Nuprl Lemma : lattice-fset-join_wf
∀[l:BoundedLattice]. ((∀x,y:Point(l). Dec(x = y ∈ Point(l))) ⇒ (∀[s:fset(Point(l))]. (\/(s) ∈ Point(l))))
Proof
Definitions occuring in Statement :
lattice-fset-join: \/(s),
bdd-lattice: BoundedLattice,
lattice-point: Point(l),
fset: fset(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fset: fset(T),
quotient: x,y:A//B[x; y],
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
lattice-fset-join: \/(s),
bfalse: ff,
deq: EqDecider(T),
squash: ↓T,
true: True,
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
eqof: eqof(d),
bnot: ¬bb,
lattice: Lattice,
l_member: (x ∈ l),
select: L[n],
cand: A c∧ B,
nat_plus: ℕ+
Lemmas referenced :
fset_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,
decidable_wf,
equal_wf,
bdd-lattice_wf,
deq-exists,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
non_neg_length,
set-equal_wf,
length_wf,
list_wf,
subtract-1-ge-0,
istype-nat,
add_nat_wf,
length_wf_nat,
istype-void,
istype-le,
decidable__le,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable__lt,
list-cases,
product_subtype_list,
set-equal-nil,
null_nil_lemma,
length_of_nil_lemma,
reduce_nil_lemma,
lattice-0_wf,
null_cons_lemma,
set-equal-cons2,
filter_wf5,
bnot_wf,
l_member_wf,
less_than_wf,
squash_wf,
true_wf,
length-filter-bnot,
subtract_wf,
subtype_rel_self,
iff_weakening_equal,
length_of_cons_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
colength-cons-not-zero,
colength_wf_list,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
le_wf,
reduce_cons_lemma,
filter_cons_lemma,
cons_wf,
filter_nil_lemma,
lattice-join_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
reduce_wf,
istype-universe,
lattice_wf,
lattice-join-idempotent,
lattice_properties,
cons_member,
bool_wf,
add_nat_plus,
nat_plus_properties,
select_wf,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
independent_isectElimination,
functionIsType,
because_Cache,
dependent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
independent_functionElimination,
rename,
promote_hyp,
pointwiseFunctionalityForEquality,
pertypeElimination,
setElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
dependent_set_memberEquality_alt,
addEquality,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
equalityIstype,
productIsType,
sqequalBase,
hypothesis_subsumption,
setIsType,
imageElimination,
imageMemberEquality,
universeEquality,
intEquality,
equalityElimination,
hyp_replacement
Latex:
\mforall{}[l:BoundedLattice]. ((\mforall{}x,y:Point(l). Dec(x = y)) {}\mRightarrow{} (\mforall{}[s:fset(Point(l))]. (\mbackslash{}/(s) \mmember{} Point(l))))
Date html generated:
2020_05_20-AM-08_43_40
Last ObjectModification:
2020_01_06-PM-00_12_04
Theory : lattices
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