Nuprl Lemma : dM-hom-basis
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. ∀[l:BoundedLattice].
∀eq:EqDecider(Point(l)). ∀[h:Hom(dM(I);l)]. ((h x) = \/(λs./\(λx.(h free-dl-inc(x))"(s))"(x)) ∈ Point(l))
Proof
Definitions occuring in Statement :
dM: dM(I),
names-deq: NamesDeq,
names: names(I),
free-dl-inc: free-dl-inc(x),
lattice-fset-join: \/(s),
lattice-fset-meet: /\(s),
bounded-lattice-hom: Hom(l1;l2),
bdd-lattice: BoundedLattice,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
prop: ℙ,
squash: ↓T,
top: Top,
true: True,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
bdd-lattice: BoundedLattice,
and: P ∧ Q,
guard: {T},
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dml-deq: free-dml-deq(T;eq),
lattice-point: Point(l),
record-select: r.x,
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),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
compose: f o g
Lemmas referenced :
dM-basis,
equal_wf,
squash_wf,
true_wf,
dM-point,
bounded-lattice-hom_wf,
dM_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
DeMorgan-algebra-structure-subtype,
deq_wf,
lattice-point_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
bdd-lattice_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_transitivity,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
fset_wf,
nat_wf,
free-DeMorgan-lattice_wf,
names_wf,
names-deq_wf,
bdd-distributive-lattice_wf,
mk-DeMorgan-algebra-equal-bounded-lattice,
lattice-hom-fset-join,
bdd-distributive-lattice-subtype-bdd-lattice,
free-dml-deq_wf,
lattice-hom-fset-meet,
deq-implies,
iff_weakening_equal,
lattice-fset-join_wf,
fset-image_wf,
deq-fset_wf,
union-deq_wf,
lattice-fset-meet_wf,
free-dl-inc_wf,
all_wf,
decidable_wf,
fset-image-compose,
assert_wf,
fset-antichain_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
subtype_rel-equal,
free-dl-point
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
applyLambdaEquality,
applyEquality,
setElimination,
rename,
hyp_replacement,
equalitySymmetry,
sqequalRule,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productEquality,
cumulativity,
dependent_functionElimination,
axiomEquality,
independent_functionElimination,
productElimination,
unionEquality,
setEquality,
functionEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[x:Point(dM(I))]. \mforall{}[l:BoundedLattice].
\mforall{}eq:EqDecider(Point(l)). \mforall{}[h:Hom(dM(I);l)]. ((h x) = \mbackslash{}/(\mlambda{}s./\mbackslash{}(\mlambda{}x.(h free-dl-inc(x))"(s))"(x)))
Date html generated:
2017_10_05-AM-01_00_18
Last ObjectModification:
2017_07_28-AM-09_25_42
Theory : cubical!type!theory
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