Nuprl Lemma : bag-drop-append
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs,cs:bag(T)].
(bag-drop(eq;bs + cs;x) = if ((#x in bs) =z 0) then bs + bag-drop(eq;cs;x) else bag-drop(eq;bs;x) + cs fi ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-drop: bag-drop(eq;bs;a),
bag-count: (#x in bs),
bag-append: as + bs,
bag: bag(T),
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sq_or: a ↓∨ b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
ge: i ≥ j ,
eq_int: (i =z j),
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag-drop-property,
bag-append_wf,
bag_wf,
bag-append-cancel,
single-bag_wf,
bag-drop_wf,
ifthenelse_wf,
eq_int_wf,
bag-count_wf,
nat_wf,
bag-member_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
bag-member-append,
bag-member-single,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bag-member-count,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
bag-append-assoc-comm,
bag-append-assoc2,
decidable__le,
nat_properties,
intformnot_wf,
int_formula_prop_not_lemma,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_nat,
false_wf,
decidable__equal_int,
equal-wf-T-base
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
productElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
inlFormation,
lambdaFormation,
equalityElimination,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_set_memberEquality,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs,cs:bag(T)].
(bag-drop(eq;bs + cs;x)
= if ((\#x in bs) =\msubz{} 0) then bs + bag-drop(eq;cs;x) else bag-drop(eq;bs;x) + cs fi )
Date html generated:
2018_05_21-PM-09_48_30
Last ObjectModification:
2018_05_19-PM-04_20_23
Theory : bags_2
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