Nuprl Lemma : lattice-extend-join
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f:T ⟶ Point(L)].
∀[a,b:Point(free-dist-lattice(T; eq))].
  lattice-extend(L;eq;eqL;f;a ∨ b) ≤ lattice-extend(L;eq;eqL;f;a) ∨ lattice-extend(L;eq;eqL;f;b)
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-le: a ≤ b
, 
lattice-join: a ∨ b
, 
lattice-point: Point(l)
, 
deq: EqDecider(T)
, 
uall: ∀[x:A]. B[x]
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
lattice-le: a ≤ b
, 
subtype_rel: A ⊆r B
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
and: P ∧ Q
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
top: Top
, 
lattice-extend: lattice-extend(L;eq;eqL;f;ac)
, 
lattice-extend': lattice-extend'(L;eq;eqL;f;ac)
, 
squash: ↓T
, 
implies: P 
⇒ Q
, 
bdd-lattice: BoundedLattice
, 
all: ∀x:A. B[x]
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
fset-ac-lub: fset-ac-lub(eq;ac1;ac2)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
f-subset: xs ⊆ ys
, 
uiff: uiff(P;Q)
Lemmas referenced : 
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, 
free-dl-join, 
free-dl-point, 
lattice-le_transitivity, 
bdd-distributive-lattice-subtype-lattice, 
lattice-extend'_wf, 
fset-ac-lub_wf, 
assert_wf, 
fset-antichain_wf, 
fset_wf, 
fset-union_wf, 
deq-fset_wf, 
lattice-le_wf, 
squash_wf, 
true_wf, 
lattice-fset-join_wf, 
all_wf, 
decidable_wf, 
bdd-lattice_wf, 
bdd-distributive-lattice-subtype-bdd-lattice, 
fset-image-union, 
lattice-fset-meet_wf, 
decidable-equal-deq, 
fset-image_wf, 
iff_weakening_equal, 
lattice-fset-join-union, 
lattice-le_weakening, 
lattice-fset-join_functionality_wrt_subset, 
fset-minimals_wf, 
f-proper-subset-dec_wf, 
fset-image_functionality_wrt_subset, 
member-fset-minimals, 
fset-member_witness, 
fset-member_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalRule, 
sqequalHypSubstitution, 
axiomEquality, 
hypothesis, 
extract_by_obid, 
isectElimination, 
thin, 
cumulativity, 
hypothesisEquality, 
applyEquality, 
instantiate, 
lambdaEquality, 
productEquality, 
universeEquality, 
because_Cache, 
independent_isectElimination, 
isect_memberEquality, 
functionEquality, 
voidElimination, 
voidEquality, 
functionExtensionality, 
setElimination, 
rename, 
dependent_set_memberEquality, 
setEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
lambdaFormation, 
dependent_functionElimination, 
imageMemberEquality, 
baseClosed, 
natural_numberEquality, 
productElimination
Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[L:BoundedDistributiveLattice].  \mforall{}[eqL:EqDecider(Point(L))].
\mforall{}[f:T  {}\mrightarrow{}  Point(L)].  \mforall{}[a,b:Point(free-dist-lattice(T;  eq))].
    lattice-extend(L;eq;eqL;f;a  \mvee{}  b)  \mleq{}  lattice-extend(L;eq;eqL;f;a)  \mvee{}  lattice-extend(L;eq;eqL;f;b)
Date html generated:
2017_10_05-AM-00_35_18
Last ObjectModification:
2017_07_28-AM-09_14_38
Theory : lattices
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