Nuprl Lemma : finite-quotient
∀A:Type. ∀R:A ⟶ A ⟶ ℙ. (finite(A) ⇒ EquivRel(A;x,y.x R y) ⇒ (∀x,y:A. Dec(x R y)) ⇒ finite(x,y:A//(x R y)))
Proof
Definitions occuring in Statement :
finite: finite(T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
decidable: Dec(P),
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
squash: ↓T,
guard: {T},
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
sq_stable: SqStable(P)
Lemmas referenced :
finite-iff-listable,
quotient_wf,
all_wf,
decidable_wf,
equiv_rel_wf,
finite_wf,
decidable__quotient_equal,
remove-repeats_wf,
infix_ap_wf,
mk_deq_wf,
subtype_rel_list,
subtype_quotient,
no_repeats_wf,
l_member_wf,
remove-repeats-no_repeats,
squash_wf,
equal-wf-base,
equal_wf,
remove-repeats_property,
less_than_wf,
length_wf,
select_wf,
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,
sq_stable__l_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
functionEquality,
universeEquality,
rename,
dependent_pairFormation,
instantiate,
because_Cache,
productEquality,
independent_pairFormation,
dependent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
setElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}A:Type. \mforall{}R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}.
(finite(A) {}\mRightarrow{} EquivRel(A;x,y.x R y) {}\mRightarrow{} (\mforall{}x,y:A. Dec(x R y)) {}\mRightarrow{} finite(x,y:A//(x R y)))
Date html generated:
2017_04_17-AM-09_34_11
Last ObjectModification:
2017_02_27-PM-05_33_55
Theory : equipollence!!cardinality!
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