Nuprl Lemma : fdl-is-1_wf
∀[X:Type]. ∀[x:Point(free-dl(X))]. (fdl-is-1(x) ∈ 𝔹)
Proof
Definitions occuring in Statement :
fdl-is-1: fdl-is-1(x),
free-dl: free-dl(X),
lattice-point: Point(l),
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
dlattice-eq: dlattice-eq(X;as;bs),
dlattice-order: as ⇒ bs,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
nil: [],
it: ⋅,
assert: ↑b,
cons: [a / b],
false: False,
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
fdl-is-1: fdl-is-1(x)
Lemmas referenced :
dlattice-eq-equiv,
list_wf,
dlattice-eq_wf,
bool_wf,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
lattice-point_wf,
free-dl_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,
assert-bl-exists,
isaxiom_wf_list,
l_member_wf,
assert_wf,
bl-exists_wf,
l_exists_wf,
dlattice-order_wf,
l_all_iff,
l_contains_wf,
l_exists_iff,
exists_wf,
all_wf,
list-cases,
product_subtype_list,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
length_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
select_wf,
cons_wf,
non_neg_length,
intformle_wf,
int_formula_prop_le_lemma,
btrue_neq_bfalse,
iff_imp_equal_bool
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
cumulativity,
hypothesis,
promote_hyp,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_functionElimination,
pointwiseFunctionality,
pertypeElimination,
productElimination,
independent_functionElimination,
productEquality,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
setElimination,
rename,
setEquality,
functionEquality,
allFunctionality,
levelHypothesis,
dependent_pairFormation,
unionElimination,
hypothesis_subsumption,
voidElimination,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
applyLambdaEquality,
baseApply,
closedConclusion,
int_eqEquality,
intEquality,
computeAll,
addEquality
Latex:
\mforall{}[X:Type]. \mforall{}[x:Point(free-dl(X))]. (fdl-is-1(x) \mmember{} \mBbbB{})
Date html generated:
2020_05_20-AM-08_42_42
Last ObjectModification:
2017_07_28-AM-09_13_37
Theory : lattices
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