Nuprl Lemma : bag-dickson-lemma
∀p:ℕ. ∀[T:Type]. (T ~ ℕp ⇒ (∀A:ℕ ⟶ bag(T). ∃j:ℕ. ∃i:ℕj. sub-bag(T;A[i];A[j])))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
squash: ↓T,
true: True
Lemmas referenced :
Dickson's lemma,
bag-size_wf,
assert_wf,
eq_int_wf,
int_seg_wf,
bag-filter_wf,
nat_wf,
sub-bag_wf,
int_seg_subtype_nat,
false_wf,
exists_wf,
bag_wf,
equipollent_wf,
assert_of_eq_int,
equal_wf,
all_wf,
iff_wf,
sub-bag-iff,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
bag-count-sqequal,
le_wf,
squash_wf,
true_wf,
set_wf,
iff_imp_equal_bool
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isectElimination,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
because_Cache,
dependent_pairFormation,
independent_isectElimination,
independent_pairFormation,
functionEquality,
universeEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
addLevel,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
levelHypothesis,
hyp_replacement,
imageElimination,
impliesFunctionality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}p:\mBbbN{}. \mforall{}[T:Type]. (T \msim{} \mBbbN{}p {}\mRightarrow{} (\mforall{}A:\mBbbN{} {}\mrightarrow{} bag(T). \mexists{}j:\mBbbN{}. \mexists{}i:\mBbbN{}j. sub-bag(T;A[i];A[j])))
Date html generated:
2016_10_25-AM-11_31_24
Last ObjectModification:
2016_07_12-AM-07_36_37
Theory : bags_2
Home
Index