Nuprl Lemma : Moessner-theorem
∀[x,y:Atom].
∀[n:ℕ]. ∀[k:ℕ+].
(Moessner(ℤ-rng;x;y;1;λi.if (i =z 0) then 0 if (i =z 1) then n else 0 fi ;k)[bag-rep(n;x)] = k^n ∈ ℤ)
supposing ¬(x = y ∈ Atom)
Proof
Definitions occuring in Statement :
Moessner: Moessner(r;x;y;h;d;k),
fps-one: 1,
fps-coeff: f[b],
bag-rep: bag-rep(n;x),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
lambda: λx.A[x],
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
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...},
prop: ℙ,
squash: ↓T,
true: True,
int_ring: ℤ-rng,
pi1: fst(t),
rng_car: |r|,
nequal: a ≠ b ∈ T ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat_plus: ℕ+,
integ_dom: IntegDom{i},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
lelt: i ≤ j < k,
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
int-prod: Π(f[x] | x < k),
eq_int: (i =z j),
subtract: n - m
Lemmas referenced :
KozenSilva-corollary2,
eq_int_wf,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nat_properties,
nequal-le-implies,
zero-add,
le_wf,
nat_plus_subtype_nat,
equal_wf,
squash_wf,
true_wf,
istype-universe,
nat_plus_wf,
nat_wf,
atom_subtype_base,
istype-void,
istype-atom,
subtype_rel_self,
bag-rep_wf,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
itermSubtract_wf,
itermVar_wf,
intformand_wf,
subtract_wf,
false_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
nat_plus_properties,
assert_of_eq_int,
eqtt_to_assert,
fps-one_wf,
Moessner_wf,
integ_dom_wf,
int_ring_wf,
crng_wf,
power-series_wf,
bag_wf,
fps-coeff_wf,
and_wf,
iff_weakening_equal,
btrue_wf,
eq_int_eq_true,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
equal-wf-T-base,
not_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_seg_wf,
ifthenelse_wf,
singleton_support_sum,
int-prod-split,
exp_wf2,
exp0_lemma,
itermAdd_wf,
istype-int,
int_term_value_add_lemma,
istype-less_than,
primrec1_lemma,
minus-zero,
one-mul,
add-zero,
primrec0_lemma,
less_than_wf,
ge_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
subtract-add-cancel,
assert_of_lt_int,
lt_int_wf,
primrec-unroll
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
closedConclusion,
natural_numberEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIsType4,
baseApply,
baseClosed,
applyEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
independent_pairFormation,
dependent_set_memberEquality_alt,
universeIsType,
equalityIsType1,
hyp_replacement,
imageElimination,
universeEquality,
intEquality,
imageMemberEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsType,
int_eqEquality,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
approximateComputation,
lambdaFormation,
functionExtensionality,
atomEquality,
lambdaEquality,
applyLambdaEquality,
addEquality,
productIsType,
intWeakElimination
Latex:
\mforall{}[x,y:Atom].
\mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
(Moessner(\mBbbZ{}-rng;x;y;1;\mlambda{}i.if (i =\msubz{} 0) then 0
if (i =\msubz{} 1) then n
else 0
fi ;k)[bag-rep(n;x)]
= k\^{}n)
supposing \mneg{}(x = y)
Date html generated:
2019_10_16-AM-11_37_07
Last ObjectModification:
2018_10_16-PM-03_15_16
Theory : power!series
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