Nuprl Lemma : fps-set-to-one-one
∀[r:CRng]. ∀[y:Atom]. ∀[n:ℕ]. ([1]_n(y:=1) = if (n =z 0) then 1 else 0 fi ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1),
fps-one: 1,
fps-zero: 0,
power-series: PowerSeries(X;r),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
natural_number: $n,
atom: Atom,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-zero: 0,
fps-one: 1,
fps-coeff: f[b],
fps-set-to-one: [f]_n(y:=1),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
crng: CRng,
rng: Rng,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
bag-count: (#x in bs),
count: count(P;L),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-bag: {},
nil: [],
bag-size: #(bs),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
true: True,
band: p ∧b q
Lemmas referenced :
fps-ext,
fps-set-to-one_wf,
fps-one_wf,
ifthenelse_wf,
eq_int_wf,
power-series_wf,
fps-zero_wf,
lt_int_wf,
bag-count_wf,
atom-deq_wf,
eqtt_to_assert,
assert_of_lt_int,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nat_properties,
nequal-le-implies,
zero-add,
istype-le,
rng_zero_wf,
bool_wf,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
bag-size_wf,
bag_wf,
istype-nat,
istype-atom,
crng_wf,
bag-null_wf,
assert-bag-null,
equal-wf-T-base,
length_of_nil_lemma,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-empty-bag,
empty_bag_append_lemma,
bag_size_empty_lemma,
bag-null-rep,
subtract_wf,
decidable__le,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_subtype_base,
decidable__equal_int,
bag_null_empty_lemma,
rng_one_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
bag-null-append,
bag-rep_wf,
list-subtype-bag,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
iff_imp_equal_bool,
bool_cases,
band_wf,
btrue_wf,
iff_functionality_wrt_iff,
false_wf,
iff_transitivity,
assert_of_band,
istype-assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
atomEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
productElimination,
independent_isectElimination,
lambdaFormation_alt,
sqequalRule,
applyEquality,
because_Cache,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
independent_pairFormation,
dependent_set_memberEquality_alt,
cumulativity,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
hyp_replacement,
applyLambdaEquality,
imageElimination,
baseClosed,
sqequalBase,
approximateComputation,
int_eqEquality,
intEquality,
universeEquality,
closedConclusion,
imageMemberEquality,
productEquality,
productIsType
Latex:
\mforall{}[r:CRng]. \mforall{}[y:Atom]. \mforall{}[n:\mBbbN{}]. ([1]\_n(y:=1) = if (n =\msubz{} 0) then 1 else 0 fi )
Date html generated:
2019_10_16-AM-11_36_29
Last ObjectModification:
2018_11_26-PM-03_09_16
Theory : power!series
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