Nuprl Lemma : int-bag-product-positive
∀[b:bag(ℤ)]. 0 < Π(b) supposing ∀[x:ℤ]. (x ↓∈ b ⇒ 0 < x)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
int-bag-product: Π(b),
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
int-bag-product: Π(b),
bag-product: Πx ∈ b. f[x],
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat_plus: ℕ+,
guard: {T},
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
subtype_rel: A ⊆r B,
l_member: (x ∈ l),
select: L[n],
cand: A c∧ B,
uiff: uiff(P;Q),
true: True
Lemmas referenced :
istype-int,
bag-member_wf,
istype-less_than,
member-less_than,
int-bag-product_wf,
bag_wf,
bag_to_squash_list,
less_than_wf,
bag-member-list,
decidable__equal_int,
l_member_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
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,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
list_accum_nil_lemma,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
nat_plus_wf,
nil_wf,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-le,
list_wf,
list_accum_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
list_accum_cons_lemma,
cons_wf,
cons_member,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
add-is-int-iff,
false_wf,
length_wf,
list_subtype_base,
mul_nat_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
sqequalRule,
isectIsType,
extract_by_obid,
functionIsType,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
natural_numberEquality,
isect_memberEquality_alt,
independent_isectElimination,
isectIsTypeImplies,
inhabitedIsType,
imageElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
isectEquality,
functionEquality,
rename,
lambdaFormation_alt,
independent_functionElimination,
dependent_functionElimination,
setElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
equalityTransitivity,
functionIsTypeImplies,
unionElimination,
because_Cache,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
multiplyEquality,
instantiate,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
inrFormation_alt,
pointwiseFunctionality,
productIsType,
imageMemberEquality,
dependent_set_memberEquality
Latex:
\mforall{}[b:bag(\mBbbZ{})]. 0 < \mPi{}(b) supposing \mforall{}[x:\mBbbZ{}]. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} 0 < x)
Date html generated:
2020_05_20-AM-08_01_51
Last ObjectModification:
2019_11_27-PM-03_08_09
Theory : bags
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