Nuprl Lemma : finite-quotient-bound
∀A:Type. ∀R:A ⟶ A ⟶ ℙ. ∀n:ℕ.
(A ~ ℕn ⇒ EquivRel(A;x,y.x R y) ⇒ (∀x,y:A. Dec(x R y)) ⇒ (∃m:ℕ. ((m ≤ n) ∧ x,y:A//(x R y) ~ ℕm)))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
infix_ap: x f y,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
finite: finite(T),
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
infix_ap: x f y,
guard: {T},
equipollent: A ~ B,
subtype_rel: A ⊆r B,
preima_of_rel: R_f,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
pi1: fst(t),
quotient: x,y:A//B[x; y],
inject: Inj(A;B;f),
biject: Bij(A;B;f)
Lemmas referenced :
finite-quotient,
equipollent_wf,
int_seg_wf,
le_wf,
quotient_wf,
infix_ap_wf,
decidable_wf,
equiv_rel_wf,
nat_wf,
equipollent_inversion,
biject-quotient,
preima_of_equiv_rel,
quo-lift_wf,
biject_wf,
preima_of_rel_wf,
equipollent_transitivity,
pigeon-hole,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
less_than_wf,
subtract_wf,
subtype_rel_self,
not_wf,
subtype_rel_function,
int_seg_subtype,
false_wf,
sq_stable__le,
le_weakening2,
primrec-wf2,
all_wf,
exists_wf,
decidable__exists_int_seg,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
lelt_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_dep_function,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
equal-wf-base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
inject_wf,
quotient-member-eq,
compose_wf,
injection-composition
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
Error :dependent_pairFormation_alt,
hypothesis,
Error :universeIsType,
isectElimination,
natural_numberEquality,
setElimination,
rename,
productElimination,
independent_pairFormation,
sqequalRule,
Error :productIsType,
because_Cache,
cumulativity,
Error :lambdaEquality_alt,
instantiate,
universeEquality,
functionExtensionality,
applyEquality,
Error :inhabitedIsType,
independent_isectElimination,
Error :functionIsType,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :setIsType,
functionEquality,
productEquality,
Error :dependent_set_memberEquality_alt,
unionElimination,
addEquality,
promote_hyp,
applyLambdaEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
hyp_replacement
Latex:
\mforall{}A:Type. \mforall{}R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}. \mforall{}n:\mBbbN{}.
(A \msim{} \mBbbN{}n
{}\mRightarrow{} EquivRel(A;x,y.x R y)
{}\mRightarrow{} (\mforall{}x,y:A. Dec(x R y))
{}\mRightarrow{} (\mexists{}m:\mBbbN{}. ((m \mleq{} n) \mwedge{} x,y:A//(x R y) \msim{} \mBbbN{}m)))
Date html generated:
2019_06_20-PM-02_19_08
Last ObjectModification:
2018_09_30-PM-02_47_40
Theory : equipollence!!cardinality!
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