Nuprl Lemma : count-by-decidable-equiv
∀[A:Type]. ∀[E:A ⟶ A ⟶ ℙ].
(EquivRel(A;x,y.E[x;y])
⇒ (∀x,y:A. Dec(E[x;y]))
⇒ (∀k:ℕ
(A ~ ℕk
⇒ (∃L:A List
((∀a,b∈L. ¬E[a;b])
∧ (∀a:A. (∃b∈L. E[a;b]))
∧ (∃f:ℕ||L|| ⟶ ℕ. ((∀i:ℕ||L||. {a:A| E[a;L[i]]} ~ ℕf i) ∧ A ~ i:ℕ||L|| × ℕf i ∧ (k = Σ(f i | i < ||L||) ∈\000C ℤ))))))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
pairwise: (∀x,y∈L. P[x; y]),
sum: Σ(f[x] | x < k),
l_exists: (∃x∈L. P[x]),
select: L[n],
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],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
distinct-representatives,
pairwise_wf2,
not_wf,
all_wf,
l_exists_wf,
l_member_wf,
exists_wf,
int_seg_wf,
length_wf,
nat_wf,
equipollent_wf,
select_wf,
int_seg_properties,
nat_properties,
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,
equal_wf,
sum_wf,
length_wf_nat,
decidable_wf,
equiv_rel_wf,
equipollent-partition,
count-by-equiv,
id-biject,
product_functionality_wrt_equipollent_dependent,
equipollent_transitivity,
equipollent_functionality_wrt_equipollent2,
equipollent-sum,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-nsub,
non_neg_sum,
non_neg_length,
lelt_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_functionElimination,
hypothesis,
productElimination,
dependent_pairFormation,
independent_pairFormation,
sqequalRule,
productEquality,
instantiate,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
universeEquality,
because_Cache,
setElimination,
rename,
setEquality,
functionEquality,
natural_numberEquality,
independent_isectElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;x,y.E[x;y])
{}\mRightarrow{} (\mforall{}x,y:A. Dec(E[x;y]))
{}\mRightarrow{} (\mforall{}k:\mBbbN{}
(A \msim{} \mBbbN{}k
{}\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{} (\mexists{}f:\mBbbN{}||L|| {}\mrightarrow{} \mBbbN{}
((\mforall{}i:\mBbbN{}||L||. \{a:A| E[a;L[i]]\} \msim{} \mBbbN{}f i)
\mwedge{} A \msim{} i:\mBbbN{}||L|| \mtimes{} \mBbbN{}f i
\mwedge{} (k = \mSigma{}(f i | i < ||L||)))))))))
Date html generated:
2017_04_17-AM-09_33_23
Last ObjectModification:
2017_02_27-PM-05_33_02
Theory : equipollence!!cardinality!
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