Nuprl Lemma : KozenSilva-corollary1
∀[x,y:Atom].
∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;1;λi.if (i =z 0) then 0 else d (i - 1) fi ;k)
= Π(i∈upto(k)).(((k - i)*atom(x)+atom(y)))^(d i)
∈ PowerSeries(ℤ-rng))
supposing ¬(x = y ∈ Atom)
Proof
Definitions occuring in Statement :
Moessner: Moessner(r;x;y;h;d;k),
fps-exp: (f)^(n),
fps-scalar-mul: (c)*f,
fps-product: Π(x∈b).f[x],
fps-add: (f+g),
fps-atom: atom(x),
fps-one: 1,
power-series: PowerSeries(X;r),
upto: upto(n),
atom-deq: AtomDeq,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
atom: Atom,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
integ_dom: IntegDom{i},
so_lambda: λ2x.t[x],
int_seg: {i..j-},
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
crng: CRng,
rng: Rng,
subtract: n - m,
rng_one: 1,
pi2: snd(t),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
upto_wf,
list-subtype-bag,
int_seg_wf,
subtype_rel_self,
bag_wf,
power-series_wf,
int_ring_wf,
integ_dom_wf,
KozenSilva-corollary0,
fps-product_wf,
atom-valueall-type,
atom-deq_wf,
fps-exp_wf,
fps-add_wf,
fps-scalar-mul_wf,
subtract_wf,
fps-atom_wf,
nat_wf,
int_seg_subtype_nat,
false_wf,
iff_weakening_equal,
rng_nat_op-int,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rng_one_wf,
rng_car_wf,
not_wf,
equal-wf-base,
atom_subtype_base,
lifting-strict-spread,
strict4-spread,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
setElimination,
rename,
natural_numberEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
atomEquality,
intEquality,
functionExtensionality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
axiomEquality,
functionEquality
Latex:
\mforall{}[x,y:Atom].
\mforall{}[d:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[k:\mBbbN{}].
(Moessner(\mBbbZ{}-rng;x;y;1;\mlambda{}i.if (i =\msubz{} 0) then 0 else d (i - 1) fi ;k)
= \mPi{}(i\mmember{}upto(k)).(((k - i)*atom(x)+atom(y)))\^{}(d i))
supposing \mneg{}(x = y)
Date html generated:
2018_05_21-PM-10_14_22
Last ObjectModification:
2017_07_26-PM-06_35_34
Theory : power!series
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