Nuprl Lemma : quotient-equipollent-nat
∀[A:Type]. ∀[k:ℕ].
(finite-type(A)
⇒ (∀E:A ⟶ A ⟶ ℙ. x,y:A//E[x;y] ~ ℕk ⇐⇒ A ~ ℕk mod (a1,a2.E[a1;a2]) supposing EquivRel(A;x,y.E[x;y])))
Proof
Definitions occuring in Statement :
equiv-equipollent: A ~ B mod (a1,a2.E[a1; a2]),
equipollent: A ~ B,
finite-type: finite-type(T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q
Lemmas referenced :
equiv-equipollent_wf,
int_seg_wf,
quotient-equipollent,
decidable__equal-int_seg,
equipollent_wf,
quotient_wf,
iff_wf,
equiv_rel_wf,
finite-type_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[k:\mBbbN{}].
(finite-type(A)
{}\mRightarrow{} (\mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}
x,y:A//E[x;y] \msim{} \mBbbN{}k \mLeftarrow{}{}\mRightarrow{} A \msim{} \mBbbN{}k mod (a1,a2.E[a1;a2]) supposing EquivRel(A;x,y.E[x;y])))
Date html generated:
2018_05_21-PM-00_53_12
Last ObjectModification:
2018_05_19-AM-06_40_09
Theory : equipollence!!cardinality!
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