Nuprl Lemma : Pascal-completion-property
∀[r:CRng]. ∀[f,g:PowerSeries(r)]. ∀[x,y:Atom].
((Pascal-completion(r;f;g;x;y)(x:=0) = f ∈ PowerSeries(r))
∧ (Pascal-completion(r;f;g;x;y)(y:=0) = g ∈ PowerSeries(r))
∧ fps-Pascal(r;x;y;Pascal-completion(r;f;g;x;y)))
∧ (∀h:PowerSeries(r)
(fps-Pascal(r;x;y;h)
⇒ (h(x:=0) = f ∈ PowerSeries(r))
⇒ (h(y:=0) = g ∈ PowerSeries(r))
⇒ (h = Pascal-completion(r;f;g;x;y) ∈ PowerSeries(r))))
supposing (¬(1 = 0 ∈ |r|))
∧ (¬(x = y ∈ Atom))
∧ (f(x:=0) = f ∈ PowerSeries(r))
∧ (g(y:=0) = g ∈ PowerSeries(r))
∧ (f(y:=0) = g(x:=0) ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
Pascal-completion: Pascal-completion(r;f;g;x;y),
fps-Pascal: fps-Pascal(r;x;y;f),
fps-elim-x: f(x:=0),
power-series: PowerSeries(X;r),
atom-deq: AtomDeq,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
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,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
fps-Pascal: fps-Pascal(r;x;y;f),
crng: CRng,
rng: Rng,
Pascal-completion: Pascal-completion(r;f;g;x;y),
true: True,
atom-deq: AtomDeq,
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),
guard: {T},
bnot: ¬bb,
assert: ↑b,
fps-sub: (f-g),
nequal: a ≠ b ∈ T ,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
empty-bag: {},
fps-atom: atom(x),
fps-neg: -(f),
fps-one: 1,
fps-add: (f+g),
fps-coeff: f[b],
single-bag: {x},
fps-single: <c>,
bag-null: bag-null(bs),
null: null(as),
nil: [],
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),
top: Top,
lt_int: i <z j,
band: p ∧b q
Lemmas referenced :
equal_wf,
power-series_wf,
fps-elim-x_wf,
atom-deq_wf,
fps-Pascal_wf,
bag_wf,
not_wf,
rng_car_wf,
rng_one_wf,
rng_zero_wf,
equal-wf-base,
crng_wf,
atom-valueall-type,
fps-sub_wf,
fps-add_wf,
fps-mul_wf,
fps-one_wf,
fps-atom_wf,
fps-pascal_wf,
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,
neg_id_fps,
fps-neg_wf,
fps-pascal-elim,
iff_weakening_equal,
mon_ident_fps,
fps-div_wf,
squash_wf,
true_wf,
fps-elim-x-mul,
valueall-type_wf,
deq_wf,
fps-elim-x-sub,
fps-elim-x-add,
fps-elim-x-one,
fps-elim-x-atom,
fps-elim-x-elim-x,
mul_over_plus_fps,
mul_over_minus_fps,
mul_assoc_fps,
mul_one_fps,
mul_ac_1_fps,
mul_comm_fps,
mon_assoc_fps,
abmonoid_ac_1_fps,
abmonoid_comm_fps,
iabgrp_op_inv_assoc_fps,
fps-div-property,
rng_times_wf,
fps-coeff_wf,
empty-bag_wf,
rng_minus_wf,
rng_plus_wf,
infix_ap_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_zero,
rng_plus_zero,
fps-pascal-symmetry,
fps-Pascal-iff,
Pascal-completion_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
lambdaFormation,
extract_by_obid,
isectElimination,
atomEquality,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
axiomEquality,
lambdaEquality,
dependent_functionElimination,
productEquality,
setElimination,
rename,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
unionElimination,
equalityElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
universeEquality,
equalityUniverse,
levelHypothesis,
voidEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(r)]. \mforall{}[x,y:Atom].
((Pascal-completion(r;f;g;x;y)(x:=0) = f)
\mwedge{} (Pascal-completion(r;f;g;x;y)(y:=0) = g)
\mwedge{} fps-Pascal(r;x;y;Pascal-completion(r;f;g;x;y)))
\mwedge{} (\mforall{}h:PowerSeries(r)
(fps-Pascal(r;x;y;h)
{}\mRightarrow{} (h(x:=0) = f)
{}\mRightarrow{} (h(y:=0) = g)
{}\mRightarrow{} (h = Pascal-completion(r;f;g;x;y))))
supposing (\mneg{}(1 = 0)) \mwedge{} (\mneg{}(x = y)) \mwedge{} (f(x:=0) = f) \mwedge{} (g(y:=0) = g) \mwedge{} (f(y:=0) = g(x:=0))
Date html generated:
2018_05_21-PM-10_12_32
Last ObjectModification:
2017_07_26-PM-06_34_58
Theory : power!series
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