Nuprl Lemma : quotient-dl_wf
∀[l:BoundedDistributiveLattice]. ∀[eq:Point(l) ⟶ Point(l) ⟶ ℙ].
(l//x,y.eq[x;y] ∈ BoundedDistributiveLattice) supposing
((∀a,c,b,d:Point(l). (eq[a;c] ⇒ eq[b;d] ⇒ eq[a ∨ b;c ∨ d])) and
(∀a,c,b,d:Point(l). (eq[a;c] ⇒ eq[b;d] ⇒ eq[a ∧ b;c ∧ d])) and
EquivRel(Point(l);x,y.eq[x;y]))
Proof
Definitions occuring in Statement :
quotient-dl: l//x,y.eq[x; y],
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
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],
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
all: ∀x:A. B[x],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T,
true: True,
lattice-join: a ∨ b,
quotient-dl: l//x,y.eq[x; y],
lattice-0: 0,
lattice-1: 1,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
lattice-axioms: lattice-axioms(l),
sq_stable: SqStable(P),
bounded-lattice-axioms: bounded-lattice-axioms(l)
Lemmas referenced :
all_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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
equiv_rel_wf,
bdd-distributive-lattice_wf,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
quotient_wf,
mk-bounded-distributive-lattice_wf,
lattice-0_wf,
subtype_quotient,
lattice-1_wf,
sq_stable__and,
sq_stable__uall,
sq_stable__equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
functionEquality,
functionExtensionality,
isect_memberEquality,
promote_hyp,
lambdaFormation,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
pointwiseFunctionality,
pertypeElimination,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
independent_pairEquality,
pointwiseFunctionalityForEquality
Latex:
\mforall{}[l:BoundedDistributiveLattice]. \mforall{}[eq:Point(l) {}\mrightarrow{} Point(l) {}\mrightarrow{} \mBbbP{}].
(l//x,y.eq[x;y] \mmember{} BoundedDistributiveLattice) supposing
((\mforall{}a,c,b,d:Point(l). (eq[a;c] {}\mRightarrow{} eq[b;d] {}\mRightarrow{} eq[a \mvee{} b;c \mvee{} d])) and
(\mforall{}a,c,b,d:Point(l). (eq[a;c] {}\mRightarrow{} eq[b;d] {}\mRightarrow{} eq[a \mwedge{} b;c \mwedge{} d])) and
EquivRel(Point(l);x,y.eq[x;y]))
Date html generated:
2020_05_20-AM-08_58_59
Last ObjectModification:
2017_07_28-AM-09_18_07
Theory : lattices
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