Nuprl Lemma : cut-order-iff1
∀[Info:Type]
∀es:EO+(Info). ∀X:EClass(Top). ∀f:sys-antecedent(es;X). ∀a,b:E(X).
(a ≤(X;f) b ⇐⇒ (a = b ∈ E(X)) ∨ ((f b < b) ∧ a ≤(X;f) f b) ∨ ((↑b ∈b prior(X)) ∧ a ≤(X;f) prior(X)(b)))
Proof
Definitions occuring in Statement :
cut-order: a ≤(X;f) b,
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-causl: (e < e'),
assert: ↑b,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
cut-order: a ≤(X;f) b,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
true: True,
prop: ℙ,
and: P ∧ Q,
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
uimplies: b supposing a,
top: Top,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
or: P ∨ Q,
cand: A c∧ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
es-cut: Cut(X;f),
label: ...$L... t,
es-causle: e c≤ e',
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info). \mforall{}X:EClass(Top). \mforall{}f:sys-antecedent(es;X). \mforall{}a,b:E(X).
(a \mleq{}(X;f) b
\mLeftarrow{}{}\mRightarrow{} (a = b) \mvee{} ((f b < b) \mwedge{} a \mleq{}(X;f) f b) \mvee{} ((\muparrow{}b \mmember{}\msubb{} prior(X)) \mwedge{} a \mleq{}(X;f) prior(X)(b)))
Date html generated:
2016_05_17-AM-07_42_19
Last ObjectModification:
2016_01_17-PM-02_59_11
Theory : event-ordering
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