Nuprl Lemma : sum-reindex
∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ]. ∀[m:ℕ]. ∀[b:ℕm ⟶ ℤ].
Σ(a[i] | i < n) = Σ(b[j] | j < m) ∈ ℤ
supposing ∃f:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ⟶ {j:ℕm| ¬(b[j] = 0 ∈ ℤ)} . (Bij({i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ;{j:ℕm| ¬(b[j] = 0 ∈ ℤ\000C)} ;f) ∧ (∀i:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} . (a[i] = b[f i] ∈ ℤ)))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
biject: Bij(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
ge: i ≥ j ,
exists: ∃x:A. B[x],
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
bag-no-repeats: bag-no-repeats(T;bs),
cand: A c∧ B,
squash: ↓T,
int_seg: {i..j-},
istype: istype(T),
lelt: i ≤ j < k,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
less_than: a < b,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
guard: {T},
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q),
surject: Surj(A;B;f),
less_than': less_than'(a;b),
compose: f o g
Lemmas referenced :
sum-as-bag-accum,
int_seg_wf,
bag-accum_wf,
assert_wf,
bnot_wf,
eq_int_wf,
istype-assert,
istype-int,
bag-filter_wf,
bag-map_wf,
list-subtype-bag,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
biject_wf,
not_wf,
bag-filter-map,
upto_wf,
subtype_rel_self,
no_repeats_upto,
equal_wf,
bag_wf,
no_repeats_wf,
bag-no-repeats_wf,
le_wf,
lelt_wf,
set_subtype_base,
list_subtype_base,
bag-extensionality-no-repeats,
decidable__equal_int_seg,
subtype_rel_bag,
subtype_rel_dep_function,
subtype_rel_sets_simple,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
bag-member_wf,
bag-filter-no-repeats,
bag-map-no-repeats,
assert_elim,
bfalse_wf,
squash_wf,
true_wf,
istype-universe,
bool_wf,
eq_int_eq_true,
iff_weakening_equal,
btrue_neq_bfalse,
subtype_base_sq,
bag-filter-no-repeats2,
subtype_rel_sets,
bag-member-map,
bag-member-filter,
bag-member-filter-set,
istype-less_than,
istype-le,
l_member_wf,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformand_wf,
decidable__lt,
int_seg_properties,
istype-false,
int_seg_subtype_nat,
nat_wf,
subtype_rel_set,
member_upto2,
int_term_value_constant_lemma,
itermConstant_wf,
list_wf,
bag-map-map,
set_wf,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
universeIsType,
natural_numberEquality,
setElimination,
rename,
hypothesis,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
closedConclusion,
setEquality,
intEquality,
addEquality,
setIsType,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
productIsType,
functionIsType,
baseClosed,
sqequalBase,
axiomEquality,
isectIsTypeImplies,
dependent_set_memberEquality,
dependent_pairFormation,
independent_pairFormation,
productEquality,
imageMemberEquality,
baseApply,
equalityIsType4,
dependent_set_memberEquality_alt,
imageElimination,
instantiate,
universeEquality,
cumulativity,
equalityIsType1,
lambdaFormation,
isect_memberEquality,
voidEquality,
lambdaEquality,
functionExtensionality,
functionEquality,
impliesFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}]. \mforall{}[b:\mBbbN{}m {}\mrightarrow{} \mBbbZ{}].
\mSigma{}(a[i] | i < n) = \mSigma{}(b[j] | j < m)
supposing \mexists{}f:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} {}\mrightarrow{} \{j:\mBbbN{}m| \mneg{}(b[j] = 0)\} . (Bij(\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} ;\{j:\mBbbN{}m| \mneg{}(b[j]\000C = 0)\} ;f) \mwedge{} (\mforall{}i:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} . (a[i] = b[f i])))
Date html generated:
2019_10_15-AM-11_33_28
Last ObjectModification:
2019_06_26-PM-04_41_32
Theory : general
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