Nuprl Lemma : f-lattice_wf
∀[X:Type]. (f-lattice(X) ∈ BoundedDistributiveLattice)
Proof
Definitions occuring in Statement :
f-lattice: f-lattice(X),
bdd-distributive-lattice: BoundedDistributiveLattice,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
lattice-point: Point(l),
record-select: r.x,
free-dl: free-dl(X),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
free-dl-type: free-dl-type(X),
quotient: x,y:A//B[x; y],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
f-lattice: f-lattice(X),
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
flattice-equiv: flattice-equiv(X;x;y),
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
free-dl-meet: free-dl-meet(as;bs),
list_accum: list_accum,
lattice-meet: a ∧ b,
true: True,
append: as @ bs,
list_ind: list_ind,
lattice-join: a ∨ b,
free-dl-join: free-dl-join(as;bs)
Lemmas referenced :
subtype_quotient,
list_wf,
dlattice-eq_wf,
dlattice-eq-equiv,
flattice-equiv-equiv,
quotient-dl_wf,
free-dl_wf,
flattice-equiv_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,
free-dl-meet_wf_list,
flattice-order_wf,
exists_wf,
squash_wf,
true_wf,
flattice-order-meet,
append_wf,
flattice-order-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
independent_isectElimination,
because_Cache,
applyEquality,
instantiate,
productEquality,
universeEquality,
lambdaFormation,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]. (f-lattice(X) \mmember{} BoundedDistributiveLattice)
Date html generated:
2020_05_20-AM-09_00_03
Last ObjectModification:
2017_07_28-AM-09_18_30
Theory : lattices
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