classrel(es;S;A;f;init;X;pred(e2);s2)
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2]))
⇒ (e1 <loc e2)
⇒ a ∈ X(e1)
⇒ 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[a;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,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
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)
Lemmas :
es-pred_wf,
event-ordering+_subtype,
es-pred-locl,
es-causl_weakening,
es-locl_transitivity1,
es-le_weakening,
classrel_wf,
iterated_classrel_wf,
es-le_wf,
es-locl_wf,
es-E_wf,
exists_wf,
es-le-self,
all_wf,
not_wf,
decidable__exists-classrel-between3-sv,
iff_weakening_equal,
and_wf,
equal_wf,
member_wf,
es-le-pred,
iterated_classrel-single-val,
es-locl-trichotomy,
es-loc_wf,
es-loc-pred,
es-pred_property,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
es-causle_weakening_eq,
es-causle_wf,
es-causl_irreflexivity
\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{}R:A {}\mrightarrow{} S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}e1,e2:E. \mforall{}v1,v2:S. \mforall{}a:A.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (\mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} e' \mleq{}loc e2
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e;s)
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} R[a1;s;f a2 s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E.
((e1 <loc e2)
{}\mRightarrow{} a1 \mmember{} X(e1)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;s1)
{}\mRightarrow{} a2 \mmember{} X(e2)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e2);s2)
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} a \mmember{} X(e1)
{}\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[a;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:
2015_07_17-PM-00_25_18
Last ObjectModification:
2015_02_04-PM-05_37_46
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