Nuprl Lemma : fps-moebius-eq
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. (fps-moebius(eq;r) = (1÷λb.1) ∈ PowerSeries(X;r)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-moebius: fps-moebius(eq;r),
fps-div: (f÷g),
fps-one: 1,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-div: (f÷g),
fps-moebius: fps-moebius(eq;r),
power-series: PowerSeries(X;r),
has-value: (a)↓,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
le: A ≤ B,
subtype_rel: A ⊆r B,
fps-div-coeff: fps-div-coeff(eq;r;f;g;x;b),
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
crng: CRng,
rng: Rng,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
infix_ap: x f y,
true: True,
fps-one: 1,
fps-coeff: f[b],
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
bag-filter: [x∈b|p[x]],
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
bag-partitions: bag-partitions(eq;bs),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bag-splits: bag-splits(b),
empty-bag: {},
nil: [],
single-bag: {x},
cons: [a / b],
bag-to-set: bag-to-set(eq;bs),
bag-remove-repeats: bag-remove-repeats(eq;bs),
list-to-set: list-to-set(eq;L),
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
bag-null: bag-null(bs),
null: null(as),
assoc: Assoc(T;op),
comm: Comm(T;op),
integ_dom: IntegDom{i},
int_ring: ℤ-rng,
rng_car: |r|,
rng_zero: 0,
rng_plus: +r,
less_than: a < b
Lemmas referenced :
value-type-has-value,
int-value-type,
bag-moebius_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
bag-size_wf,
subtract-1-ge-0,
nat_wf,
add_nat_wf,
istype-false,
le_wf,
decidable__le,
add-is-int-iff,
set_subtype_base,
int_subtype_base,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable__lt,
rng_car_wf,
rng_one_wf,
fps-coeff_wf,
fps-one_wf,
rng_minus_wf,
bag-summation_wf,
assert_wf,
bnot_wf,
bag-null_wf,
rng_plus_wf,
rng_zero_wf,
pi1_wf_top,
infix_ap_wf,
rng_times_wf,
fps-div-coeff_wf,
bag-filter_wf,
bag-partitions_wf,
crng_all_properties,
rng_plus_comm2,
int-to-ring_wf,
equal_wf,
rng_times_over_plus,
rng_times_over_minus,
rng_times_one,
rng_plus_comm,
iff_weakening_equal,
bag-moebius-property,
eqtt_to_assert,
assert-bag-null,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
equal-wf-T-base,
ifthenelse_wf,
empty-bag_wf,
int-to-ring-one,
bag-summation-empty,
squash_wf,
true_wf,
rng_minus_zero,
subtype_rel_self,
rng_plus_zero,
decidable__equal_int,
subtype_rel_product,
top_wf,
pi2_wf,
int-to-ring-minus,
assoc_wf,
comm_wf,
bag-summation-hom,
int_ring_wf,
int-to-ring-hom,
bag-settype,
subtract_wf,
bag-member_wf,
bag-member-filter,
bag-member-partitions,
bag-size-append,
itermSubtract_wf,
int_term_value_subtract_lemma,
null-bag-size,
eq_int_wf,
not_wf,
equal-wf-base,
iff_transitivity,
assert_of_bnot,
assert_of_eq_int,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
applyEquality,
because_Cache,
dependent_set_memberEquality_alt,
addEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseClosed,
equalityIsType1,
setEquality,
productEquality,
independent_pairEquality,
productIsType,
setIsType,
imageElimination,
imageMemberEquality,
equalityElimination,
equalityIsType3,
baseApply,
closedConclusion,
instantiate,
cumulativity,
equalityIsType4,
hyp_replacement,
minusEquality,
functionIsType
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. (fps-moebius(eq;r) = (1\mdiv{}\mlambda{}b.1)) supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_34_39
Last ObjectModification:
2018_10_16-AM-09_34_31
Theory : power!series
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