Nuprl Lemma : bag-summation_functionality_wrt_le_1
∀[T:Type]. ∀[b:bag(T)]. ∀[f,g:T ⟶ ℤ]. Σ(x∈b). f[x] ≤ Σ(x∈b). g[x] supposing ∀x:T. (f[x] ≤ g[x])
Proof
Definitions occuring in Statement :
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],
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
so_lambda: λ2x.t[x],
le: A ≤ B,
all: ∀x:A. B[x],
so_apply: x[s],
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
prop: ℙ,
not: ¬A,
false: False,
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
assoc: Assoc(T;op),
comm: Comm(T;op),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
guard: {T}
Lemmas referenced :
bag_to_squash_list,
sq_stable__le,
bag-summation_wf,
le_wf,
less_than'_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_accum_wf,
list_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
false_wf,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
le_functionality,
add_functionality_wrt_le,
le_weakening
Rules used in proof :
cut,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
isect_memberFormation,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
imageElimination,
because_Cache,
sqequalRule,
lambdaEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
promote_hyp,
rename,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
cumulativity,
intEquality,
addEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
applyEquality,
functionExtensionality,
isect_memberEquality,
functionEquality,
voidElimination,
universeEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[f,g:T {}\mrightarrow{} \mBbbZ{}]. \mSigma{}(x\mmember{}b). f[x] \mleq{} \mSigma{}(x\mmember{}b). g[x] supposing \mforall{}x:T. (f[x] \mleq{} g[x])
Date html generated:
2016_10_25-AM-10_36_00
Last ObjectModification:
2016_07_12-AM-06_50_35
Theory : bags
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