Nuprl Lemma : count-related-pairs
∀[T,S:Type]. ∀[K:T List]. ∀[L:S List]. ∀[R:T ⟶ S ⟶ 𝔹].
(Σ(||filter(R t;L)|| | t ∈ K) = Σ(||filter(λt.(R t s);K)|| | s ∈ L) ∈ ℤ)
Proof
Definitions occuring in Statement :
lsum: Σ(f[x] | x ∈ L),
length: ||as||,
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
istype: istype(T),
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
bool_wf,
list_wf,
istype-universe,
subtype_rel_dep_function,
l_member_wf,
double-lsum-swap,
ifthenelse_wf,
equal_wf,
squash_wf,
true_wf,
lsum_wf,
istype-int,
length-filter-lsum,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
inhabitedIsType,
instantiate,
universeEquality,
intEquality,
applyEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality_alt,
setEquality,
setIsType,
independent_isectElimination,
lambdaFormation_alt,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T,S:Type]. \mforall{}[K:T List]. \mforall{}[L:S List]. \mforall{}[R:T {}\mrightarrow{} S {}\mrightarrow{} \mBbbB{}].
(\mSigma{}(||filter(R t;L)|| | t \mmember{} K) = \mSigma{}(||filter(\mlambda{}t.(R t s);K)|| | s \mmember{} L))
Date html generated:
2020_05_19-PM-09_48_21
Last ObjectModification:
2019_11_12-PM-11_50_49
Theory : list_1
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