Nuprl Lemma : fdl-eq-1
∀[X:Type]. ∀x:Point(free-dl(X)). (x = 1 ∈ Point(free-dl(X)) ⇐⇒ ↑fdl-is-1(x))
Proof
Definitions occuring in Statement :
fdl-is-1: fdl-is-1(x),
free-dl: free-dl(X),
lattice-1: 1,
lattice-point: Point(l),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
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],
member: t ∈ T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
lattice-1: 1,
cons: [a / b],
fdl-is-1: fdl-is-1(x),
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
assert: ↑b,
true: True,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A,
dlattice-eq: dlattice-eq(X;as;bs),
dlattice-order: as ⇒ bs,
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),
less_than: a < b,
squash: ↓T,
select: L[n],
l_exists: (∃x∈L. P[x]),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
nil: [],
l_contains: A ⊆ B,
nat_plus: ℕ+,
ge: i ≥ j
Lemmas referenced :
free-dl-type_wf,
iff_wf,
assert_wf,
equal-wf-base,
list_wf,
dlattice-eq_wf,
fdl-is-1_wf,
member_wf,
equal-wf-T-base,
subtype_quotient,
dlattice-eq-equiv,
quotient-member-eq,
cons_wf,
nil_wf,
bl-exists_wf,
isaxiom_wf_list,
l_member_wf,
bool_wf,
eqtt_to_assert,
assert-bl-exists,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_exists_wf,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list-cases,
l_contains_wf,
product_subtype_list,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
non_neg_length,
btrue_neq_bfalse,
l_all_iff,
l_contains_nil,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
pointwiseFunctionalityForEquality,
because_Cache,
pertypeElimination,
productElimination,
productEquality,
independent_pairEquality,
lambdaEquality,
baseClosed,
applyEquality,
independent_isectElimination,
axiomEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
rename,
setEquality,
unionElimination,
equalityElimination,
natural_numberEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
hypothesis_subsumption,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality
Latex:
\mforall{}[X:Type]. \mforall{}x:Point(free-dl(X)). (x = 1 \mLeftarrow{}{}\mRightarrow{} \muparrow{}fdl-is-1(x))
Date html generated:
2020_05_20-AM-08_42_47
Last ObjectModification:
2017_07_28-AM-09_13_39
Theory : lattices
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