Nuprl Lemma : iterated-classrel-as-prior
∀[Info,A,S:Type]. ∀[init:Id ─→ bag(S)]. ∀[f:A ─→ S ─→ S]. ∀[X:EClass(A)].
∀es:EO+(Info). ∀e:E.
∀[v:S]
(iterated-classrel(es;S;A;f;init;X;e;v)
⇐⇒ ∃z:S
(prior-iterated-classrel(es;A;S;z;X;f;init;e)
∧ ((∃a:A. (a ∈ X(e) ∧ (v = (f a z) ∈ S))) ∨ ((∀a:A. (¬a ∈ X(e))) ∧ (v = z ∈ S)))))
Proof
Definitions occuring in Statement :
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
iterated-classrel: iterated-classrel(es;S;A;f;init;X;e;v),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Lemmas :
iterated-classrel_wf,
exists_wf,
prior-iterated-classrel_wf,
or_wf,
classrel_wf,
all_wf,
not_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
eclass_wf,
Id_wf,
bag_wf,
es-first_wf2,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_wf,
es-pred_wf,
eqff_to_assert,
assert_of_bnot,
bag-member_wf,
es-loc_wf,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
equal_wf,
bool_cases_sqequal,
assert-bnot
\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{}e:E.
\mforall{}[v:S]
(iterated-classrel(es;S;A;f;init;X;e;v)
\mLeftarrow{}{}\mRightarrow{} \mexists{}z:S
(prior-iterated-classrel(es;A;S;z;X;f;init;e)
\mwedge{} ((\mexists{}a:A. (a \mmember{} X(e) \mwedge{} (v = (f a z)))) \mvee{} ((\mforall{}a:A. (\mneg{}a \mmember{} X(e))) \mwedge{} (v = z)))))
Date html generated:
2015_07_17-PM-00_27_42
Last ObjectModification:
2015_01_27-PM-11_34_48
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