Nuprl Lemma : free-dist-lattice-with-constraints-property
∀[T:Type]
∀eq:EqDecider(T)
∀[Cs:T ⟶ fset(fset(T))]
∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀f:T ⟶ Point(L).
∃g:Hom(free-dist-lattice-with-constraints(T;eq;x.Cs[x]);L)
(f = (g o (λx.free-dlwc-inc(eq;a.Cs[a];x))) ∈ (T ⟶ Point(L)))
supposing ∀x:T. ∀c:fset(T). (c ∈ Cs[x] ⇒ (/\(f"(c)) = 0 ∈ Point(L)))
Proof
Definitions occuring in Statement :
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-fset-meet: /\(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
compose: f o g,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice
Lemmas referenced :
lattice-extend-is-hom-constrained,
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
lattice-extend-dlwc-inc,
subtype_rel_self,
iff_weakening_equal,
compose_wf,
free-dist-lattice-with-constraints_wf,
free-dlwc-inc_wf,
fset_wf,
fset-member_wf,
deq-fset_wf,
lattice-fset-meet_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
decidable-equal-deq,
fset-image_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
bounded-lattice-structure-subtype,
lattice-0_wf,
deq_wf,
bdd-distributive-lattice_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
dependent_pairFormation_alt,
extract_by_obid,
isectElimination,
applyEquality,
universeIsType,
independent_isectElimination,
functionExtensionality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
independent_functionElimination,
equalityIstype,
setElimination,
functionIsType,
productEquality,
cumulativity,
isectEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T)
\mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]
\mforall{}L:BoundedDistributiveLattice. \mforall{}eqL:EqDecider(Point(L)). \mforall{}f:T {}\mrightarrow{} Point(L).
\mexists{}g:Hom(free-dist-lattice-with-constraints(T;eq;x.Cs[x]);L)
(f = (g o (\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x))))
supposing \mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (/\mbackslash{}(f"(c)) = 0))
Date html generated:
2020_05_20-AM-08_50_28
Last ObjectModification:
2020_01_03-PM-09_26_08
Theory : lattices
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