Nuprl Lemma : free-DeMorgan-algebra-property
∀T:Type. ∀eq:EqDecider(T). ∀dm:DeMorganAlgebra. ∀eq2:EqDecider(Point(dm)). ∀f:T ⟶ Point(dm).
(∃g:dma-hom(free-DeMorgan-algebra(T;eq);dm) [(∀i:T. ((g <i>) = (f i) ∈ Point(dm)))])
Proof
Definitions occuring in Statement :
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dma-hom: dma-hom(dma1;dma2),
DeMorgan-algebra: DeMorganAlgebra,
dminc: <i>,
lattice-point: Point(l),
deq: EqDecider(T),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
btrue: tt,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
free-dist-lattice: free-dist-lattice(T; eq),
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
record-select: r.x,
lattice-point: Point(l),
lattice-hom: Hom(l1;l2),
bounded-lattice-hom: Hom(l1;l2),
dma-hom: dma-hom(dma1;dma2),
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
sq_exists: ∃x:A [B[x]],
dminc: <i>,
compose: f o g,
exists: ∃x:A. B[x],
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
prop: ℙ,
DeMorgan-algebra: DeMorganAlgebra,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
dmopp: <1-i>,
top: Top,
dma-neg: ¬(x),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
bdd-distributive-lattice: BoundedDistributiveLattice,
cand: A c∧ B
Lemmas referenced :
DeMorgan-algebra_wf,
deq_wf,
dminc_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
lattice-point_wf,
all_wf,
equal_wf,
dma-neg_wf,
DeMorgan-algebra-subtype,
union-deq_wf,
free-dist-lattice-property,
free-dma-hom-is-lattice-hom,
free-dma-point,
free-dist-lattice-hom-unique,
rec_select_update_lemma,
subtype_rel_self,
true_wf,
squash_wf,
DeMorgan-algebra-laws,
iff_weakening_equal,
lattice-1_wf,
bdd-distributive-lattice_wf,
free-DeMorgan-lattice_wf,
lattice-0_wf,
dm-neg-properties,
dm-neg_wf,
istype-universe,
dmopp_wf,
dm-neg-opp,
dm-neg-inc,
subtype_rel-equal,
free-DeMorgan-algebra_wf
Rules used in proof :
universeEquality,
functionEquality,
functionExtensionality,
independent_isectElimination,
productEquality,
instantiate,
cumulativity,
dependent_set_memberFormation,
inlEquality,
applyLambdaEquality,
productElimination,
independent_functionElimination,
rename,
setElimination,
unionElimination,
because_Cache,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesis,
isectElimination,
hypothesisEquality,
unionEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrEquality,
dependent_set_memberEquality,
hyp_replacement,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
imageMemberEquality,
imageElimination,
natural_numberEquality,
independent_pairFormation,
axiomEquality,
independent_pairEquality,
isect_memberFormation,
lambdaEquality_alt,
isectEquality,
universeIsType
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}dm:DeMorganAlgebra. \mforall{}eq2:EqDecider(Point(dm)). \mforall{}f:T {}\mrightarrow{} Point(dm).
(\mexists{}g:dma-hom(free-DeMorgan-algebra(T;eq);dm) [(\mforall{}i:T. ((g <i>) = (f i)))])
Date html generated:
2020_05_20-AM-08_56_45
Last ObjectModification:
2020_02_04-PM-04_57_53
Theory : lattices
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