Nuprl Lemma : bag-combine-no-repeats
∀[T1,T2:Type]. ∀[f:T1 ⟶ bag(T2)]. ∀[eq1:EqDecider(T1)]. ∀[eq2:EqDecider(T2)]. ∀[bs:bag(T1)].
(bag-no-repeats(T2;⋃x∈bs.f[x])) supposing
(bag-no-repeats(T1;bs) and
((∀x,y:T1. ∀z:T2. (z ↓∈ f[x] ⇒ z ↓∈ f[y] ⇒ (x = y ∈ T1))) ∧ (∀x:T1. bag-no-repeats(T2;f[x]))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
bag: bag(T),
quotient: x,y:A//B[x; y],
sq_type: SQType(T),
guard: {T},
true: True,
istype: istype(T),
bag-filter: [x∈b|p[x]],
bag-sum: bag-sum(ba;x.f[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
less_than: a < b,
less_than': less_than'(a;b),
cand: A c∧ B,
rev_implies: P ⇐ Q,
bag-size: #(bs),
ge: i ≥ j ,
deq: EqDecider(T),
colength: colength(L),
nil: [],
it: ⋅,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
eqof: eqof(d),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+
Lemmas referenced :
bag-no-repeats-count,
iff_weakening_uiff,
bag-no-repeats_wf,
uall_wf,
uiff_wf,
le_wf,
bag-count_wf,
equal-wf-T-base,
istype-universe,
bag-combine_wf,
decidable__equal_int,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
bag-count-combine,
bag-sum-count,
decidable__le,
le_witness_for_triv,
set_subtype_base,
int_subtype_base,
bag-member_wf,
bag_wf,
deq_wf,
list_wf,
permutation_wf,
permutation_weakening,
equal_wf,
subtype_base_sq,
list-subtype-bag,
filter_wf5,
l_member_wf,
le_int_wf,
list-cases,
list_accum_nil_lemma,
product_subtype_list,
list_accum_cons_lemma,
less_than_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
list_accum_wf,
length_wf_nat,
nat_wf,
cons_member,
cons_wf,
member_filter,
assert_of_le_int,
bag-member-count,
list-member-bag-member,
subtype_rel_dep_function,
bool_wf,
bag-count-sqequal,
nat_properties,
ge_wf,
member-less_than,
filter_nil_lemma,
length_of_nil_lemma,
colength-cons-not-zero,
colength_wf_list,
istype-false,
length_wf,
subtract-1-ge-0,
spread_cons_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
filter_cons_lemma,
null_nil_lemma,
btrue_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
eqtt_to_assert,
safe-assert-deq,
length_of_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
assert_wf,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
length_cons_ge_one,
eqof_wf,
reduce_tl_cons_lemma,
tl_wf,
nil_wf,
add_nat_plus,
nat_plus_properties,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
because_Cache,
independent_functionElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
equalityTransitivity,
equalitySymmetry,
equalityIsType4,
intEquality,
baseClosed,
independent_pairEquality,
axiomEquality,
imageElimination,
imageMemberEquality,
productIsType,
functionIsType,
inhabitedIsType,
equalityIsType1,
universeEquality,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
instantiate,
cumulativity,
isectIsType,
setElimination,
rename,
closedConclusion,
setIsType,
hypothesis_subsumption,
addEquality,
baseApply,
dependent_set_memberEquality_alt,
inlFormation_alt,
inrFormation_alt,
hyp_replacement,
applyLambdaEquality,
productEquality,
functionEquality,
setEquality,
intWeakElimination,
functionIsTypeImplies,
equalityIsType3,
equalityIsType2,
equalityElimination
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} bag(T2)]. \mforall{}[eq1:EqDecider(T1)]. \mforall{}[eq2:EqDecider(T2)]. \mforall{}[bs:bag(T1)].
(bag-no-repeats(T2;\mcup{}x\mmember{}bs.f[x])) supposing
(bag-no-repeats(T1;bs) and
((\mforall{}x,y:T1. \mforall{}z:T2. (z \mdownarrow{}\mmember{} f[x] {}\mRightarrow{} z \mdownarrow{}\mmember{} f[y] {}\mRightarrow{} (x = y))) \mwedge{} (\mforall{}x:T1. bag-no-repeats(T2;f[x]))))
Date html generated:
2019_10_16-AM-11_30_31
Last ObjectModification:
2018_10_11-PM-03_22_09
Theory : bags_2
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