Nuprl Lemma : KozenSilva-theorem
∀[r:CRng]. ∀[x,y:Atom].
∀[h:PowerSeries(r)]. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(r;x;y;h;d;k)
= ([h]_d 0(y:=((k ⋅r 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅r 1)*atom(x)+atom(y)))^(d (i + 1)))
∈ PowerSeries(r))
supposing ¬(x = y ∈ Atom)
Proof
Definitions occuring in Statement :
Moessner: Moessner(r;x;y;h;d;k),
fps-compose: g(x:=f),
fps-exp: (f)^(n),
fps-scalar-mul: (c)*f,
fps-product: Π(x∈b).f[x],
fps-slice: [f]_n,
fps-mul: (f*g),
fps-add: (f+g),
fps-atom: atom(x),
power-series: PowerSeries(X;r),
upto: upto(n),
atom-deq: AtomDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
atom: Atom,
equal: s = t ∈ T,
rng_nat_op: n ⋅r e,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
Moessner: Moessner(r;x;y;h;d;k),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
Moessner-aux: Moessner-aux(r;x;y;h;d;k),
eq_int: (i =z j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
upto: upto(n),
from-upto: [n, m),
lt_int: i <z j,
bfalse: ff,
subtype_rel: A ⊆r B,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
nequal: a ≠ b ∈ T ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-summation: Σ(x∈b). f[x],
bag-product: Πx ∈ b. f[x],
fps-product: Π(x∈b).f[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
rng_zero: 0,
add_grp_of_rng: r↓+gp,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
nat_op: n x(op;id) e,
mon_nat_op: n ⋅ e,
rng_nat_op: n ⋅r e,
less_than': less_than'(a;b),
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
decidable: Dec(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
crng: CRng,
rng: Rng,
istype: istype(T),
nat_plus: ℕ+,
cand: A c∧ B,
atom-deq: AtomDeq,
empty-bag: {},
fps-atom: atom(x),
fps-add: (f+g),
fps-coeff: f[b],
single-bag: {x},
fps-single: <c>,
bag-eq: bag-eq(eq;as;bs),
bag-count: (#x in bs),
bag-all: bag-all(x.p[x];bs),
count: count(P;L),
bag-map: bag-map(f;bs),
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
band: p ∧b q,
infix_ap: x f y,
bag-append: as + bs
Lemmas referenced :
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,
istype-less_than,
btrue_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
eq_int_wf,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
subtract-1-ge-0,
subtract-add-cancel,
int_subtype_base,
istype-nat,
power-series_wf,
atom_subtype_base,
istype-atom,
crng_wf,
list_accum_nil_lemma,
sum-unroll,
le_wf,
false_wf,
nat_wf,
zero-add,
iff_weakening_equal,
fps-compose-identity,
equal_wf,
fps-zero_wf,
fps-add_wf,
fps-compose_wf,
fps-one_wf,
atom-valueall-type,
fps-atom_wf,
atom-deq_wf,
fps-slice_wf,
mul_one_fps,
abmonoid_comm_fps,
mon_ident_fps,
squash_wf,
true_wf,
fps-mul_wf,
valueall-type_wf,
deq_wf,
fps-scalar-mul-zero,
subtype_rel_self,
decidable__le,
istype-le,
sum_wf,
int_seg_subtype_nat,
istype-false,
int_seg_wf,
non_neg_sum,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
fps-pascal_wf,
fps-scalar-mul_wf,
rng_nat_op_wf,
rng_one_wf,
fps-product_wf,
fps-exp_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
list-subtype-bag,
Moessner-aux_wf,
fps-set-to-one_wf,
fps-set-to-one-slice,
istype-universe,
fps-slice-slice,
sum_split1,
decidable__lt,
not-lt-2,
not-equal-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
add-subtract-cancel,
upto_wf,
KozenSilva-lemma,
fps-compose-mul,
fps-mul-assoc,
fps-compose-compose,
fps-compose-add,
fps-compose-scalar-mul,
neg_assert_of_eq_atom,
assert_of_eq_atom,
eq_atom_wf,
rng_zero_wf,
rng_car_wf,
fps-compose-atom,
reduce_nil_lemma,
reduce_cons_lemma,
map_nil_lemma,
map_cons_lemma,
rng_plus_zero,
fps-add-assoc,
fps-scalar-mul-one,
rng_nat_op_one,
fps-scalar-mul-rng-add,
rng_nat_op_add,
upto_decomp1,
decidable__equal_int,
fps-compose-fps-product,
bag-append_wf,
subtype_rel_bag,
int_seg_subtype,
not-le-2,
add-swap,
add-mul-special,
zero-mul,
le-add-cancel2,
single-bag_wf,
fps-product-append,
fps-product-single,
fps-compose-exp,
nat_plus_properties,
minus-minus
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
equalityIsType4,
baseClosed,
baseApply,
closedConclusion,
applyEquality,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
isectIsTypeImplies,
functionIsType,
atomEquality,
voidEquality,
isect_memberEquality,
lambdaFormation,
dependent_set_memberEquality,
functionExtensionality,
imageMemberEquality,
imageElimination,
lambdaEquality,
universeEquality,
dependent_set_memberEquality_alt,
applyLambdaEquality,
addEquality,
hyp_replacement,
intEquality,
dependent_pairFormation,
equalityIsType3,
minusEquality,
multiplyEquality,
productIsType
Latex:
\mforall{}[r:CRng]. \mforall{}[x,y:Atom].
\mforall{}[h:PowerSeries(r)]. \mforall{}[d:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[k:\mBbbN{}].
(Moessner(r;x;y;h;d;k)
= ([h]\_d 0(y:=((k \mcdot{}r 1)*atom(x)+atom(y)))*\mPi{}(i\mmember{}upto(k)).((((k - i) \mcdot{}r 1)*atom(x)
+atom(y)))\^{}(d (i + 1))))
supposing \mneg{}(x = y)
Date html generated:
2019_10_16-AM-11_36_49
Last ObjectModification:
2018_10_19-AM-00_13_17
Theory : power!series
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