Nuprl Lemma : mk-lattice_wf
∀[T:Type]. ∀[m,j:T ⟶ T ⟶ T].
mk-lattice(T;m;j) ∈ Lattice
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))
Proof
Definitions occuring in Statement :
mk-lattice: mk-lattice(T;m;j),
lattice: Lattice,
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-lattice: mk-lattice(T;m;j),
lattice: Lattice,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
lattice-structure: LatticeStructure,
record+: record+,
record-update: r[x := v],
record: record(x.T[x]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
record-select: r.x,
top: Top,
eq_atom: x =a y,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
lattice-axioms: lattice-axioms(l),
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
cand: A c∧ B
Lemmas referenced :
uall_wf,
equal_wf,
lattice-axioms_wf,
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
assert_wf,
atom_subtype_base,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
because_Cache,
applyEquality,
functionExtensionality,
isect_memberEquality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
dependentIntersection_memberEquality,
tokenEquality,
lambdaFormation,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
atomEquality,
independent_functionElimination,
independent_isectElimination,
instantiate,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[m,j:T {}\mrightarrow{} T {}\mrightarrow{} T].
mk-lattice(T;m;j) \mmember{} Lattice
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))
Date html generated:
2020_05_20-AM-08_23_56
Last ObjectModification:
2017_07_28-AM-09_12_34
Theory : lattices
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