Nuprl Lemma : general-pigeon-hole
∀[n,m,k:ℕ]. ∀[f:ℕn ⟶ ℕm].
n ≤ (k * m) supposing ∀L:ℕn List. (no_repeats(ℕn;L) ⇒ (∃i:ℕm. (∀x∈L.f[x] = i ∈ ℕm)) ⇒ (||L|| ≤ k))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
gt: i > j,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
l_all: (∀x∈L.P[x]),
lelt: i ≤ j < k,
guard: {T},
cand: A c∧ B,
no_repeats: no_repeats(T;l),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l)
Lemmas referenced :
finite-partition,
sum_bound,
length_wf,
nat_wf,
int_seg_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
less_than'_wf,
all_wf,
list_wf,
no_repeats_wf,
exists_wf,
l_all_wf2,
l_member_wf,
equal_wf,
le_wf,
list-set-type2,
subtype_rel_list,
lelt_wf,
non_neg_length,
int_seg_properties,
select_wf,
length_wf_nat,
itermConstant_wf,
int_term_value_constant_lemma,
not_wf,
less_than_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
equal-wf-base,
int_subtype_base,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
lambdaFormation,
multiplyEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_pairEquality,
axiomEquality,
functionEquality,
setEquality,
dependent_set_memberEquality,
applyLambdaEquality,
independent_functionElimination,
hyp_replacement
Latex:
\mforall{}[n,m,k:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m].
n \mleq{} (k * m) supposing \mforall{}L:\mBbbN{}n List. (no\_repeats(\mBbbN{}n;L) {}\mRightarrow{} (\mexists{}i:\mBbbN{}m. (\mforall{}x\mmember{}L.f[x] = i)) {}\mRightarrow{} (||L|| \mleq{} k))
Date html generated:
2018_05_21-PM-06_51_35
Last ObjectModification:
2017_07_26-PM-04_57_48
Theory : general
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