Nuprl Lemma : bag-count-cons
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ∀[b:bag(T)].
((#x in y.b) = if eq x y then (#x in b) + 1 else (#x in b) fi ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
cons-bag: x.b,
bag: bag(T),
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
single-bag: {x},
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
deq: EqDecider(T),
true: True,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cons-bag-as-append,
bag_wf,
deq_wf,
cons_wf,
nil_wf,
list-subtype-bag,
bag-count_wf,
nat_wf,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
bag-count-append,
add_functionality_wrt_eq,
bag-count-single,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
axiomEquality,
because_Cache,
universeEquality,
intEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
addEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. \mforall{}[b:bag(T)].
((\#x in y.b) = if eq x y then (\#x in b) + 1 else (\#x in b) fi )
Date html generated:
2018_05_21-PM-09_45_52
Last ObjectModification:
2018_05_19-PM-04_19_20
Theory : bags_2
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