Nuprl Lemma : member-count-repeats3
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[x:T]. ∀[n:ℕ+].
n = ||filter(λy.(eq y x);L)|| ∈ ℤ supposing (<x, n> ∈ count-repeats(L,eq))
Proof
Definitions occuring in Statement :
count-repeats: count-repeats(L,eq),
l_member: (x ∈ l),
length: ||as||,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
all: ∀x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat_plus: ℕ+,
deq: EqDecider(T)
Lemmas referenced :
l_member_wf,
nat_plus_wf,
count-repeats_wf,
list_wf,
deq_wf,
member-count-repeats2,
lelt_wf,
length_wf,
equal_wf,
filter_wf5
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
productEquality,
cumulativity,
hypothesisEquality,
independent_pairEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
hyp_replacement,
Error :applyLambdaEquality,
spreadEquality,
intEquality,
lambdaEquality,
applyEquality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[x:T]. \mforall{}[n:\mBbbN{}\msupplus{}].
n = ||filter(\mlambda{}y.(eq y x);L)|| supposing (<x, n> \mmember{} count-repeats(L,eq))
Date html generated:
2016_10_21-AM-10_37_54
Last ObjectModification:
2016_07_12-AM-05_48_57
Theory : decidable!equality
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