Nuprl Lemma : sum-count-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. (Σ(snd(count-repeats(L,eq)[i]) | i < ||count-repeats(L,eq)||) = ||L|| ∈ ℤ)
Proof
Definitions occuring in Statement :
count-repeats: count-repeats(L,eq),
sum: Σ(f[x] | x < k),
select: L[n],
length: ||as||,
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
pi2: snd(t),
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
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,
all: ∀x:A. B[x],
sum_aux: sum_aux(k;v;i;x.f[x]),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sum: Σ(f[x] | x < k),
top: Top,
subtype_rel: A ⊆r B,
pi1: fst(t),
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
squash: ↓T,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
deq: EqDecider(T),
nat_plus: ℕ+,
pi2: snd(t),
no_repeats: no_repeats(T;l),
cons: [a / b],
colength: colength(L),
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
eqof: eqof(d),
l_all: (∀x∈L.P[x])
Lemmas referenced :
istype-universe,
deq_wf,
list_wf,
istype-nat,
subtract-1-ge-0,
istype-less_than,
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,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
length_wf,
le_wf,
length_of_nil_lemma,
l_member_wf,
false_wf,
not_wf,
l_all_iff,
bfalse_wf,
filter_is_nil,
deq_member_nil_lemma,
top_wf,
subtype_rel_list,
count-repeats_wf,
nat_plus_wf,
map_wf,
first0,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
decidable__lt,
map-length,
pi1_wf,
istype-le,
firstn_decomp,
int_subtype_base,
subtype_base_sq,
istype-top,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
sum-unroll,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
select-map,
filter_wf5,
length_wf_nat,
set_subtype_base,
select_wf,
member-count-repeats2,
member_firstn,
nat_plus_properties,
istype-void,
no_repeats-count-repeats1,
intformeq_wf,
int_formula_prop_eq_lemma,
firstn_wf,
list-cases,
filter_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
spread_cons_lemma,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
filter_cons_lemma,
deq-member_wf,
equal-wf-T-base,
istype-assert,
equal_wf,
append_wf,
cons_wf,
nil_wf,
bnot_wf,
member_append,
eqof_wf,
length_of_cons_lemma,
add-is-int-iff,
cons_member,
iff_transitivity,
assert-deq-member,
assert_of_bnot,
uiff_transitivity,
safe-assert-deq,
member_singleton,
int_seg_wf,
iff_weakening_equal,
subtype_rel_self,
map_length_nat,
true_wf,
squash_wf,
firstn_all,
int_seg_properties,
filter_trivial,
select_member,
member-count-repeats1
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
universeIsType,
hypothesis,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionIsTypeImplies,
voidElimination,
independent_pairFormation,
Error :memTop,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation_alt,
lambdaFormation,
setEquality,
voidEquality,
isect_memberEquality,
applyEquality,
productElimination,
lambdaEquality,
because_Cache,
cumulativity,
productEquality,
promote_hyp,
equalityIstype,
productIsType,
dependent_set_memberEquality_alt,
intEquality,
imageElimination,
baseClosed,
imageMemberEquality,
axiomSqEquality,
lessCases,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
sqequalBase,
setIsType,
closedConclusion,
applyLambdaEquality,
baseApply,
hypothesis_subsumption,
functionIsType,
inlFormation_alt,
pointwiseFunctionality,
inrFormation_alt,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List].
(\mSigma{}(snd(count-repeats(L,eq)[i]) | i < ||count-repeats(L,eq)||) = ||L||)
Date html generated:
2020_05_19-PM-09_52_21
Last ObjectModification:
2020_03_11-PM-07_57_43
Theory : decidable!equality
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