Nuprl Lemma : rng_lsum_when_swap
∀r:Rng. ∀A:Type. ∀f:A ⟶ |r|. ∀b:𝔹. ∀as:A List.
((Σ{A,r} x ∈ as. (when b. f[x])) = (when b. (Σ{A,r} x ∈ as. f[x])) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{A,r} x ∈ as. f[x],
list: T List,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng_when: rng_when,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
rng: Rng,
so_apply: x[s],
implies: P ⇒ Q,
rng_lsum: Σ{A,r} x ∈ as. f[x],
top: Top,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
grp_op: *,
infix_ap: x f y
Lemmas referenced :
list_induction,
equal_wf,
rng_car_wf,
rng_lsum_wf,
rng_when_wf,
list_wf,
mon_for_nil_lemma,
squash_wf,
true_wf,
rng_zero_wf,
rng_when_of_zero,
iff_weakening_equal,
mon_for_cons_lemma,
rng_plus_wf,
infix_ap_wf,
rng_when_thru_plus,
bool_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
cumulativity,
applyEquality,
functionExtensionality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
functionEquality
Latex:
\mforall{}r:Rng. \mforall{}A:Type. \mforall{}f:A {}\mrightarrow{} |r|. \mforall{}b:\mBbbB{}. \mforall{}as:A List.
((\mSigma{}\{A,r\} x \mmember{} as. (when b. f[x])) = (when b. (\mSigma{}\{A,r\} x \mmember{} as. f[x])))
Date html generated:
2017_10_01-AM-10_00_59
Last ObjectModification:
2017_03_03-PM-01_03_28
Theory : list_3
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