Nuprl Lemma : rng_lsum_swap
∀[r:Rng]. ∀[A,B:Type]. ∀[F:A ⟶ B ⟶ |r|]. ∀[as:A List]. ∀[bs:B List].
(Σ{r} a ∈ as. Σ{r} b ∈ bs. F[a;b] = Σ{r} b ∈ bs. Σ{r} a ∈ as. F[a;b] ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
infix_ap: x f y,
and: P ∧ Q,
true: True,
prop: ℙ,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
so_apply: x[s1;s2],
rng: Rng,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
infix_ap_wf,
rng_plus_zero,
rng_plus_wf,
rng_wf,
cons_wf,
rng_lsum_cons_lemma,
rng_zero_wf,
rng_lsum_nil_lemma,
nil_wf,
rng_lsum_wf,
rng_car_wf,
equal_wf,
list_wf,
uall_wf,
list_induction,
squash_wf,
true_wf,
rng_lsum_0,
iff_weakening_equal,
rng_lsum_plus,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm
Rules used in proof :
productElimination,
equalitySymmetry,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
functionEquality,
axiomEquality,
lambdaFormation,
isect_memberEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_functionElimination,
functionExtensionality,
applyEquality,
because_Cache,
rename,
setElimination,
hypothesis,
cumulativity,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[A,B:Type]. \mforall{}[F:A {}\mrightarrow{} B {}\mrightarrow{} |r|]. \mforall{}[as:A List]. \mforall{}[bs:B List].
(\mSigma{}\{r\} a \mmember{} as. \mSigma{}\{r\} b \mmember{} bs. F[a;b] = \mSigma{}\{r\} b \mmember{} bs. \mSigma{}\{r\} a \mmember{} as. F[a;b])
Date html generated:
2018_05_21-PM-09_32_57
Last ObjectModification:
2017_12_14-PM-11_11_59
Theory : matrices
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