Nuprl Lemma : lattice-extend-dlwc-inc
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))].
∀[f:T ⟶ Point(L)].
∀[x:T]. (lattice-extend-wc(L;eq;eqL;f;free-dlwc-inc(eq;a.Cs[a];x)) = (f x) ∈ Point(L))
supposing ∀x:T. ∀c:fset(T). (c ∈ Cs[x] ⇒ (/\(f"(c)) = 0 ∈ Point(L)))
Proof
Definitions occuring in Statement :
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
lattice-fset-meet: /\(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-0: 0,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bdd-distributive-lattice: BoundedDistributiveLattice,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
top: Top,
lattice-fset-join: \/(s),
empty-fset: {},
not: ¬A,
decidable: Dec(P),
lattice-fset-meet: /\(s)
Lemmas referenced :
bdd-distributive-lattice_wf,
deq_wf,
lattice-0_wf,
fset-image_wf,
decidable-equal-deq,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-meet_wf,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
deq-fset_wf,
fset-member_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
eqtt_to_assert,
fset-singleton_wf,
assert_wf,
f-subset_wf,
iff_wf,
all_wf,
bool_wf,
deq-f-subset_wf,
fset-filter_wf,
fset_wf,
fset-null_wf,
lattice-fset-meet-singleton,
iff_weakening_equal,
lattice-fset-join-singleton,
true_wf,
squash_wf,
fset-image-singleton,
reduce_nil_lemma,
fset-image-empty,
assert-deq-f-subset,
exists_wf,
not_wf,
fset-filter-is-empty,
equal-wf-T-base,
assert-fset-null,
f-subset-singleton,
lattice-0-equal-lattice-1-implies
Rules used in proof :
universeEquality,
productEquality,
axiomEquality,
isect_memberEquality,
voidElimination,
because_Cache,
independent_functionElimination,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation,
functionExtensionality,
functionEquality,
setEquality,
rename,
setElimination,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
voidEquality,
existsLevelFunctionality,
andLevelFunctionality,
independent_pairFormation,
existsFunctionality,
impliesFunctionality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]. \mforall{}[L:BoundedDistributiveLattice].
\mforall{}[eqL:EqDecider(Point(L))]. \mforall{}[f:T {}\mrightarrow{} Point(L)].
\mforall{}[x:T]. (lattice-extend-wc(L;eq;eqL;f;free-dlwc-inc(eq;a.Cs[a];x)) = (f x))
supposing \mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (/\mbackslash{}(f"(c)) = 0))
Date html generated:
2020_05_20-AM-08_49_06
Last ObjectModification:
2020_02_03-PM-03_02_04
Theory : lattices
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