Nuprl Lemma : free-dl-le
∀[T:Type]. ∀eq:EqDecider(T). ∀x,y:Point(free-dist-lattice(T; eq)). (x ≤ y ⇐⇒ fset-ac-le(eq;x;y))
Proof
Definitions occuring in Statement :
free-dist-lattice: free-dist-lattice(T; eq),
lattice-le: a ≤ b,
lattice-point: Point(l),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
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-le: a ≤ b,
lattice-meet: a ∧ b,
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
order: Order(T;x,y.R[x; y]),
refl: Refl(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
fset-ac-order,
fset-ac-glb_wf,
f-subset_wf,
iff_wf,
all_wf,
bool_wf,
deq-f-subset_wf,
bnot_wf,
fset-filter_wf,
fset-null_wf,
assert_witness,
iff_weakening_equal,
fset-antichain_wf,
assert_wf,
fset_wf,
true_wf,
squash_wf,
fset-ac-le_wf,
fset-ac-glb-is-glb,
rec_select_update_lemma,
free-dl-point,
deq_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
free-dist-lattice_wf,
lattice-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
because_Cache,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
productElimination,
independent_pairFormation,
introduction,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x,y:Point(free-dist-lattice(T; eq)). (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} fset-ac-le(eq;x;y))
Date html generated:
2020_05_20-AM-08_45_17
Last ObjectModification:
2016_01_17-PM-00_40_14
Theory : lattices
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