Nuprl Lemma : bag-count-rep
∀[T:Type]. ∀[n:ℕ]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ((#x in bag-rep(n;y)) = if eq x y then n else 0 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-rep: bag-rep(n;x),
deq: EqDecider(T),
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
bag-rep: bag-rep(n;x),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
eqof: eqof(d),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
cons-bag: x.b,
bag-count: (#x in bs),
count: count(P;L),
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
deq_wf,
primrec-unroll,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
bag-count-empty,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
reduce_cons_lemma,
nat_wf,
ifthenelse_wf,
bag-count_wf,
bag-rep_wf,
le_wf,
list-subtype-bag,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
cumulativity,
applyEquality,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
universeEquality,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T].
((\#x in bag-rep(n;y)) = if eq x y then n else 0 fi )
Date html generated:
2018_05_21-PM-09_46_12
Last ObjectModification:
2017_07_26-PM-06_29_57
Theory : bags_2
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