Nuprl Lemma : concat-lifting-strict
∀[n:ℕ]. ∀[bags:k:ℕn ⟶ bag(Top)]. ∀[f:Top]. concat-lifting(n;f;bags) ~ {} supposing ∃k:ℕn. ((bags k) = {} ∈ bag(Top))
Proof
Definitions occuring in Statement :
concat-lifting: concat-lifting(n;f;bags),
empty-bag: {},
bag: bag(T),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
concat-lifting: concat-lifting(n;f;bags),
concat-lifting-list: concat-lifting-list(n;bags),
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lifting-gen-list-rev: lifting-gen-list-rev(n;bags),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
lelt: i ≤ j < k,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-combine: ⋃x∈bs.f[x],
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
empty-bag: {},
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nil: [],
less_than: a < b,
squash: ↓T,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
subtract_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
equal-empty-bag,
int_seg_wf,
lelt_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
bag_combine_empty_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
nat_wf,
int_seg_subtype_nat,
false_wf,
bag_union_empty_lemma,
exists_wf,
equal-wf-T-base,
bag_wf,
top_wf,
list_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
map_nil_lemma,
reduce_nil_lemma,
product_subtype_list,
spread_cons_lemma,
set_subtype_base,
int_subtype_base,
map_cons_lemma,
reduce_cons_lemma,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
lambdaFormation,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
baseClosed,
functionEquality,
hypothesis_subsumption,
applyLambdaEquality,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[bags:k:\mBbbN{}n {}\mrightarrow{} bag(Top)]. \mforall{}[f:Top].
concat-lifting(n;f;bags) \msim{} \{\} supposing \mexists{}k:\mBbbN{}n. ((bags k) = \{\})
Date html generated:
2017_10_01-AM-09_05_03
Last ObjectModification:
2017_07_26-PM-04_45_00
Theory : bags
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