Nuprl Lemma : bag-subtract-size
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)].
#(bag-subtract(eq;bs;as)) = (#(bs) - #(as)) ∈ ℤ supposing sub-bag(T;as;bs)
Proof
Definitions occuring in Statement :
bag-subtract: bag-subtract(eq;bs;as),
sub-bag: sub-bag(T;as;bs),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
top: Top,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
bag-size_wf,
bag-subtract_wf,
nat_wf,
subtract_wf,
sub-bag_wf,
bag_wf,
deq_wf,
bag-size-append,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
bag-subtract-append,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
extract_by_obid,
isectElimination,
intEquality,
cumulativity,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
universeEquality,
natural_numberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)].
\#(bag-subtract(eq;bs;as)) = (\#(bs) - \#(as)) supposing sub-bag(T;as;bs)
Date html generated:
2018_05_21-PM-09_49_27
Last ObjectModification:
2017_07_26-PM-06_31_05
Theory : bags_2
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