Nuprl Lemma : mk-distributive-lattice_wf
∀[T:Type]. ∀[m,j:T ⟶ T ⟶ T].
mk-distributive-lattice(T; m; j) ∈ DistributiveLattice
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,b,c:T]. (m[a;j[b;c]] = j[m[a;b];m[a;c]] ∈ T))
Proof
Definitions occuring in Statement :
mk-distributive-lattice: mk-distributive-lattice(T; m; j),
distributive-lattice: DistributiveLattice,
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,
and: P ∧ Q,
mk-distributive-lattice: mk-distributive-lattice(T; m; j),
distributive-lattice: DistributiveLattice,
cand: A c∧ B,
subtype_rel: A ⊆r B,
mk-lattice: mk-lattice(T;m;j),
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],
record-select: r.x,
record-update: r[x := v],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
mk-lattice_wf,
rec_select_update_lemma,
subtype_rel_self,
lattice-point_wf,
and_wf,
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,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
lemma_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_pairFormation,
applyEquality,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[m,j:T {}\mrightarrow{} T {}\mrightarrow{} T].
mk-distributive-lattice(T; m; j) \mmember{} DistributiveLattice
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,b,c:T]. (m[a;j[b;c]] = j[m[a;b];m[a;c]]))
Date html generated:
2020_05_20-AM-08_25_02
Last ObjectModification:
2015_12_28-PM-02_03_53
Theory : lattices
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