### 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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