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:
2020_05_20-AM-08_45_54
Last ObjectModification:
2017_07_28-AM-09_14_38
Theory : lattices
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