Nuprl Lemma : bag-summation-equal-implies-all-equal-1
∀[T:Type]. ∀[b:bag(T)]. ∀[f,g:T ⟶ ℤ].
(∀x:T. (x ↓∈ b ⇒ (f[x] = g[x] ∈ ℤ))) supposing ((Σ(x∈b). g[x] ≤ Σ(x∈b). f[x]) and (∀x:T. (x ↓∈ b ⇒ (f[x] ≤ g[x]))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
assoc: Assoc(T;op),
comm: Comm(T;op),
subtype_rel: A ⊆r B,
le: A ≤ B,
empty-bag: {},
uiff: uiff(P;Q),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons-bag: x.b,
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b
Lemmas referenced :
bag_to_squash_list,
bag-member_wf,
le_wf,
bag-summation_wf,
all_wf,
bag_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
list_induction,
list-subtype-bag,
equal_wf,
list_wf,
bag-member-empty-iff,
empty-bag_wf,
list_accum_nil_lemma,
bag-member-cons,
cons-bag_wf,
itermConstant_wf,
int_term_value_constant_lemma,
squash_wf,
true_wf,
bag-summation-cons,
iff_weakening_equal,
decidable__le,
add-is-int-iff,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
false_wf,
bag-summation_functionality_wrt_le,
and_wf
Rules used in proof :
cut,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
isect_memberFormation,
introduction,
lambdaFormation,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
imageElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
intEquality,
lambdaEquality,
addEquality,
natural_numberEquality,
sqequalRule,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
functionEquality,
rename,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
universeEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
independent_pairEquality,
imageMemberEquality,
baseClosed,
inlFormation,
pointwiseFunctionality,
baseApply,
closedConclusion,
inrFormation,
setElimination,
setEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[f,g:T {}\mrightarrow{} \mBbbZ{}].
(\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (f[x] = g[x]))) supposing
((\mSigma{}(x\mmember{}b). g[x] \mleq{} \mSigma{}(x\mmember{}b). f[x]) and
(\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (f[x] \mleq{} g[x]))))
Date html generated:
2017_10_01-AM-09_02_38
Last ObjectModification:
2017_07_26-PM-04_43_44
Theory : bags
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