Nuprl Lemma : free-dist-lattice-hom-unique
∀T:Type. ∀eq:EqDecider(T). ∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀f:T ⟶ Point(L).
∀g,h:Hom(free-dist-lattice(T; eq);L).
((f = (g o (λx.free-dl-inc(x))) ∈ (T ⟶ Point(L)))
⇒ (f = (h o (λx.free-dl-inc(x))) ∈ (T ⟶ Point(L)))
⇒ (g = h ∈ Hom(free-dist-lattice(T; eq);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],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
compose: f o g
Lemmas referenced :
free-dist-lattice-hom-unique2,
equal_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,
compose_wf,
free-dist-lattice_wf,
free-dl-inc_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
functionEquality,
cumulativity,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
setElimination,
rename,
equalityUniverse,
levelHypothesis,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:BoundedDistributiveLattice. \mforall{}eqL:EqDecider(Point(L)).
\mforall{}f:T {}\mrightarrow{} Point(L). \mforall{}g,h:Hom(free-dist-lattice(T; eq);L).
((f = (g o (\mlambda{}x.free-dl-inc(x)))) {}\mRightarrow{} (f = (h o (\mlambda{}x.free-dl-inc(x)))) {}\mRightarrow{} (g = h))
Date html generated:
2020_05_20-AM-08_46_55
Last ObjectModification:
2015_12_28-PM-01_59_45
Theory : lattices
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