Nuprl Lemma : cardinality-le-list
∀[T:Type]. ∀n:ℕ. (|T| ≤ n ⇒ (∃L:T List. ((||L|| = n ∈ ℤ) ∧ (∀x:T. (x ∈ L)))))
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
l_member: (x ∈ l),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
cardinality-le: |T| ≤ n,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
top: Top,
surject: Surj(A;B;f),
l_member: (x ∈ l),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
exists_wf,
int_seg_wf,
surject_wf,
nat_wf,
mklist_wf,
mklist_length,
equal_wf,
length_wf,
all_wf,
l_member_wf,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
less_than_wf,
select_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
squash_wf,
true_wf,
mklist_select,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
cumulativity,
lambdaEquality,
because_Cache,
functionExtensionality,
applyEquality,
universeEquality,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productEquality,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
unionElimination,
int_eqEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}n:\mBbbN{}. (|T| \mleq{} n {}\mRightarrow{} (\mexists{}L:T List. ((||L|| = n) \mwedge{} (\mforall{}x:T. (x \mmember{} L)))))
Date html generated:
2017_04_17-AM-07_45_16
Last ObjectModification:
2017_02_27-PM-04_17_04
Theory : list_1
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