Nuprl Lemma : iterated_classrel_mem2
∀[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]))
⇒ (∀a:A. ∀s:S. ∀e:E. (e1 ≤loc e ⇒ e ≤loc e2 ⇒ a ∈ X(e) ⇒ s ∈ prior(X*(f,init,e)) ⇒ R[a;s;f a s]))
⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ e' ≤loc e2
⇒ a1 ∈ X(e)
⇒ s1 ∈ prior(X*(f,init,e))
⇒ a2 ∈ X(e')
⇒ iterated_classrel(es;S;A;f;init;X;pred(e');s2)
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2]))
⇒ e1 ≤loc e2
⇒ a ∈ X(e1)
⇒ v1 ∈ prior(X*(f,init,e1))
⇒ iterated_classrel(es;S;A;f;init;X;e2;v2)
⇒ R[a;v1;v2])
Proof
Definitions occuring in Statement :
prior_iterated_classrel: s ∈ prior(X*(f,init,e)),
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],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
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],
member: t ∈ T,
subtype_rel: A ⊆r B,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
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_stable: SqStable(P),
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[s1;s2;s3],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prior_iterated_classrel: s ∈ prior(X*(f,init,e)),
single-valued-bag: single-valued-bag(b;T),
single-valued-classrel: single-valued-classrel(es;X;T),
cand: A c∧ B,
es-le: e ≤loc e' ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
es-E: E,
es-base-E: es-base-E(es)
Latex:
\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{}a:A. \mforall{}s:S. \mforall{}e:E.
(e1 \mleq{}loc e {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} s \mmember{} prior(X*(f,init,e)) {}\mRightarrow{} R[a;s;f a s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} e' \mleq{}loc e2
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} s1 \mmember{} prior(X*(f,init,e))
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e');s2)
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} e1 \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e1)
{}\mRightarrow{} v1 \mmember{} prior(X*(f,init,e1))
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e2;v2)
{}\mRightarrow{} R[a;v1;v2])
Date html generated:
2016_05_16-PM-01_53_30
Last ObjectModification:
2016_01_17-PM-07_44_37
Theory : event-ordering
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