Nuprl Lemma : lattice-extend-meet
∀[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) ∧ lattice-extend(L;eq;eqL;f;b) ≤ lattice-extend(L;eq;eqL;f;a ∧ 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-meet: 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,
top: Top,
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-extend': lattice-extend'(L;eq;eqL;f;ac),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
lattice-le: a ≤ b,
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
uiff: uiff(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
cand: A c∧ B,
lattice-fset-meet: /\(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
compose: f o g,
true: True,
bdd-lattice: BoundedLattice,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
free-dl-meet,
free-dl-point,
lattice-le_transitivity,
bdd-distributive-lattice-subtype-lattice,
lattice-meet_wf,
lattice-extend'_wf,
f-union_wf,
fset_wf,
deq-fset_wf,
fset-image_wf,
fset-union_wf,
fset-ac-glb_wf,
assert_wf,
fset-antichain_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-join_wf,
deq_wf,
bdd-distributive-lattice_wf,
fset-minimals-ac-le,
fset-minimals_wf,
f-proper-subset-dec_wf,
decidable-equal-deq,
fset-ac-le-implies2,
fset-ac-le_wf,
lattice-fset-join-is-lub,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-meet_wf,
lattice-fset-join_wf,
member-fset-image-iff,
fset-member_wf,
sq_stable_from_decidable,
lattice-le_wf,
lattice-fset-meet_functionality_wrt_subset,
fset-image_functionality_wrt_subset,
fset-image-compose,
squash_wf,
true_wf,
all_wf,
decidable_wf,
bdd-lattice_wf,
iff_weakening_equal,
fset-extensionality,
fset-member_witness,
exists_wf,
uiff_wf,
iff_weakening_uiff,
member-f-union,
fset-image-union,
lattice-le_weakening,
lattice-meet-join-images-distrib
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
because_Cache,
hypothesisEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
cumulativity,
lambdaEquality,
dependent_set_memberEquality,
setEquality,
independent_isectElimination,
axiomEquality,
instantiate,
productEquality,
universeEquality,
functionEquality,
lambdaFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
imageElimination,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
independent_pairFormation,
natural_numberEquality,
independent_pairEquality,
addLevel,
existsFunctionality,
andLevelFunctionality
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) \mwedge{} lattice-extend(L;eq;eqL;f;b) \mleq{} lattice-extend(L;eq;eqL;f;a \mwedge{} b)
Date html generated:
2020_05_20-AM-08_46_09
Last ObjectModification:
2017_07_28-AM-09_14_43
Theory : lattices
Home
Index