Nuprl Lemma : fps-exp-linear-coeff
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[x,y:X].
∀[r:CRng]. ∀[k:|r|]. ∀[m,n:ℕ].
((((k)*atom(x)+atom(y)))^(m)[bag-rep(n;x)] = if (n =z m) then k ↑r m else 0 fi ∈ |r|)
supposing ¬(x = y ∈ X)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
fps-scalar-mul: (c)*f,
fps-add: (f+g),
fps-atom: atom(x),
fps-coeff: f[b],
bag-rep: bag-rep(n;x),
deq: EqDecider(T),
nat: ℕ,
valueall-type: valueall-type(T),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T,
rng_nexp: e ↑r n,
crng: CRng,
rng_zero: 0,
rng_car: |r|
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,
crng: CRng,
rng: Rng,
fps-coeff: f[b],
fps-one: 1,
int_upper: {i...},
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
less_than': less_than'(a;b),
le: A ≤ B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
subtype_rel: A ⊆r B,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
fps-mul: (f*g),
bag-partitions: bag-partitions(eq;bs),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bag-splits: bag-splits(b),
list_ind: list_ind,
bag-rep: bag-rep(n;x),
primrec: primrec(n;b;c),
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,
reduce: reduce(f;k;as),
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
cand: A c∧ B,
so_lambda: λ2x.t[x],
power-series: PowerSeries(X;r),
pi1: fst(t),
pi2: snd(t),
so_apply: x[s],
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
fps-atom: atom(x),
fps-scalar-mul: (c)*f,
fps-add: (f+g),
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),
lt_int: i <z j,
band: p ∧b q,
infix_ap: x f y,
respects-equality: respects-equality(S;T),
bag-size: #(bs),
length: ||as||,
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
less_than: a < b
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,
subtract-1-ge-0,
istype-nat,
rng_car_wf,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe,
rng_nexp_zero,
bag-null-rep,
rng_zero_wf,
zero-add,
nequal-le-implies,
upper_subtype_nat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_wf,
false_wf,
rng_nexp_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
list-subtype-bag,
bag-rep_wf,
fps-atom_wf,
fps-scalar-mul_wf,
fps-add_wf,
squash_wf,
true_wf,
fps-coeff_wf,
bag_wf,
power-series_wf,
fps-exp-zero,
subtype_rel_self,
iff_weakening_equal,
subtract-add-cancel,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
istype-le,
istype-false,
intformeq_wf,
int_formula_prop_eq_lemma,
fps-exp_wf,
fps-exp-add,
fps-mul_wf,
fps-exp-one,
decidable__equal_int,
int_subtype_base,
bag-summation-single,
rng_plus_wf,
rng_plus_comm2,
rng_times_wf,
empty-bag_wf,
crng_properties,
rng_properties,
reduce_nil_lemma,
reduce_cons_lemma,
map_nil_lemma,
map_cons_lemma,
nil_wf,
rng_times_over_plus,
rng_times_zero,
rng_plus_zero,
bag-summation-single-non-zero-no-repeats,
product-deq_wf,
bag-deq_wf,
bag-partitions_wf,
bag-member_wf,
bag-member-partitions,
bag-size_wf,
bag-append-equal-bag-rep,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
respects-equality-product,
list_wf,
subtype-respects-equality,
bag_qinc,
bag-eq_wf,
single-bag_wf,
assert-bag-eq,
equal-wf-base,
set_subtype_base,
iff_weakening_uiff,
assert_wf,
no-repeats-bag-partitions,
bag-rep-add,
bag-summation_wf,
crng_all_properties,
int_upper_properties,
rng_nexp_unroll,
primrec1_lemma,
cons_bag_empty_lemma,
member_wf,
rng_one_wf,
rng_times_one,
rng_plus_comm,
single-valued-bag-single,
single-valued-bag_wf,
bag-only_wf2,
bag_only_single_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
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,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
functionIsType,
equalityIstype,
instantiate,
universeEquality,
voidEquality,
isect_memberEquality,
hypothesis_subsumption,
cumulativity,
promote_hyp,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
lambdaEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
addEquality,
closedConclusion,
isectIsType,
intEquality,
productEquality,
productIsType,
independent_pairEquality,
inlFormation_alt,
pointwiseFunctionality,
baseApply,
inrFormation_alt,
sqequalBase,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[x,y:X].
\mforall{}[r:CRng]. \mforall{}[k:|r|]. \mforall{}[m,n:\mBbbN{}].
((((k)*atom(x)+atom(y)))\^{}(m)[bag-rep(n;x)] = if (n =\msubz{} m) then k \muparrow{}r m else 0 fi )
supposing \mneg{}(x = y)
supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_34_54
Last ObjectModification:
2018_11_26-PM-03_09_25
Theory : power!series
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