Nuprl Lemma : decidable__exists-pred-iterated-classrel-between3-sv
∀[Info,T,S:Type].
∀init:Id ⟶ bag(S). ∀f:T ⟶ S ⟶ S. ∀X:EClass(T). ∀es:EO+(Info). ∀P:T ⟶ S ⟶ ℙ. ∀e1,e2:E.
(single-valued-classrel(es;X;T)
⇒ single-valued-bag(init loc(e1);S)
⇒ (∀v:T. ∀s:S. Dec(P[v;s]))
⇒ Dec(∃e:E
∃s:S
((e1 <loc e) ∧ e ≤loc e2 ∧ iterated-classrel(es;S;T;f;init;X;pred(e);s) ∧ (∃v:T. (v ∈ X(e) ∧ P[v;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,
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],
implies: P ⇒ Q,
and: 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],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
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,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
so_lambda: λ2x y.t[x; y],
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
es-E: E,
es-base-E: es-base-E(es),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
es-locl: (e <loc e'),
es-le: e ≤loc e' ,
single-valued-classrel: single-valued-classrel(es;X;T),
sq_type: SQType(T),
assert: ↑b,
bfalse: ff
Latex:
\mforall{}[Info,T,S:Type].
\mforall{}init:Id {}\mrightarrow{} bag(S). \mforall{}f:T {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X:EClass(T). \mforall{}es:EO+(Info). \mforall{}P:T {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}e1,e2:E.
(single-valued-classrel(es;X;T)
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (\mforall{}v:T. \mforall{}s:S. Dec(P[v;s]))
{}\mRightarrow{} Dec(\mexists{}e:E
\mexists{}s:S
((e1 <loc e)
\mwedge{} e \mleq{}loc e2
\mwedge{} iterated-classrel(es;S;T;f;init;X;pred(e);s)
\mwedge{} (\mexists{}v:T. (v \mmember{} X(e) \mwedge{} P[v;s])))))
Date html generated:
2016_05_16-PM-01_56_43
Last ObjectModification:
2016_01_17-PM-07_49_08
Theory : event-ordering
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