Nuprl Lemma : lattice-extend-dl-inc
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f:T ⟶ Point(L)]. ∀[x:T].
(lattice-extend(L;eq;eqL;f;free-dl-inc(x)) = (f x) ∈ Point(L))
Proof
Definitions occuring in Statement :
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
free-dl-inc: free-dl-inc(x),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
free-dl-inc: free-dl-inc(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
top: Top,
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,
squash: ↓T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-image-singleton,
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,
deq_wf,
bdd-distributive-lattice_wf,
squash_wf,
true_wf,
lattice-fset-join-singleton,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-meet_wf,
decidable-equal-deq,
fset-singleton_wf,
iff_weakening_equal,
lattice-fset-meet-singleton
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,
axiomEquality,
functionEquality,
cumulativity,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:BoundedDistributiveLattice]. \mforall{}[eqL:EqDecider(Point(L))].
\mforall{}[f:T {}\mrightarrow{} Point(L)]. \mforall{}[x:T].
(lattice-extend(L;eq;eqL;f;free-dl-inc(x)) = (f x))
Date html generated:
2020_05_20-AM-08_45_38
Last ObjectModification:
2017_07_28-AM-09_14_33
Theory : lattices
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