Nuprl Lemma : lattice-extend_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f:T ⟶ Point(L)].
∀[ac:Point(free-dist-lattice(T; eq))].
(lattice-extend(L;eq;eqL;f;ac) ∈ Point(L))
Proof
Definitions occuring in Statement :
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
free-dist-lattice: free-dist-lattice(T; eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
free-dl-point,
lattice-fset-join_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
decidable-equal-deq,
fset-image_wf,
fset_wf,
deq-fset_wf,
lattice-fset-meet_wf,
lattice-point_wf,
free-dist-lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
deq_wf,
bdd-distributive-lattice_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
hypothesisEquality,
applyEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_functionElimination,
lambdaEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:BoundedDistributiveLattice]. \mforall{}[eqL:EqDecider(Point(L))].
\mforall{}[f:T {}\mrightarrow{} Point(L)]. \mforall{}[ac:Point(free-dist-lattice(T; eq))].
(lattice-extend(L;eq;eqL;f;ac) \mmember{} Point(L))
Date html generated:
2020_05_20-AM-08_45_32
Last ObjectModification:
2015_12_28-PM-02_00_10
Theory : lattices
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