Nuprl Lemma : fps-pascal-elim
∀[r:CRng]. ∀[x,y:Atom]. Δ(x,y)(x:=0) = (1÷(1-atom(y))) ∈ PowerSeries(r) supposing (¬(x = y ∈ Atom)) ∧ (¬(1 = 0 ∈ |r|))
Proof
Definitions occuring in Statement :
fps-pascal: Δ(x,y),
fps-elim-x: f(x:=0),
fps-div: (f÷g),
fps-sub: (f-g),
fps-atom: atom(x),
fps-one: 1,
power-series: PowerSeries(X;r),
atom-deq: AtomDeq,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
atom: Atom,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
fps-pascal: Δ(x,y),
fps-elim-x: f(x:=0),
fps-atom: atom(x),
squash: ↓T,
prop: ℙ,
crng: CRng,
rng: Rng,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
all: ∀x:A. B[x],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
fps-sub: (f-g),
empty-bag: {},
fps-add: (f+g),
fps-one: 1,
fps-coeff: f[b],
infix_ap: x f y,
single-bag: {x},
fps-single: <c>,
fps-neg: -(f),
bag-eq: bag-eq(eq;as;bs),
bag-null: bag-null(bs),
null: null(as),
nil: [],
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,
atom-deq: AtomDeq,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-elim-div,
atom-valueall-type,
atom-deq_wf,
fps-one_wf,
fps-sub_wf,
fps-add_wf,
fps-atom_wf,
rng_one_wf,
fps-div_wf,
iff_weakening_equal,
not_wf,
equal-wf-base,
atom_subtype_base,
rng_car_wf,
rng_zero_wf,
crng_wf,
fps-zero_wf,
atomdeq_reduce_lemma,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
fps-neg_wf,
fps-elim-x-one,
fps-elim-x-sub,
fps-elim-x-add,
fps-elim-x-atom,
neg_thru_op_fps,
abmonoid_ac_1_fps,
abmonoid_comm_fps,
fps-non-trivial,
rng_minus_wf,
rng_plus_wf,
infix_ap_wf,
rng_times_wf,
reduce_nil_lemma,
reduce_cons_lemma,
map_nil_lemma,
map_cons_lemma,
rng_times_over_plus,
rng_times_over_minus,
rng_times_zero,
rng_times_one,
rng_minus_over_plus,
rng_minus_zero,
rng_plus_zero,
valueall-type_wf,
deq_wf,
neg_id_fps,
mon_ident_fps
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
independent_isectElimination,
atomEquality,
setElimination,
rename,
independent_pairFormation,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
natural_numberEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[r:CRng]. \mforall{}[x,y:Atom]. \mDelta{}(x,y)(x:=0) = (1\mdiv{}(1-atom(y))) supposing (\mneg{}(x = y)) \mwedge{} (\mneg{}(1 = 0))
Date html generated:
2018_05_21-PM-10_12_00
Last ObjectModification:
2017_07_26-PM-06_34_51
Theory : power!series
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