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)
Lemmas :
es-pred_wf,
event-ordering+_subtype,
es-pred-locl,
es-causl_weakening,
es-locl_transitivity1,
es-le_weakening,
iterated-classrel_wf,
assert_elim,
es-first_wf2,
es-locl-first,
btrue_neq_bfalse,
assert_wf,
classrel_wf,
es-le_wf,
es-locl_wf,
es-E_wf,
exists_wf,
all_wf,
not_wf,
es-le-pred,
es-le-self,
and_wf,
equal_wf,
member_wf,
decidable__exists-classrel-between3-sv,
iterated-classrel-single-val,
squash_wf,
true_wf,
event-ordering+_wf,
eclass_wf,
Id_wf,
bag_wf,
iff_weakening_equal,
es-pred_property,
es-loc_wf,
es-loc-pred,
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{}[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:
2015_07_17-PM-00_28_13
Last ObjectModification:
2015_02_04-PM-05_34_27
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