Nuprl Lemma : finite-type-equipollent
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (finite-type(T) ⇐⇒ ∃n:ℕ. ℕn ~ T))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
finite-type: finite-type(T),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
equipollent: A ~ B,
finite-type: finite-type(T),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
rev_implies: P ⇐ Q,
biject: Bij(A;B;f),
all: ∀x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
inject: Inj(A;B;f),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
less_than: a < b,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
surject: Surj(A;B;f)
Lemmas referenced :
exists_wf,
nat_wf,
int_seg_wf,
surject_wf,
biject_wf,
all_wf,
decidable_wf,
equal_wf,
subtract_wf,
set_wf,
less_than_wf,
primrec-wf2,
false_wf,
le_wf,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__inject-finite-type,
finite-type-int_seg,
decidable__equal_int_seg,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
not-inject,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
add-member-int_seg2,
intformeq_wf,
int_formula_prop_eq_lemma,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
dependent_pairFormation,
universeEquality,
intEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (finite-type(T) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. \mBbbN{}n \msim{} T))
Date html generated:
2017_04_17-AM-09_34_51
Last ObjectModification:
2017_02_27-PM-05_34_06
Theory : equipollence!!cardinality!
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