Nuprl Lemma : mk-bounded-distributive-lattice_wf
∀[T:Type]. ∀[m,j:T ⟶ T ⟶ T]. ∀[z,o:T].
{points=T;
meet=m;
join=j;
0=z;
1=o} ∈ BoundedDistributiveLattice
supposing (∀[a,b:T]. (m[a;b] = m[b;a] ∈ T))
∧ (∀[a,b:T]. (j[a;b] = j[b;a] ∈ T))
∧ (∀[a,b,c:T]. (m[a;m[b;c]] = m[m[a;b];c] ∈ T))
∧ (∀[a,b,c:T]. (j[a;j[b;c]] = j[j[a;b];c] ∈ T))
∧ (∀[a,b:T]. (j[a;m[a;b]] = a ∈ T))
∧ (∀[a,b:T]. (m[a;j[a;b]] = a ∈ T))
∧ (∀[a:T]. (m[a;o] = a ∈ T))
∧ (∀[a:T]. (j[a;z] = a ∈ T))
∧ (∀[a,b,c:T]. (m[a;j[b;c]] = j[m[a;b];m[a;c]] ∈ T))
Proof
Definitions occuring in Statement :
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
and: P ∧ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
cand: A c∧ B,
subtype_rel: A ⊆r B,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
lattice-axioms: lattice-axioms(l),
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
all: ∀x:A. B[x],
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
so_apply: x[s1;s2],
bounded-lattice-axioms: bounded-lattice-axioms(l),
lattice-1: 1,
lattice-0: 0,
record-select: r.x,
record-update: r[x := v],
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
mk-bounded-lattice_wf,
rec_select_update_lemma,
subtype_rel_self,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
and_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
lemma_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
independent_pairFormation,
applyEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
instantiate,
lambdaEquality,
cumulativity,
axiomEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[m,j:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[z,o:T].
\{points=T;
meet=m;
join=j;
0=z;
1=o\} \mmember{} BoundedDistributiveLattice
supposing (\mforall{}[a,b:T]. (m[a;b] = m[b;a]))
\mwedge{} (\mforall{}[a,b:T]. (j[a;b] = j[b;a]))
\mwedge{} (\mforall{}[a,b,c:T]. (m[a;m[b;c]] = m[m[a;b];c]))
\mwedge{} (\mforall{}[a,b,c:T]. (j[a;j[b;c]] = j[j[a;b];c]))
\mwedge{} (\mforall{}[a,b:T]. (j[a;m[a;b]] = a))
\mwedge{} (\mforall{}[a,b:T]. (m[a;j[a;b]] = a))
\mwedge{} (\mforall{}[a:T]. (m[a;o] = a))
\mwedge{} (\mforall{}[a:T]. (j[a;z] = a))
\mwedge{} (\mforall{}[a,b,c:T]. (m[a;j[b;c]] = j[m[a;b];m[a;c]]))
Date html generated:
2020_05_20-AM-08_25_13
Last ObjectModification:
2015_12_28-PM-02_04_08
Theory : lattices
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