Nuprl Lemma : cut-of-singleton
∀[Info:Type]. ∀[es:EO+(Info)]. ∀[X:EClass(Top)]. ∀[f:sys-antecedent(es;X)]. ∀[e:E(X)].
(cut(X;f;{e})
= if e ∈b prior(X) then if f e = e then {e} else {e} ⋃ cut(X;f;{f e}) fi ⋃ cut(X;f;{prior(X)(e)})
if f e = e then {e}
else {e} ⋃ cut(X;f;{f e})
fi
∈ Cut(X;f))
Proof
Definitions occuring in Statement :
cut-of: cut(X;f;s),
es-cut: Cut(X;f),
es-prior-interface: prior(X),
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-eq-E: e = e',
es-eq: es-eq(es),
fset-singleton: {x},
fset-union: x ⋃ y,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
es-cut: Cut(X;f),
sys-antecedent: sys-antecedent(es;Sys),
prop: ℙ,
es-E-interface: E(X),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
top: Top,
es-cut-add: c+e,
f-subset: xs ⊆ ys,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
true: True,
es-E: E,
or: P ∨ Q,
cand: A c∧ B,
squash: ↓T,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
fset-singleton: {x},
cons: [a / b],
sq_stable: SqStable(P)
Latex:
\mforall{}[Info:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[X:EClass(Top)]. \mforall{}[f:sys-antecedent(es;X)]. \mforall{}[e:E(X)].
(cut(X;f;\{e\})
= if e \mmember{}\msubb{} prior(X) then if f e = e then \{e\} else \{e\} \mcup{} cut(X;f;\{f e\}) fi \mcup{} cut(X;f;\{prior(X)(e)\})
if f e = e then \{e\}
else \{e\} \mcup{} cut(X;f;\{f e\})
fi )
Date html generated:
2016_05_17-AM-07_41_51
Last ObjectModification:
2016_01_17-PM-03_09_17
Theory : event-ordering
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