Nuprl Lemma : finite-partition
∀n,k:ℕ. ∀c:ℕn ⟶ ℕk.
∃p:ℕk ⟶ (ℕ List)
((Σ(||p j|| | j < k) = n ∈ ℤ)
∧ (∀j:ℕk. ∀x,y:ℕ||p j||. p j[x] > p j[y] supposing x < y)
∧ (∀j:ℕk. ∀x:ℕ||p j||. (p j[x] < n c∧ ((c p j[x]) = j ∈ ℤ))))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
cand: A c∧ B,
gt: i > j,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
int_seg: {i..j-},
prop: ℙ,
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
gt: i > j,
cand: A c∧ B,
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
less_than': less_than'(a;b),
uiff: uiff(P;Q),
subtract: n - m,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
cons: [a / b]
Lemmas referenced :
nat_wf,
int_seg_wf,
subtract_wf,
list_wf,
sum_wf,
length_wf,
int_subtype_base,
less_than_wf,
gt_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
le_wf,
intformless_wf,
int_formula_prop_less_lemma,
non_neg_length,
length_wf_nat,
set_subtype_base,
lelt_wf,
primrec-wf2,
all_wf,
exists_wf,
equal-wf-base,
isect_wf,
equal-wf-T-base,
member-less_than,
satisfiable-full-omega-tt,
iff_weakening_equal,
empty_support,
true_wf,
squash_wf,
equal_wf,
base_wf,
stuck-spread,
length_of_nil_lemma,
nil_wf,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
itermSubtract_wf,
int_term_value_subtract_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
sum_functionality,
length_of_cons_lemma,
sum-ite,
singleton_support_sum,
intformeq_wf,
int_formula_prop_eq_lemma,
not_wf,
equal-wf-base-T,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
select_cons_tl,
subtype_rel_list,
add-is-int-iff,
false_wf,
assert_wf,
bnot_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
select-cons-tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
Error :productIsType,
Error :equalityIsType4,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
applyEquality,
functionExtensionality,
because_Cache,
Error :isectIsType,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
imageElimination,
Error :equalityIsType3,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
intEquality,
Error :setIsType,
functionEquality,
productEquality,
dependent_set_memberEquality,
computeAll,
isect_memberFormation,
imageMemberEquality,
universeEquality,
voidEquality,
isect_memberEquality,
baseClosed,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
addEquality,
minusEquality,
multiplyEquality,
equalityElimination,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
hyp_replacement,
Error :isect_memberFormation_alt,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}n,k:\mBbbN{}. \mforall{}c:\mBbbN{}n {}\mrightarrow{} \mBbbN{}k.
\mexists{}p:\mBbbN{}k {}\mrightarrow{} (\mBbbN{} List)
((\mSigma{}(||p j|| | j < k) = n)
\mwedge{} (\mforall{}j:\mBbbN{}k. \mforall{}x,y:\mBbbN{}||p j||. p j[x] > p j[y] supposing x < y)
\mwedge{} (\mforall{}j:\mBbbN{}k. \mforall{}x:\mBbbN{}||p j||. (p j[x] < n c\mwedge{} ((c p j[x]) = j))))
Date html generated:
2019_06_20-PM-01_32_14
Last ObjectModification:
2018_10_05-AM-09_40_06
Theory : list_1
Home
Index