Nuprl Lemma : count-quotient
∀k:ℕ
∀[A:Type]
(A ~ ℕk
⇒ (∀[E:A ⟶ A ⟶ ℙ]. (EquivRel(A;x,y.E[x;y]) ⇒ (∀x,y:A. Dec(E[x;y])) ⇒ (∃j:ℕ. ((j ≤ k) ∧ x,y:A//E[x;y] ~ ℕj)))))
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),
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],
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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
distinct-representatives,
equipollent-distinct-representatives,
length_wf_nat,
and_wf,
le_wf,
equipollent_wf,
quotient_wf,
int_seg_wf,
all_wf,
decidable_wf,
equiv_rel_wf,
nat_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
isectElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
natural_numberEquality,
functionEquality,
cumulativity,
universeEquality
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{}j:\mBbbN{}. ((j \mleq{} k) \mwedge{} x,y:A//E[x;y] \msim{} \mBbbN{}j)))))
Date html generated:
2016_05_14-PM-04_04_53
Last ObjectModification:
2015_12_26-PM-07_41_53
Theory : equipollence!!cardinality!
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