Nuprl Lemma : pigeon-hole-implies
∀n:ℕ. ∀[m:ℕ]. ∀f:ℕn ⟶ ℕm. ∃i:ℕn. (∃j:{ℕi| ((f i) = (f j) ∈ ℤ)}) supposing m < n
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
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],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
inject: Inj(A;B;f),
le: A ≤ B,
less_than: a < b,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
member-less_than,
decidable__exists_int_seg,
exists_wf,
int_seg_wf,
equal_wf,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__equal_int,
pigeon-hole,
less_than_wf,
nat_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
instantiate,
dependent_functionElimination,
natural_numberEquality,
because_Cache,
sqequalRule,
lambdaEquality,
intEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}[m:\mBbbN{}]. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m. \mexists{}i:\mBbbN{}n. (\mexists{}j:\{\mBbbN{}i| ((f i) = (f j))\}) supposing m < n
Date html generated:
2017_09_29-PM-05_57_59
Last ObjectModification:
2017_07_26-PM-02_04_07
Theory : list_1
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