Nuprl Lemma : bag-parts-no-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. bag-no-repeats(bag(T) List+;bag-parts(eq;bs)) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts: bag-parts(eq;bs),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
listp: A List+,
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
empty-bag: {},
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons-bag: x.b,
istype: istype(T),
bag-size: #(bs),
bag-parts: bag-parts(eq;bs),
bag-partitions: bag-partitions(eq;bs),
bag-splits: bag-splits(b),
list_ind: list_ind,
bag-append: as + bs,
append: as @ bs,
bag-map: bag-map(f;bs),
map: map(f;as),
single-bag: {x},
pi1: fst(t),
pi2: snd(t),
bottom: ⊥,
bag-to-set: bag-to-set(eq;bs),
bag-remove-repeats: bag-remove-repeats(eq;bs),
list-to-set: list-to-set(eq;L),
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
insert: insert(a;L),
eval_list: eval_list(t),
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
bor: p ∨bq,
proddeq: proddeq(a;b),
band: p ∧b q,
bag-deq: bag-deq(eq),
bag-eq: bag-eq(eq;as;bs),
bag-all: bag-all(x.p[x];bs),
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
btrue: tt,
eq_int: (i =z j),
bag-count: (#x in bs),
count: count(P;L),
bfalse: ff,
lt_int: i <z j,
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
bag-null: bag-null(bs),
null: null(as),
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
has-value: (a)↓,
uiff: uiff(P;Q),
has-valueall: has-valueall(a),
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
listp: A List+,
cand: A c∧ B,
length: ||as||,
true: True,
inject: Inj(A;B;f),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag_wf,
deq_wf,
valueall-type_wf,
istype-universe,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
bag_to_squash_list,
le_wf,
bag-size_wf,
bag-no-repeats_wf,
listp_wf,
bag-parts_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
list_wf,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
equal_wf,
list-subtype-bag,
no_repeats_wf,
istype-nat,
empty-bag-no-repeats,
length_of_nil_lemma,
lifting-strict-callbyvalueall,
has-valueall-if-has-value-callbyvalueall,
has-valueall-has-value,
has-value_wf_base,
istype-base,
is-exception_wf,
is-exception-evalall,
cbv_sqequal,
lifting-strict-ifthenelse,
strict4-decide,
lifting-strict-ispair,
has-value-ifthenelse-type,
lifting-strict-spread,
strict4-ispair,
value-type-has-value,
int-value-type,
lifting-strict-isaxiom,
cons-bag_wf,
length_of_cons_lemma,
length_wf,
add-is-int-iff,
false_wf,
valueall-type-has-valueall,
bag-valueall-type,
product-valueall-type,
bag-partitions_wf,
evalall-reduce,
bag-combine-no-repeats,
bag-member_wf,
bag-null_wf,
empty-bag_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
pi1_wf_top,
assert-bag-null,
subtype_rel_product,
top_wf,
single-bag_wf,
cons_wf_listp,
nil_wf,
pi2_wf,
set-valueall-type,
less_than_wf,
list-valueall-type,
bag-map_wf,
cons-listp,
product-deq_wf,
bag-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
list-deq_wf,
strong-subtype-set3,
strong-subtype-self,
bag-subtype,
subtype_rel_set,
bag_size_cons_lemma,
bool_cases,
eqtt_to_assert,
bag-member-empty-iff,
iff_transitivity,
bnot_wf,
not_wf,
assert_of_bnot,
istype-assert,
bag-member-single,
reduce_hd_cons_lemma,
hd_wf,
istype-false,
bag-member-map,
bag-member-parts,
listp_properties,
true_wf,
iff_weakening_equal,
tl_wf,
reduce_tl_cons_lemma,
bag-union_wf,
reduce_nil_lemma,
length_cons_ge_one,
subtype_rel_list,
bag_qinc,
bag-single-no-repeats,
bag-map-no-repeats,
easy-member-int_seg,
bag-member-partitions,
bag-size-append,
bag-size-zero,
no-repeats-bag-partitions,
bag-no-repeats-settype,
decidable__equal_product,
decidable__equal_bag,
decidable-equal-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
universeIsType,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
functionIsTypeImplies,
productElimination,
unionElimination,
applyEquality,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
promote_hyp,
hypothesis_subsumption,
hyp_replacement,
functionEquality,
equalityIstype,
baseApply,
closedConclusion,
sqequalBase,
productEquality,
addEquality,
callbyvalueExceptionCases,
inrFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
sqleReflexivity,
decideExceptionCases,
exceptionSqequal,
inlFormation_alt,
callbyvalueAdd,
addExceptionCases,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
spreadExceptionCases,
axiomSqleEquality,
callbyvalueDecide,
pointwiseFunctionality,
setEquality,
equalityElimination,
independent_pairEquality,
setIsType,
functionIsType
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. bag-no-repeats(bag(T) List\msupplus{};bag-parts(eq;bs))
supposing valueall-type(T)
Date html generated:
2020_05_20-AM-09_04_42
Last ObjectModification:
2020_01_01-PM-00_11_16
Theory : bags_2
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