Nuprl Lemma : equipollent-iff-list
∀[T:Type]. ∀n:ℕ. (T ~ ℕn ⇐⇒ ∃L:T List. (no_repeats(T;L) ∧ (||L|| = n ∈ ℤ) ∧ (∀x:T. (x ∈ L))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
nat: ℕ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
biject: Bij(A;B;f),
exists: ∃x:A. B[x],
equipollent: A ~ B,
top: Top,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
inject: Inj(A;B;f),
surject: Surj(A;B;f),
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
l_member: (x ∈ l),
pi1: fst(t),
true: True,
squash: ↓T,
no_repeats: no_repeats(T;l)
Lemmas referenced :
nat_wf,
l_member_wf,
all_wf,
length_wf,
equal_wf,
no_repeats_wf,
list_wf,
exists_wf,
int_seg_wf,
equipollent_wf,
equipollent_inversion,
length_upto,
map-length,
upto_wf,
map_wf,
set_wf,
subtype_rel_dep_function,
no_repeats_upto,
no_repeats_map,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
lelt_wf,
member_map,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
int_seg_properties,
false_wf,
int_seg_subtype_nat,
member_upto,
select_wf,
less_than_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
and_wf,
length_wf_nat,
non_neg_length,
biject_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
le_wf,
iff_weakening_equal,
true_wf,
squash_wf,
decidable__equal_int_seg
Rules used in proof :
universeEquality,
because_Cache,
intEquality,
productEquality,
lambdaEquality,
sqequalRule,
hypothesis,
rename,
setElimination,
natural_numberEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
applyEquality,
functionExtensionality,
dependent_pairFormation,
setEquality,
independent_isectElimination,
instantiate,
dependent_set_memberEquality,
dependent_functionElimination,
equalitySymmetry,
computeAll,
int_eqEquality,
unionElimination,
equalityTransitivity,
applyLambdaEquality,
hyp_replacement,
promote_hyp,
baseClosed,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}n:\mBbbN{}. (T \msim{} \mBbbN{}n \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. (no\_repeats(T;L) \mwedge{} (||L|| = n) \mwedge{} (\mforall{}x:T. (x \mmember{} L))))
Date html generated:
2018_05_21-PM-00_52_41
Last ObjectModification:
2017_12_07-PM-06_15_42
Theory : equipollence!!cardinality!
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