Nuprl Lemma : fl-filter-decomp
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:Point(face-lattice(T;eq))].
∀[Q:{x:fset(T + T)| ↑fset-contains-none(union-deq(T;T;eq;eq);x;x.face-lattice-constraints(x))} ⟶ 𝔹].
(s = fl-filter(s;x.Q[x]) ∨ fl-filter(s;x.¬bQ[x]) ∈ Point(face-lattice(T;eq)))
Proof
Definitions occuring in Statement :
fl-filter: fl-filter(s;x.Q[x]),
face-lattice: face-lattice(T;eq),
face-lattice-constraints: face-lattice-constraints(x),
lattice-join: a ∨ b,
lattice-point: Point(l),
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
fl-filter: fl-filter(s;x.Q[x])
Lemmas referenced :
deq_wf,
fset_wf,
face-lattice-constraints_wf,
fset-contains-none_wf,
union-deq_wf,
cal-filter-decomp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:Point(face-lattice(T;eq))].
\mforall{}[Q:\{x:fset(T + T)| \muparrow{}fset-contains-none(union-deq(T;T;eq;eq);x;x.face-lattice-constraints(x))\}
{}\mrightarrow{} \mBbbB{}].
(s = fl-filter(s;x.Q[x]) \mvee{} fl-filter(s;x.\mneg{}\msubb{}Q[x]))
Date html generated:
2020_05_20-AM-08_52_23
Last ObjectModification:
2016_01_18-PM-07_59_45
Theory : lattices
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