Nuprl Lemma : KozenSilva-corollary2
∀[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)[bag-rep(Σ(d i | i < k);x)]
= Π((k - i)^(d i) | i < k)
∈ ℤ)
supposing ¬(x = y ∈ Atom)
Proof
Definitions occuring in Statement :
Moessner: Moessner(r;x;y;h;d;k),
fps-one: 1,
fps-coeff: f[b],
bag-rep: bag-rep(n;x),
exp: i^n,
int-prod: Π(f[x] | x < k),
sum: Σ(f[x] | x < k),
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,
int: ℤ,
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,
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,
squash: ↓T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
so_apply: x[s],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
int-prod: Π(f[x] | x < k),
int_ring: ℤ-rng,
pi2: snd(t),
pi1: fst(t),
rng_one: 1,
btrue: tt,
empty-bag: {},
sum_aux: sum_aux(k;v;i;x.f[x]),
sum: Σ(f[x] | x < k),
primrec: primrec(n;b;c),
bag-rep: bag-rep(n;x),
null: null(as),
bag-null: bag-null(bs),
fps-one: 1,
it: ⋅,
nil: [],
bfalse: ff,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
from-upto: [n, m),
upto: upto(n),
list_accum: list_accum,
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],
fps-coeff: f[b],
integ_dom: IntegDom{i},
nat_plus: ℕ+,
rng_car: |r|,
crng: CRng,
rng: Rng,
istype: istype(T),
subtract: n - m,
uiff: uiff(P;Q),
sq_type: SQType(T),
rng_times: *,
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
assert: ↑b,
rng_zero: 0,
nequal: a ≠ b ∈ T
Lemmas referenced :
KozenSilva-corollary1,
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,
subtract-1-ge-0,
equal_wf,
squash_wf,
true_wf,
istype-universe,
int-prod_wf,
exp_wf2,
int_seg_subtype_nat,
istype-false,
subtract_wf,
int_seg_wf,
subtype_rel_self,
iff_weakening_equal,
fps-coeff_wf,
bag-rep_wf,
sum_wf,
non_neg_sum,
int_seg_properties,
decidable__le,
istype-le,
intformnot_wf,
int_formula_prop_not_lemma,
list-subtype-bag,
istype-nat,
atom_subtype_base,
istype-atom,
nat_wf,
le_wf,
false_wf,
itermAdd_wf,
int_term_value_add_lemma,
power-series_wf,
int_ring_wf,
fps-product-upto,
atom-valueall-type,
atom-deq_wf,
less_than_wf,
fps-exp_wf,
fps-add_wf,
fps-scalar-mul_wf,
rng_car_wf,
fps-atom_wf,
upto_wf,
fps-mul_wf,
fps-product_wf,
minus-zero,
add-zero,
minus-add,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-one-mul-top,
add-mul-special,
zero-mul,
le-add-cancel2,
itermSubtract_wf,
int_term_value_subtract_lemma,
sum_split,
decidable__lt,
subtype_base_sq,
int_subtype_base,
sum1,
int-prod-split,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
primrec1_lemma,
one-mul,
fps-mul-coeff-bag-rep-simple,
set_subtype_base,
lelt_wf,
fps-exp-linear-coeff,
ite_rw_false,
eq_int_wf,
rng_nexp_wf,
rng_zero_wf,
iff_weakening_uiff,
assert_wf,
equal-wf-base,
assert_of_eq_int,
bag_qinc,
eqtt_to_assert,
rng_nexp-int,
eqff_to_assert,
assert_elim,
bnot_wf,
bool_wf,
eq_int_eq_true,
bfalse_wf,
bool_subtype_base,
btrue_neq_bfalse,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
intEquality,
because_Cache,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
applyLambdaEquality,
closedConclusion,
dependent_set_memberEquality_alt,
unionElimination,
functionIsType,
isectIsTypeImplies,
equalityIsType4,
baseApply,
functionEquality,
functionExtensionality,
lambdaEquality,
dependent_set_memberEquality,
lambdaFormation,
addEquality,
atomEquality,
equalityIsType1,
multiplyEquality,
minusEquality,
productIsType,
cumulativity,
equalityIsType3,
equalityElimination,
equalityIsType2,
promote_hyp
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)[bag-rep(\mSigma{}(d i | i < k);x)]
= \mPi{}((k - i)\^{}(d i) | i < k))
supposing \mneg{}(x = y)
Date html generated:
2019_10_16-AM-11_36_59
Last ObjectModification:
2018_10_18-PM-11_53_18
Theory : power!series
Home
Index