Nuprl Lemma : es-local-le-pred-property
∀[Info:Type]
∀P:es:EO+(Info) ⟶ E ⟶ 𝔹. ∀es:EO+(Info). ∀e:E.
((↑e ∈b ≤(P) ⇐⇒ ∃a:E. (a ≤loc e ∧ (↑(P es a))))
∧ ≤(P)(e) ≤loc e ∧ (↑(P es ≤(P)(e))) ∧ (∀e'':E. (e'' ≤loc e ⇒ (≤(P)(e) <loc e'') ⇒ (¬↑(P es e''))))
supposing ↑e ∈b ≤(P))
Proof
Definitions occuring in Statement :
es-local-le-pred: ≤(P),
eclass-val: X(e),
in-eclass: e ∈b X,
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-E: E,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
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],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
es-local-le-pred: ≤(P),
eclass-val: X(e),
in-eclass: e ∈b X,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
assert: ↑b,
cand: A c∧ B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
es-le: e ≤loc e' ,
sq_type: SQType(T)
Latex:
\mforall{}[Info:Type]
\mforall{}P:es:EO+(Info) {}\mrightarrow{} E {}\mrightarrow{} \mBbbB{}. \mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} \mleq{}(P) \mLeftarrow{}{}\mRightarrow{} \mexists{}a:E. (a \mleq{}loc e \mwedge{} (\muparrow{}(P es a))))
\mwedge{} \mleq{}(P)(e) \mleq{}loc e
\mwedge{} (\muparrow{}(P es \mleq{}(P)(e)))
\mwedge{} (\mforall{}e'':E. (e'' \mleq{}loc e {}\mRightarrow{} (\mleq{}(P)(e) <loc e'') {}\mRightarrow{} (\mneg{}\muparrow{}(P es e''))))
supposing \muparrow{}e \mmember{}\msubb{} \mleq{}(P))
Date html generated:
2016_05_16-PM-11_29_47
Last ObjectModification:
2016_01_17-PM-07_31_07
Theory : event-ordering
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