Nuprl Lemma : finite-partition-property
∀k:ℕ. ∀f:ℕ ⟶ ℕk. (¬¬(∃i:ℕk. ∀n:ℕ. (¬¬(∃m:ℕ. (n < m ∧ ((f m) = i ∈ ℤ))))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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],
not: ¬A,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
exists: ∃x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
ge: i ≥ j ,
cand: A c∧ B,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
all_wf,
int_seg_wf,
not_wf,
nat_wf,
exists_wf,
less_than_wf,
equal_wf,
not_over_exists,
finite-double-negation-shift,
false_wf,
subtract_wf,
set_wf,
primrec-wf2,
le_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__lt,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lelt_wf,
nat_properties,
decidable__le,
ifthenelse_wf,
lt_int_wf,
assert_wf,
bnot_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
int_subtype_base,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
intEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
addLevel,
impliesFunctionality,
productElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
functionEquality,
allFunctionality,
levelHypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
unionElimination,
promote_hyp,
cumulativity,
baseApply,
closedConclusion,
baseClosed,
addEquality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}k. (\mneg{}\mneg{}(\mexists{}i:\mBbbN{}k. \mforall{}n:\mBbbN{}. (\mneg{}\mneg{}(\mexists{}m:\mBbbN{}. (n < m \mwedge{} ((f m) = i))))))
Date html generated:
2017_10_01-AM-09_10_39
Last ObjectModification:
2017_07_26-PM-04_46_59
Theory : general
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