Nuprl Lemma : bag-member-decomp
∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T]. ∀[b:bag(T)]. uiff(<x, b> ↓∈ bag-decomp(bs);({x} + b) = bs ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-decomp: bag-decomp(bs),
bag-member: x ↓∈ bs,
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bag-member: x ↓∈ bs,
squash: ↓T,
exists: ∃x:A. B[x],
bag-decomp: bag-decomp(bs),
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
remove-nth: remove-nth(n;L),
pi1: fst(t),
pi2: snd(t),
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
int_iseg: {i...j},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
subtract: n - m,
less_than: a < b,
le: A ≤ B,
nat: ℕ,
true: True,
ge: i ≥ j ,
l_member: (x ∈ l),
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
tl: tl(l),
cons: [a / b]
Lemmas referenced :
uiff_wf,
bag-member_wf,
bag_wf,
bag-decomp_wf,
equal_wf,
bag-append_wf,
single-bag_wf,
list_wf,
list-subtype-bag,
permutation_wf,
equal-wf-base,
member-permutation,
member_map,
int_seg_wf,
length_wf,
upto_wf,
remove-nth_wf,
subtype_rel_product,
member_wf,
map_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
quotient-member-eq,
permutation-equiv,
cons_wf,
permutation-cons,
firstn_wf,
nth_tl_wf,
append_wf,
length-append,
length_firstn,
subtype_rel_sets,
lelt_wf,
le_wf,
length_of_cons_lemma,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
exists_wf,
append_firstn_lastn_sq,
subtype_rel_list,
top_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
itermConstant_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
firstn_decomp2,
append_assoc_sq,
squash_wf,
true_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
iff_weakening_equal,
non_neg_length,
less_than_wf,
l_member_wf,
member_upto,
length_wf_nat,
nat_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
select_wf,
and_wf,
nat_wf,
select_append_back,
select-cons-hd,
add_nat_wf,
false_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
firstn_append,
nth_tl-append,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
firstn_all,
permutation_weakening,
subtype_rel_self
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
cut,
extract_by_obid,
isectElimination,
thin,
productEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_pairEquality,
sqequalRule,
dependent_functionElimination,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
applyEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
axiomEquality,
independent_functionElimination,
universeEquality,
isect_memberFormation,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
independent_pairFormation,
applyLambdaEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
setElimination,
addEquality,
intEquality,
setEquality,
unionElimination,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
minusEquality,
hyp_replacement,
equalityElimination,
promote_hyp,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. uiff(<x, b> \mdownarrow{}\mmember{} bag-decomp(bs);(\{x\} + b) = bs)
Date html generated:
2017_10_01-AM-09_01_04
Last ObjectModification:
2017_07_26-PM-04_42_16
Theory : bags
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