Nuprl Lemma : Paasche-theorem
∀[x,y:Atom].
∀[k:ℕ]. (Moessner(ℤ-rng;x;y;1;λi.if (i =z 0) then 0 else 1 fi ;k)[bag-rep(k;x)] = (k)! ∈ 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),
fact: (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: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
crng: CRng,
rng: Rng,
exp: i^n,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
lelt: i ≤ j < k,
int-prod: Π(f[x] | x < k)
Lemmas referenced :
KozenSilva-corollary2,
false_wf,
le_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
not_wf,
equal-wf-base,
atom_subtype_base,
fps-coeff_wf,
bag_wf,
power-series_wf,
crng_wf,
int_ring_wf,
Moessner_wf,
fps-one_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
bag-rep_wf,
sum_constant,
intformand_wf,
itermVar_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
iff_weakening_equal,
rng_car_wf,
integ_dom_wf,
primrec1_lemma,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
fact0_redex_lemma,
int_prod0_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
fact_unroll,
int_subtype_base,
int-prod-split,
int_seg_wf,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
int-prod_wf,
int_seg_properties
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
equalityTransitivity,
universeEquality,
intEquality,
imageMemberEquality,
baseClosed,
because_Cache,
isect_memberEquality,
axiomEquality,
atomEquality,
setElimination,
rename,
functionExtensionality,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
dependent_pairFormation,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
hypothesis_subsumption,
int_eqEquality,
multiplyEquality,
intWeakElimination,
addEquality,
functionEquality
Latex:
\mforall{}[x,y:Atom].
\mforall{}[k:\mBbbN{}]. (Moessner(\mBbbZ{}-rng;x;y;1;\mlambda{}i.if (i =\msubz{} 0) then 0 else 1 fi ;k)[bag-rep(k;x)] = (k)!)
supposing \mneg{}(x = y)
Date html generated:
2018_05_21-PM-10_15_35
Last ObjectModification:
2017_07_26-PM-06_35_44
Theory : power!series
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