Nuprl Lemma : bag-size-as-summation
∀T:Type. ∀bs:bag(T). (#(bs) = Σ(x∈bs). 1 ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-size: #(bs),
bag: bag(T),
all: ∀x:A. B[x],
lambda: λx.A[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag-size_wf,
nat_wf,
bag-summation-constant-int,
iff_weakening_equal,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
intEquality,
cumulativity,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}T:Type. \mforall{}bs:bag(T). (\#(bs) = \mSigma{}(x\mmember{}bs). 1)
Date html generated:
2018_05_21-PM-09_48_53
Last ObjectModification:
2017_07_26-PM-06_30_51
Theory : bags_2
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