Nuprl Lemma : iterated_classrel_progress
∀[Info,A,S:Type]. ∀[init:Id ─→ bag(S)]. ∀[f:A ─→ S ─→ S]. ∀[X:EClass(A)].
∀es:EO+(Info). ∀R:S ─→ S ─→ ℙ. ∀P:A ─→ S ─→ ℙ. ∀e1,e2:E. ∀v1,v2:S.
(single-valued-classrel(es;X;A)
⇒ (∀a:A. ∀s:S. Dec(P[a;s]))
⇒ (∀x,y:S. SqStable(R[x;y]))
⇒ Trans(S;x,y.R[x;y])
⇒ (∀a:A. ∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc e2
⇒ a ∈ X(e)
⇒ iterated_classrel(es;S;A;f;init;X;pred(e);s)
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ S)))))
⇒ single-valued-bag(init loc(e1);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
∃s:S
((e1 <loc e) ∧ e ≤loc e2 ∧ iterated_classrel(es;S;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[v1;v2])
∧ ((∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc e2
⇒ iterated_classrel(es;S;A;f;init;X;pred(e);s)
⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ (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]),
sq_stable: SqStable(P),
decidable: Dec(P),
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,
or: 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,
not_wf,
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,
or_wf,
es-le-pred,
es-le-self,
and_wf,
member_wf,
equal_wf,
decidable__exists-iterated-classrel-between3-sv,
squash_wf,
event_ordering_wf,
iterated_classrel-single-val,
es-loc-pred,
Id_wf,
bag_wf,
single-valued-bag_wf,
true_wf,
event-ordering+_wf,
eclass_wf,
iff_weakening_equal,
es-pred_property,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
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:S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}P:A {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}e1,e2:E. \mforall{}v1,v2:S.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}a:A. \mforall{}s:S. Dec(P[a;s]))
{}\mRightarrow{} (\mforall{}x,y:S. SqStable(R[x;y]))
{}\mRightarrow{} Trans(S;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s))))))
{}\mRightarrow{} single-valued-bag(init loc(e1);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
\mexists{}s:S
((e1 <loc e)
\mwedge{} e \mleq{}loc e2
\mwedge{} iterated\_classrel(es;S;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = v2))))
Date html generated:
2015_07_17-PM-00_25_01
Last ObjectModification:
2015_02_04-PM-05_39_19
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