Nuprl Lemma : free-dist-lattice-property
∀T:Type. ∀eq:EqDecider(T). ∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀f:T ⟶ Point(L).
∃g:Hom(free-dist-lattice(T; eq);L). (f = (g o (λx.free-dl-inc(x))) ∈ (T ⟶ Point(L)))
Proof
Definitions occuring in Statement :
free-dl-inc: free-dl-inc(x),
free-dist-lattice: free-dist-lattice(T; eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
deq: EqDecider(T),
compose: f o g,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
compose: f o g,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-extend-dl-inc,
iff_weakening_equal,
compose_wf,
free-dl-inc_wf,
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,
lattice-meet_wf,
lattice-join_wf,
deq_wf,
bdd-distributive-lattice_wf,
lattice-extend-is-hom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
functionExtensionality,
sqequalRule,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
functionEquality,
cumulativity,
instantiate,
productEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:BoundedDistributiveLattice. \mforall{}eqL:EqDecider(Point(L)).
\mforall{}f:T {}\mrightarrow{} Point(L).
\mexists{}g:Hom(free-dist-lattice(T; eq);L). (f = (g o (\mlambda{}x.free-dl-inc(x))))
Date html generated:
2017_10_05-AM-00_35_47
Last ObjectModification:
2017_07_28-AM-09_14_48
Theory : lattices
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