Nuprl Lemma : bag-decomp_wf
∀[T:Type]. ∀[bs:bag(T)]. (bag-decomp(bs) ∈ bag(T × bag(T)))
Proof
Definitions occuring in Statement :
bag-decomp: bag-decomp(bs),
bag: bag(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
bag-decomp: bag-decomp(bs),
and: P ∧ Q,
quotient: x,y:A//B[x; y],
bag: bag(T),
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
permutation: permutation(T;L1;L2),
ge: i ≥ j ,
nat: ℕ,
sq_type: SQType(T),
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
remove-nth: remove-nth(n;L),
int_iseg: {i...j},
cand: A c∧ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
inject: Inj(A;B;f),
subtract: n - m,
respects-equality: respects-equality(S;T)
Lemmas referenced :
istype-universe,
upto_wf,
list-subtype-bag,
subtype_rel_product,
remove-nth_wf,
length_wf,
int_seg_wf,
map_wf,
permutation-equiv,
permutation_wf,
list_wf,
quotient-member-eq,
bag_wf,
length_wf_nat,
length_upto,
length-map,
int_term_value_constant_lemma,
itermConstant_wf,
int_seg_properties,
member_wf,
istype-less_than,
istype-le,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
intformless_wf,
intformand_wf,
decidable__lt,
nat_properties,
le_wf,
set_subtype_base,
istype-nat,
int_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermVar_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
le_weakening,
int_seg_subtype,
subtype_rel_dep_function,
map_permute_list,
permute_list_wf,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
equal_wf,
permute_list_length,
inject_wf,
list_extensionality,
map-length,
select-map,
subtype_rel_list,
top_wf,
select-upto,
lelt_wf,
non_neg_length,
select_wf,
permute_list_select,
select_upto,
append_wf,
firstn_wf,
nth_tl_wf,
subtract_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
length-append,
add_functionality_wrt_eq,
length_firstn,
length_nth_tl,
subtype_rel_sets_simple,
add_nat_wf,
int_seg_subtype_nat,
istype-false,
istype-void,
false_wf,
permutation_inversion,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_nat_wf,
nat_wf,
add-is-int-iff,
subtype-base-respects-equality,
change-equality-type,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtract-is-int-iff,
equal-wf-base,
equal-wf-T-base,
select-append,
le_int_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
length_firstn_eq,
length_append,
select-firstn,
select-nth_tl,
zero-add,
select_nth_tl,
select_firstn,
respects-equality-list-bag,
respects-equality-trivial,
respects-equality-product,
respects-equality-list
Rules used in proof :
universeEquality,
instantiate,
isectIsTypeImplies,
isect_memberEquality_alt,
equalityTransitivity,
axiomEquality,
equalitySymmetry,
sqequalBase,
equalityIstype,
productIsType,
independent_functionElimination,
lambdaFormation_alt,
applyEquality,
natural_numberEquality,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
universeIsType,
inhabitedIsType,
lambdaEquality_alt,
productElimination,
promote_hyp,
pertypeElimination,
sqequalRule,
hypothesis,
hypothesisEquality,
productEquality,
thin,
isectElimination,
extract_by_obid,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation_alt,
Error :memTop,
independent_pairFormation,
applyLambdaEquality,
hyp_replacement,
closedConclusion,
baseApply,
dependent_set_memberEquality_alt,
cumulativity,
voidElimination,
int_eqEquality,
approximateComputation,
unionElimination,
rename,
setElimination,
baseClosed,
imageMemberEquality,
intEquality,
imageElimination,
functionIsType,
independent_pairEquality,
addEquality,
pointwiseFunctionality,
equalityElimination,
functionEquality,
minusEquality,
multiplyEquality,
functionExtensionality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. (bag-decomp(bs) \mmember{} bag(T \mtimes{} bag(T)))
Date html generated:
2020_05_20-AM-08_02_56
Last ObjectModification:
2020_01_31-PM-03_36_11
Theory : bags
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