Nuprl Lemma : distinct-representatives
∀k:ℕ
∀[A:Type]
(A ~ ℕk
⇒ (∀[E:A ⟶ A ⟶ ℙ]
(EquivRel(A;x,y.E[x;y])
⇒ (∀x,y:A. Dec(E[x;y]))
⇒ (∃L:A List. ((∀a,b∈L. ¬E[a;b]) ∧ (∀a:A. (∃b∈L. E[a;b])) ∧ (||L|| ≤ k))))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
pairwise: (∀x,y∈L. P[x; y]),
l_exists: (∃x∈L. P[x]),
length: ||as||,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
ge: i ≥ j ,
equipollent: A ~ B,
it: ⋅,
nil: [],
list_ind: list_ind,
length: ||as||,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
cand: A c∧ B,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
nat: ℕ,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
refl: Refl(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
istype: istype(T),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
pairwise: (∀x,y∈L. P[x; y]),
less_than: a < b,
select: L[n],
cons: [a / b],
uiff: uiff(P;Q),
nat_plus: ℕ+,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
istype-nat,
int_term_value_add_lemma,
itermAdd_wf,
primrec-wf2,
le_wf,
l_member_wf,
l_exists_wf,
not_wf,
list_wf,
exists_wf,
all_wf,
subtype_rel_universe1,
uall_wf,
guard_wf,
equiv_rel_wf,
decidable_wf,
equipollent-partition,
nat_properties,
equipollent_inversion,
length_wf,
pairwise_wf2,
istype-false,
pairwise-nil,
nil_wf,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
equipollent_wf,
equipollent-zero,
subtype_rel_self,
istype-less_than,
istype-le,
decidable__lt,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
subtype_rel_dep_function,
subtype_rel_list,
cons_wf,
length_of_cons_lemma,
select-cons-tl,
select_wf,
false_wf,
add-is-int-iff,
nat_plus_properties,
length_wf_nat,
add_nat_plus,
l_exists_cons
Rules used in proof :
addEquality,
Error :setIsType,
productEquality,
functionEquality,
closedConclusion,
Error :functionIsType,
baseClosed,
imageMemberEquality,
universeEquality,
Error :inhabitedIsType,
imageElimination,
intEquality,
cumulativity,
Error :isect_memberFormation_alt,
hypothesis_subsumption,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
instantiate,
applyEquality,
unionElimination,
Error :universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
productElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setEquality,
Error :equalityIstype,
baseApply,
promote_hyp,
pointwiseFunctionality,
Error :inrFormation_alt
Latex:
\mforall{}k:\mBbbN{}
\mforall{}[A:Type]
(A \msim{} \mBbbN{}k
{}\mRightarrow{} (\mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]
(EquivRel(A;x,y.E[x;y])
{}\mRightarrow{} (\mforall{}x,y:A. Dec(E[x;y]))
{}\mRightarrow{} (\mexists{}L:A List. ((\mforall{}a,b\mmember{}L. \mneg{}E[a;b]) \mwedge{} (\mforall{}a:A. (\mexists{}b\mmember{}L. E[a;b])) \mwedge{} (||L|| \mleq{} k))))))
Date html generated:
2019_06_20-PM-02_17_59
Last ObjectModification:
2019_01_25-PM-02_44_36
Theory : equipollence!!cardinality!
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