Nuprl Lemma : fps-product-upto
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[k:ℕ+]. ∀[f:ℕk ⟶ PowerSeries(X;r)].
(Π(x∈upto(k)).f[x] = (f[0]*Π(x∈upto(k - 1)).f[x + 1]) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-product: Π(x∈b).f[x],
fps-mul: (f*g),
power-series: PowerSeries(X;r),
upto: upto(n),
deq: EqDecider(T),
int_seg: {i..j-},
nat_plus: ℕ+,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat_plus: ℕ+,
single-bag: {x},
bag-append: as + bs,
upto: upto(n),
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
from-upto: [n, m),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cand: A c∧ B,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
less_than: a < b,
bag-map: bag-map(f;bs)
Lemmas referenced :
int_seg_wf,
power-series_wf,
nat_plus_wf,
crng_wf,
deq_wf,
valueall-type_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
from-upto_wf,
list-subtype-bag,
subtype_rel_sets,
le_wf,
lelt_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
single-bag_wf,
decidable__lt,
upto_wf,
subtract_wf,
subtype_rel_self,
bag-append_wf,
fps-product-append,
squash_wf,
true_wf,
fps-mul_wf,
fps-product_wf,
add-member-int_seg2,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
iff_weakening_equal,
fps-product-single,
fps-product-reindex,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
subtype_rel_dep_function,
int_seg_subtype,
from-upto-shift,
list_wf,
list_subtype_base,
set_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
cumulativity,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
callbyvalueReduce,
sqleReflexivity,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
applyEquality,
productEquality,
addEquality,
minusEquality,
dependent_set_memberEquality,
imageElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
setEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} PowerSeries(X;r)].
(\mPi{}(x\mmember{}upto(k)).f[x] = (f[0]*\mPi{}(x\mmember{}upto(k - 1)).f[x + 1]))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_22
Last ObjectModification:
2017_07_26-PM-06_33_19
Theory : power!series
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