Nuprl Lemma : free-dma-hom-is-lattice-hom
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[dm:BoundedDistributiveLattice].
(Hom(free-DeMorgan-lattice(T;eq);dm) = Hom(free-DeMorgan-algebra(T;eq);dm) ∈ Type)
Proof
Definitions occuring in Statement :
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
subtype_rel: A ⊆r B,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
bounded-lattice-structure: BoundedLatticeStructure,
record+: record+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
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,
lattice-point: Point(l),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
lattice-meet: a ∧ b,
lattice-join: a ∨ b,
lattice-1: 1,
lattice-0: 0,
record: record(x.T[x]),
record-update: r[x := v]
Lemmas referenced :
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
deq_wf,
istype-universe,
free-DeMorgan-lattice_wf,
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
atom_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
istype-void,
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,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
lattice-1_wf,
lattice-0_wf,
top_wf,
subtype_rel_self,
top-subtype-record
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependentIntersectionEqElimination,
dependentIntersection_memberEquality,
functionExtensionality,
tokenEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
atomEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
cumulativity,
dependent_functionElimination,
voidElimination,
productEquality,
independent_pairFormation,
equalityIsType4,
functionIsType,
equalityIsType1,
voidEquality,
isect_memberEquality,
lambdaFormation,
impliesFunctionality,
functionEquality,
dependentIntersectionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[dm:BoundedDistributiveLattice].
(Hom(free-DeMorgan-lattice(T;eq);dm) = Hom(free-DeMorgan-algebra(T;eq);dm))
Date html generated:
2020_05_20-AM-08_56_37
Last ObjectModification:
2018_11_10-PM-00_07_10
Theory : lattices
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