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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
DeMorgan-algebra: DeMorganAlgebra,
prop: ℙ,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
exists: ∃x:A. B[x],
compose: f o g,
dminc: <i>,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
dmopp: <1-i>,
top: Top,
dma-neg: ¬(x),
cand: A c∧ B,
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
free-dist-lattice-property,
union-deq_wf,
DeMorgan-algebra-subtype,
dma-neg_wf,
equal_wf,
all_wf,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
dminc_wf,
deq_wf,
DeMorgan-algebra_wf,
free-dma-hom-is-lattice-hom,
free-dma-point,
free-dist-lattice-hom-unique,
rec_select_update_lemma,
dm-neg-properties,
lattice-0_wf,
free-DeMorgan-lattice_wf,
bdd-distributive-lattice_wf,
lattice-1_wf,
iff_weakening_equal,
DeMorgan-algebra-laws,
squash_wf,
true_wf,
subtype_rel_self,
dm-neg_wf,
dmopp_wf,
dm-neg-opp,
dm-neg-inc,
free-DeMorgan-algebra_wf,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
unionEquality,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
setElimination,
rename,
independent_functionElimination,
productElimination,
applyLambdaEquality,
inlEquality,
dependent_set_memberFormation,
cumulativity,
instantiate,
productEquality,
independent_isectElimination,
functionExtensionality,
functionEquality,
universeEquality,
inrEquality,
dependent_set_memberEquality,
hyp_replacement,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
isect_memberFormation,
independent_pairEquality,
axiomEquality
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:
2019_10_31-AM-07_22_53
Last ObjectModification:
2018_08_21-PM-02_02_30
Theory : lattices
Home
Index