Nuprl Lemma : fset-constrained-ac-lub-is-lub
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹]. ∀[ac1,ac2:{ac:fset(fset(T))|
(↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} ].
least-upper-bound({ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} ;ac1,ac2.fset-ac-le(eq;ac1;ac2);
ac1;ac2;lub(P;ac1;ac2))
Proof
Definitions occuring in Statement :
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset-all: fset-all(s;x.P[x]),
fset: fset(T),
deq: EqDecider(T),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-constrained-ac-lub: lub(P;ac1;ac2)
Lemmas referenced :
fset-ac-lub-is-lub-constrained
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
hypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:fset(T) {}\mrightarrow{} \mBbbB{}]. \mforall{}[ac1,ac2:\{ac:fset(fset(T))|
(\muparrow{}fset-antichain(eq;ac))
\mwedge{} fset-all(ac;a.P[a])\} ].
least-upper-bound(\{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\} ;
ac1,ac2.fset-ac-le(eq;ac1;ac2);ac1;ac2;lub(P;ac1;ac2))
Date html generated:
2016_05_14-PM-03_49_11
Last ObjectModification:
2016_01_15-PM-03_27_58
Theory : finite!sets
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