Nuprl Lemma : exp_difference_factor
∀[n:ℕ+]. ∀[x,y:ℤ]. ((x^n - y^n) = (Σ(x^n - i + 1 * y^i | i < n) * (x - y)) ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
sum: Σ(f[x] | x < k),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
sq_type: SQType(T),
uiff: uiff(P;Q)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
subtract_wf,
exp_wf2,
nat_plus_subtype_nat,
left_mul_subtract_distrib,
sum_wf,
int_seg_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
iff_weakening_equal,
nat_plus_wf,
mul_com,
sum_scalar_mult,
subtype_base_sq,
int_subtype_base,
mul_assoc,
exp_step,
decidable__lt,
less_than_wf,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add-associates,
itermMinus_wf,
int_term_value_minus_lemma,
mul-swap,
not-lt-2,
condition-implies-le,
zero-add,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
add-subtract-cancel,
subtract-add-cancel,
sum_split1,
sum_split_first,
minus-zero,
exp0_lemma,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
minus-minus,
add-mul-special,
zero-mul,
subtract-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
sqequalRule,
multiplyEquality,
dependent_set_memberEquality,
setElimination,
rename,
addEquality,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
axiomEquality,
instantiate,
cumulativity,
minusEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbZ{}]. ((x\^{}n - y\^{}n) = (\mSigma{}(x\^{}n - i + 1 * y\^{}i | i < n) * (x - y)))
Date html generated:
2017_04_17-AM-09_45_23
Last ObjectModification:
2017_02_27-PM-05_40_27
Theory : num_thy_1
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