Nuprl Lemma : rng_sum_swap
∀[r:Rng]. ∀[k,m:ℕ]. ∀[F:ℕm ⟶ ℕk ⟶ |r|].
((Σ(r) 0 ≤ i < m. Σ(r) 0 ≤ j < k. F[i;j]) = (Σ(r) 0 ≤ j < k. Σ(r) 0 ≤ i < m. F[i;j]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
rng: Rng,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
rng_wf,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
rng_car_wf,
int_seg_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
rng_zero_wf,
rng_sum_wf,
equal_wf,
squash_wf,
true_wf,
rng_sum_unroll_base,
iff_weakening_equal,
rng_sum_unroll_hi,
rng_plus_zero,
rng_plus_wf,
lelt_wf,
decidable__lt,
int_seg_properties,
infix_ap_wf,
rng_sum_plus
Rules used in proof :
unionElimination,
because_Cache,
functionEquality,
axiomEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[k,m:\mBbbN{}]. \mforall{}[F:\mBbbN{}m {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} |r|].
((\mSigma{}(r) 0 \mleq{} i < m. \mSigma{}(r) 0 \mleq{} j < k. F[i;j]) = (\mSigma{}(r) 0 \mleq{} j < k. \mSigma{}(r) 0 \mleq{} i < m. F[i;j]))
Date html generated:
2018_05_21-PM-03_15_09
Last ObjectModification:
2017_12_11-PM-05_05_44
Theory : rings_1
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