Nuprl Lemma : bag-diff-property
∀[T:Type]
∀eq:EqDecider(T). ∀as,bs:bag(T).
case bag-diff(eq;bs;as) of inl(cs) => bs = (as + cs) ∈ bag(T) | inr(z) => ∀cs:bag(T). (¬(bs = (as + cs) ∈ bag(T)))
Proof
Definitions occuring in Statement :
bag-diff: bag-diff(eq;bs;as),
bag-append: as + bs,
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-diff: bag-diff(eq;bs;as),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
list_accum: list_accum,
nil: [],
it: ⋅,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
single-bag: {x},
true: True,
sq_or: a ↓∨ b,
sq_stable: SqStable(P)
Lemmas referenced :
bag_wf,
deq_wf,
bag_to_squash_list,
squash_wf,
bag-diff_wf,
unit_wf2,
equal_wf,
bag-append_wf,
all_wf,
not_wf,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
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,
lelt_wf,
subtype_rel_self,
le_wf,
length_wf,
non_neg_length,
nat_properties,
less_than_wf,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
list-cases,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
equal-wf-T-base,
list_wf,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
list_accum_wf,
bag-remove1_wf,
list-subtype-bag,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
list_ind_nil_lemma,
list_accum_append,
list_accum_cons_lemma,
list_accum_nil_lemma,
bag-remove1-property,
last_wf,
append_wf,
cons_wf,
nil_wf,
true_wf,
bag-append-assoc,
iff_weakening_equal,
single-bag_wf,
bag-append-cancel,
bag-member_wf,
bag-member-append,
bag-member-single,
bag-append-comm-assoc,
sq_stable__equal,
sq_stable__all,
sq_stable__not
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeEquality,
imageElimination,
productElimination,
promote_hyp,
rename,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
unionEquality,
equalityTransitivity,
unionElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
because_Cache,
setElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
addLevel,
applyEquality,
instantiate,
dependent_set_memberEquality,
levelHypothesis,
hypothesis_subsumption,
impliesFunctionality,
productEquality,
functionEquality,
inlEquality,
inrEquality,
addEquality,
cumulativity,
equalityUniverse,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:bag(T).
case bag-diff(eq;bs;as) of inl(cs) => bs = (as + cs) | inr(z) => \mforall{}cs:bag(T). (\mneg{}(bs = (as + cs)))
Date html generated:
2019_10_16-AM-11_31_55
Last ObjectModification:
2018_08_21-PM-01_59_55
Theory : bags_2
Home
Index