Nuprl Lemma : mk-DeMorgan-algebra_wf
∀[L:BoundedDistributiveLattice]. ∀[n:Point(L) ⟶ Point(L)].
mk-DeMorgan-algebra(L;n) ∈ DeMorganAlgebra
supposing (∀x:Point(L). ((n (n x)) = x ∈ Point(L)))
∧ ((∀x,y:Point(L). ((n x ∧ y) = n x ∨ n y ∈ Point(L))) ∨ (∀x,y:Point(L). ((n x ∨ y) = n x ∧ n y ∈ Point(L))))
Proof
Definitions occuring in Statement :
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
DeMorgan-algebra: DeMorganAlgebra,
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
or: P ∨ Q,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
bfalse: ff,
top: Top,
guard: {T},
sq_type: SQType(T),
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
record: record(x.T[x]),
record-update: r[x := v],
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
record-select: r.x,
record+: record+,
bounded-lattice-structure: BoundedLatticeStructure,
lattice-point: Point(l),
DeMorgan-algebra-structure: DeMorganAlgebraStructure,
false: False,
lattice-meet: a ∧ b,
lattice-join: a ∨ b,
lattice-1: 1,
lattice-0: 0,
true: True,
squash: ↓T,
cand: A c∧ B,
DeMorgan-algebra: DeMorganAlgebra,
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
dma-neg: ¬(x),
DeMorgan-algebra-axioms: DeMorgan-algebra-axioms(dma)
Lemmas referenced :
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
bdd-distributive-lattice_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
not_wf,
bnot_wf,
iff_transitivity,
rec_select_update_lemma,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
atom_subtype_base,
assert_wf,
bool_wf,
equal-wf-base,
uiff_transitivity,
eq_atom_wf,
subtype_rel_self,
istype-void,
istype-assert,
lattice-1_wf,
lattice-0_wf,
top-subtype-record,
DeMorgan-algebra-axioms_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bdd-distributive-lattice-subtype-distributive-lattice,
distributive-lattice-distrib,
iff_weakening_equal,
true_wf,
squash_wf,
implies-DeMorgan-algebra-axioms
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productIsType,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
inhabitedIsType,
because_Cache,
independent_isectElimination,
equalityIsType1,
unionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
rename,
setElimination,
impliesFunctionality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
atomEquality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
unionElimination,
lambdaFormation,
functionExtensionality,
dependentIntersection_memberEquality,
functionEquality,
universeEquality,
tokenEquality,
dependentIntersectionEqElimination,
dependentIntersectionElimination,
dependent_set_memberEquality_alt,
isectIsType,
lambdaFormation_alt,
equalityIsType4,
isect_memberFormation,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[L:BoundedDistributiveLattice]. \mforall{}[n:Point(L) {}\mrightarrow{} Point(L)].
mk-DeMorgan-algebra(L;n) \mmember{} DeMorganAlgebra
supposing (\mforall{}x:Point(L). ((n (n x)) = x))
\mwedge{} ((\mforall{}x,y:Point(L). ((n x \mwedge{} y) = n x \mvee{} n y)) \mvee{} (\mforall{}x,y:Point(L). ((n x \mvee{} y) = n x \mwedge{} n y)))
Date html generated:
2020_05_20-AM-08_55_52
Last ObjectModification:
2018_11_12-AM-09_22_11
Theory : lattices
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