Nuprl Lemma : equipollent-list
∀[T:Type]. ∀k:ℕ. (T ~ ℕk ⇒ (∀n:ℕ. {as:T List| ||as|| = n ∈ ℤ} ~ ℕk^n))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
exp: i^n,
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
less_than: a < b,
squash: ↓T,
surject: Surj(A;B;f)
Lemmas referenced :
equipollent-exp,
equipollent_wf,
int_seg_wf,
istype-nat,
istype-universe,
list_wf,
equal-wf-base,
length_wf_nat,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
exp_wf2,
equipollent_functionality_wrt_equipollent2,
equipollent_inversion,
function_functionality_wrt_equipollent_right,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
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,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
biject_wf,
list_extensionality,
istype-less_than,
istype-le,
map-length,
length_upto,
map_wf,
upto_wf,
select-map,
subtype_rel_list,
top_wf,
length_wf,
select-upto
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
inhabitedIsType,
universeIsType,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesis,
instantiate,
universeEquality,
setEquality,
intEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
independent_isectElimination,
because_Cache,
functionEquality,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
unionElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
equalityIstype,
sqequalBase,
functionIsType,
imageElimination,
dependent_set_memberEquality_alt,
applyLambdaEquality,
productIsType,
functionExtensionality_alt
Latex:
\mforall{}[T:Type]. \mforall{}k:\mBbbN{}. (T \msim{} \mBbbN{}k {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \{as:T List| ||as|| = n\} \msim{} \mBbbN{}k\^{}n))
Date html generated:
2020_05_19-PM-10_00_39
Last ObjectModification:
2020_01_04-PM-08_00_20
Theory : equipollence!!cardinality!
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