Nuprl Lemma : fps-single-bag-rep
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[n:ℕ]. (<bag-rep(n;x)> = (atom(x))^(n) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
fps-atom: atom(x),
fps-single: <c>,
power-series: PowerSeries(X;r),
bag-rep: bag-rep(n;x),
deq: EqDecider(T),
nat: ℕ,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
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,
bag-rep: bag-rep(n;x),
primrec: primrec(n;b;c),
empty-bag: {},
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
fps-one: 1,
fps-coeff: f[b],
fps-single: <c>,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
crng: CRng,
rng: Rng,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
fps-atom: atom(x)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-single_wf,
nil_wf,
list-subtype-bag,
fps-exp-zero,
fps-atom_wf,
subtype_rel_self,
iff_weakening_equal,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-ext,
empty-bag_wf,
fps-one_wf,
bag-eq_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
bag-null_wf,
assert-bag-null,
rng_one_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
bag_wf,
rng_zero_wf,
bag-rep_wf,
le_wf,
fps-exp-unroll,
fps-mul_wf,
fps-mul-single,
single-bag_wf,
cons-bag-as-append,
bag-append-comm,
primrec-unroll,
lt_int_wf,
assert_of_lt_int,
cons-bag_wf,
primrec_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
unionElimination,
universeEquality,
equalityElimination,
promote_hyp,
cumulativity,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[n:\mBbbN{}]. (<bag-rep(n;x)> = (atom(x))\^{}(n))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_58_37
Last ObjectModification:
2018_05_19-PM-04_15_01
Theory : power!series
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