Nuprl Lemma : fps-pascal-slice
∀[r:CRng]. ∀[x,y:Atom]. ∀[n:ℕ]. ([Δ(x,y)]_n = ((<{x}>+<{y}>))^(n) ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-pascal: Δ(x,y),
fps-exp: (f)^(n),
fps-slice: [f]_n,
fps-add: (f+g),
fps-single: <c>,
power-series: PowerSeries(X;r),
single-bag: {x},
atom-deq: AtomDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
atom: Atom,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
nat: ℕ,
crng: CRng,
rng: Rng,
fps-pascal: Δ(x,y),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
eq_int: (i =z j),
bag-size: #(bs),
length: ||as||,
list_ind: list_ind,
single-bag: {x},
cons: [a / b],
nil: [],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fps-geometric-slice1,
atom-valueall-type,
atom-deq_wf,
fps-add_wf,
fps-single_wf,
single-bag_wf,
equal_wf,
power-series_wf,
fps-slice_wf,
fps-div_wf,
rng_one_wf,
fps-one_wf,
fps-sub_wf,
fps-exp_wf,
nat_wf,
crng_wf,
eq_int_wf,
bag-size_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
fps-zero_wf,
ite_rw_true,
iff_weakening_equal,
squash_wf,
true_wf,
fps-add-slice,
fps-single-slice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
atomEquality,
dependent_functionElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
isect_memberEquality,
axiomEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[x,y:Atom]. \mforall{}[n:\mBbbN{}]. ([\mDelta{}(x,y)]\_n = ((<\{x\}>+<\{y\}>))\^{}(n))
Date html generated:
2018_05_21-PM-10_11_53
Last ObjectModification:
2017_07_26-PM-06_34_49
Theory : power!series
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